research article homogeneous triebel-lizorkin spaces on...

Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 475103 16 pageshttpdxdoiorg1011552013475103

Research ArticleHomogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

Guorong Hu

Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba Mekuro-ku Tokyo 153-8914 Japan

Correspondence should be addressed to Guorong Hu hugrmsu-tokyoacjp

Received 7 January 2013 Accepted 3 March 2013

Academic Editor Jozef Banas

Copyright copy 2013 Guorong HuThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Homogeneous Triebel-Lizorkin spaces with full range of parameters are introduced on stratified Lie groups in terms of Littlewood-Paley-type decomposition It is shown that the scale of these spaces is independent of the choice of Littlewood-Paley-typedecomposition and the sub-Laplacian used for the construction of the decomposition Some basic properties of these spaces aregiven As the main result of this paper boundedness of a class of singular integral operators on these function spaces is obtained

1 Introduction

In recent years there were several efforts of extending Besovand Triebel-Lizorkin spaces from Euclidean spaces to otherdomains and non-isotropic settings In particular Han et al[1] developed a theory of these function spaces on spacesof homogenous type with the additional reverse doublingproperty That setting is quite general and includes forexample Lie groups of polynomial growth However the highlevel of generality imposes restrictions on the possible valuesof the parameters of the function spaces

For the purpose of studying subelliptic regularity Folland[2] introduced fractional Sobolev spaces and Lipschitz spaceson stratified Lie groups Later Folland and Stein [3] estab-lished the theory of Hardy spaces on general homogeneousgroups Besov spaces on stratified Lie groups were firstintroduced by Saka [4] by means of the heat semigroupassociated to the sub-Laplacian Recently Fuhr and Mayeli[5] introduced homogeneous Besov spaces on stratified Liegroups in terms of Littlewood-Paley-type decomposition andestablished wavelet characterization of them However theintegrability parameter 119901 and the summability parameter 119902of the function spaces studied in both [4 5] are restricted tobe no less than 1 Moreover systematic treatment of Triebel-Lizorkin spaces on stratified Lie groups can not be found inthe literature to our best knowledge

Thepurpose of this paper is to introduce and study homo-geneous Triebel-Lizorkin spaceswith full range of parameterson stratified Lie groups Motivated by [5] we define these

function spaces via Littlewood-Paley-type decompositionWe find that a helpful way to treat the case that either theintegrability parameter 119901 or the summability parameter 119902is less than 1 is to take the Peetre type maximal functioninto considerationWith the help of the almost orthogonalityestimate on stratified Lie groups (see Lemma 2) we showthat our definition of homogeneous Triebel-Lizorkin spacesis independent of the choice of the Littlewood-Paley-typedecomposition and the sub-Laplacian used for the construc-tion of the decompositionThus these function spaces reflectof properties of the group not of the sub-Laplacian used forthe construction of the decomposition

Singular integral theory is a powerful tool for the studyof partial differential equationsThe 119871119901-boundedness of con-volution operators with homogeneous distribution kernelson Lie groups endowed with suitable homogeneous structurewas proved by Knapp and Stein [6] (for 119901 = 2) and Koranyiand Vagi [7] (for 1 lt 119901 lt infin) In Section 4 of this paperwe prove the boundedness on homogeneous Triebel-Lizorkinspaces of a class of convolution type singular integral oper-ators on stratified Lie groups which includes convolutionoperators with homogeneous distribution kernels

This paper is organized as follows After reviewingsome basic notions concerning stratified Lie groups andtheir associated sub-Laplaicans in Section 2 in Section 3 weintroduce homogeneous Triebel-Lizorkin spaces 120572

119901119902(119866) on

stratified Lie groups and give some basic properties of themIn Section 4 we show the 120572

119901119902(119866)-boundedness of a class

of convolution singular integral operators Throughout this

2 Journal of Function Spaces and Applications

paper the letter 119862 will denote a positive constant which isindependent of the main variables involved but whose valuemay differ from line to line The notation 119886 ≲ 119887 or 119887 ≳ 119886 forsome variable quantities 119886 and 119887means that 119886 le 119862119887 for someconstant 119862 gt 0 119886 sim 119887 stands for 119886 ≲ 119887 ≲ 119886 We agree that theset N of natural numbers contains 0

2 Preliminaries

In this section we briefly review the basic notions concerningstratified Lie groups and their associated sub-Laplacians Formore details we refer the reader to the monograph by Follandand Stein [3] A Lie group 119866 is called a stratified Lie groupif it is connected and simply connected and its Lie algebra gmay be decomposed as a direct sum g = 1198811 oplus sdot sdot sdot oplus 119881119898 with[1198811 119881119896] = 119881119896+1 for 1 le 119896 le 119898 minus 1 and [1198811 119881119898] = 0 Sucha group 119866 is clearly nilpotent and thus it may be identifiedwith g (as a manifold) via the exponential map exp g rarr 119866Examples of stratified Lie groups include Euclidean spacesR119899

and the Heisenberg group H119899The algebra g is equipped with a family of dilations 120575119905

119905 gt 0 which are the algebra automorphisms defined by

120575119905(

119898

sum

119895=1

119883119895) =

119898

sum

119895=1

119905119895119883119895 (119883119895 isin 119881119895) (1)

Under our identification of 119866 with g 120575119905 may also be viewedas a map 119866 rarr 119866 We generally write 119905119909 instead of 120575119905(119909) for119909 isin 119866 We shall denote by

Δ =

119898

sum

119895=1

119895 [dim (119881119895)] (2)

the homogeneous dimension of 119866A homogeneous norm on G is a continuous function

119909 997891rarr |119909| from 119866 to [0infin) smooth away from 0 (the groupidentity) vanishing only at 0 and satisfying |119909minus1| = |119909| and|119905119909| = 119905|119909| for all 119909 isin 119866 and 119905 gt 0 Homogeneous norms on 119866always exist and any two of them are equivalent We assume119866 is provided with a fixed homogeneous norm It satisfies atriangle inequality there exists a constant 120574 ge 1 such that|119909119910| le 120574(|119909|+|119910|) for all 119909 119910 isin 119866 If 119909 isin 119866 and 119903 gt 0we definethe ball of radius 119903 about 119909 by 119861(119909 119903) = 119910 isin 119866 |119910

minus1119909| lt

119903 The Lebesgue measure on g induces a bi-invariant Haarmeasure on 119866 As done in [3] we fix the normalization ofHaar measure by requiring that the measure of 119861(0 1) be 1We shall denote the measure of any measurable 119864 sub 119866 by |119864|Clearly we have |120575119905(119864)| = 119905

Δ|119864| All integrals on 119866 are with

respect to (the normalization of) Haar measure Convolutionis defined by

119891 lowast 119892 (119909) = int119891 (119910) 119892 (119910minus1119909) 119889119910 = int119891 (119909119910

minus1) 119892 (119910) 119889119910

(3)

We consider g as the Lie algebra of all left-invariant vectorfields on119866 and fix a basis1198831 119883119899 of g obtained as a unionof bases of the119881119895 In particular1198831 119883] with ] = dim(1198811)

is a basis of 1198811 We denote by 1198841 119884119899 the correspondingbasis for right-invariant vector fields that is

119884119895119891 (119909) =119889

119889119905119891 (exp (119905119883119895) 119909) |119905=0 (4)

If 119868 = (1198941 119894119899) isin N119899 is a multi-index we set119883119868= 119883

1198941

1sdot sdot sdot 119883

119894119899

119899

and 119884119868 = 11988411989411sdot sdot sdot 119884

119894119899

119899 Moreover we set

|119868| =

119899

sum

119896=1

119894119896 119889 (119868) =

119899

sum

119896=1

119889119896119894119896 (5)

where the integers 1198891 le sdot sdot sdot le 119889119899 are given according to that119883119896 isin 119881119889

119896

Then 119883119868 (resp 119884119868) is a left-invariant (resp right-invariant) differential operator homogeneous of degree 119889(119868)with respect to the dilations 120575119905 119905 gt 0

A complex-valued function 119875 on119866 is called a polynomialon119866 if 119875 ∘ exp is a polynomial on g Let 1205851 120585119899 be the basisfor the linear forms on g dual to the basis 1198831 119883119899 for gand set 120578119895 = 120585119895 ∘ exp

minus1 From our definition of polynomialson 119866 1205781 120578119899 are generators of the algebra of polynomialson 119866 Thus every polynomial on 119866 can be written uniquelyas

119875 = sum

119868

119886119868120578119868 119886119868 isin C (6)

where all but finitely many of the coefficients vanish and120578119868= 120578

1198941 sdot sdot sdot 120578

119894119899 A polynomial of the type (6) is called of

homogeneous degree 119871 where 119871 isin N if 119889(119868) le 119871 holds forall multi-indices 119868 with 119886119868 = 0 We let P denote the spaceof all polynomials on 119866 and let P119871 denote the space ofpolynomials on 119866 of homogeneous degree 119871 Note thatP119871 isinvariant under left and right translations (see [3 Proposition125]) A function 119891 119866 rarr C is said to have vanishingmoments of order 119871 if

forall119875 isin P119871 int119866

119891 (119909) 119875 (119909) 119889119909 = 0 (7)

with the absolute convergence of the integralThe Schwartz class on 119866 is defined by

S (119866) = 119891 isin 119862infin(119866) 119875

120597|119868|119891

1205971205781198941 sdot sdot sdot 120597120578119894119899isin 119871

infin(119866)

forall119868 isin N119899 forall119875 isin P

(8)

that is 119891 isin S(119866) if and only if 119891 ∘ exp isin S(g) As is indicatedin [3 p 35] S(119866) is a Frechet space and several differentchoices of families of norms induce the same topology onS(119866) In this paper for our purpose we use the family sdot (119871119873) 119871119873 isin N of norms given by

10038171003817100381710038171206011003817100381710038171003817(119871119873) = sup

119889(119868)le119871119909isin119866

(1 + |119909|)119873(10038161003816100381610038161003816119883119868120601 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119883119868120601 (119909)

10038161003816100381610038161003816) (9)

Here and in what follows we use the notation convention120601(119909) = 120601(119909

minus1) for any function 120601 119866 rarr C The dual space

Journal of Function Spaces and Applications 3

S1015840(119866) ofS(119866) is the space of tempered distributions on 119866 If

119891 isin S1015840(119866) and 120601 isin S(119866) we shall denote the evaluation of 119891

on 120601 by ⟨119891 120601⟩Using the above conventions for the choice of the basis

1198831 119883119899 and ] = dim(1198811) the sub-Laplacian is defined byL = minussum

]119895=1

1198832

119895 When restricted to smooth functions with

compact supportL is essentially self-adjoint Its closure hasdomainD = 119906 isin 119871

2(119866) L119906 isin 119871

2(119866) whereL119906 is taken

in the sense of distributions We denote this extension still bythe symbolL By the spectral theoremL admits a spectralresolution

L = int

infin

0

120582 119889119864 (120582) (10)

where 119889119864(120582) is the projection measure If 119898 is a boundedBorel measurable function on [0infin) the operator

119898(L) = int

infin

0

119898(120582) 119889119864 (120582) (11)

is bounded on 1198712(119866) and commutes with left translationsThus by the Schwartz kernel theorem there exists a tempereddistribution119872 on 119866 such that

119898(L) 119891 = 119891 lowast119872 forall119891 isin S (119866) (12)

Note that the point 120582 = 0 may be neglected in the spectralresolution since the projection measure of 0 is zero (see[8 p 76]) Consequently we should regard119898 as functions onR+

equiv (0infin) rather than on [0infin)Let S(R+

) denote the space of restrictions to R+ offunctions in S(R) An important fact proved by Hulanicki[9] is as in the following lemma

Lemma 1 If 119898 isin S(R+) then the distribution kernel 119872 of

119898(L) is in S(119866)

Moreover from the proof of [10 Corollary 1] we see thatif 119898 is a function in S(R+

) which vanishes identically nearthe origin then119872 is a Schwartz function with all momentsvanishing

In the sequel if not other specified we will generally useGreak alphabets with hats to denote functions inS(R+

) anduse Greek alphabets without hats to denote the associateddistribution kernels for example for 120601 isin S(R+

) we shalldenote by 120601 the distribution kernel of the operator 120601(L)whereL is a sub-Laplacian fixed in the context

3 Homogeneous Triebel-Lizorkin Spaces onStratified Lie Groups

For any function ℎ on 119866 and 119905 gt 0 we define the 1198711-normalized dilation of ℎ by

119863119905ℎ (119909) = 119905Δℎ (119905119909) (13)

Before we introduce the homogeneous Triebel-Lizorkinspaces on stratified Lie groups we prove the following basicestimate which is a generalization of [11 Lemma B1] and willbe frequently used throughout this paper

Lemma 2 Let 119871119873 isin N with 119873 ge 119871 + Δ + 2 Suppose both120601 120595 isin S(119866) have vanishing moments of order 119871 Then thereexists a constant 119862 gt 0 such that for all 119895 ℓ isin Z and all 119909 isin 119866

1003816100381610038161003816(1198632119895120601) lowast (1198632ℓ120595) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(119871+1119873)2

minus|119895minusℓ|119871 2(119895andℓ)Δ

(1 + 2119895andℓ |119909|)119873

(14)

where 119895 and ℓ = min119895 ℓ

Proof Using dilations and the facts (1198632119895120601) lowast (1198632ℓ120595)(119909) =

(1198632ℓ ) lowast (1198632119895120601)(119909minus1) and 120601(119871119873) = 120601(119871119873) (see (9)) we

may assume ℓ ge 119895 = 0 To proceed we follow the idea inthe proof of [11 Lemma B1] Let 1198631 = 119910 |119910| lt 11198632 = 119910 |119910| ge 1 and |119909119910minus1| le |119909|2120574 and 1198633 = 119910

|119910| ge 1 and |119909119910minus1| gt |119909|2120574 Let 119910 997891rarr 119875119909120601(119910) be the leftTaylor polynomial of 120601 at 119909 of homogeneous degree 119871 (see [3pp 26-27]) Then using vanishing moments of 120595

1003816100381610038161003816120601 lowast (1198632ℓ120595) (119909)1003816100381610038161003816

le int10038161003816100381610038161003816120601 (119909119910

minus1) minus 119875119909120601 (119910

minus1)100381610038161003816100381610038161003816100381610038161003816(1198632ℓ120595) (119910)

1003816100381610038161003816 119889119910

equiv int1198631

+int1198632

+int1198633

(15)

For 119910 isin 1198631 the stratified Taylor formula (cf [3 Corollary144]) yields that with 119887 a suitable positive constant

10038161003816100381610038161003816120601 (119909119910

minus1) minus 119875119909120601 (119910

minus1)10038161003816100381610038161003816

≲10038161003816100381610038161199101003816100381610038161003816119871+1 sup

|119911|le119887119871+1

|119910|119889(119868)=119871+1

10038161003816100381610038161003816(119883

119868120601) (119909119911)

10038161003816100381610038161003816

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873) sup

|119911|le119887119871+1

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + |119909119911|)119873≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + |119909|)119873

(16)

since 2119887119871+1(1 + |119909119911|) ge 2119887119871+1 + |119909119911| ge 2119887119871+1 + |119909|120574 minus |119911| ge119887119871+1

+ |119909|120574 ≳ (1 + |119909|)120574 if |119911| le 119887119871+1 Hence we have

int1198631

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2ℓΔ

(1 + |119909|)119873int1198631

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889119910

=10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ(119871+1)

(1 + |119909|)119873int119866

100381610038161003816100381610038162ℓ11991010038161003816100381610038161003816

119871+1

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889 (2

ℓ119910)

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ(119871+1)

(1 + |119909|)119873

(17)

where for the last inequality we used [3 Corollary 117] andthat119873 minus 119871 minus 1 ge Δ + 1


For119910 isin 1198632 we have |119910| ge |119909|120574minus|119909119910minus1| ge |119909|120574minus|119909|2120574 =

|119909|2120574 On the other hand |119910| le 120574(|119909| + |119909119910minus1|) le 120574(|119909| +

|119909|2120574) = (120574 + (12))|119909| Thus we have

1 + 2ℓ 1003816100381610038161003816119910

1003816100381610038161003816 ge 2ℓ 1003816100381610038161003816119910

1003816100381610038161003816 ge2ℓ(1 +

10038161003816100381610038161199101003816100381610038161003816)

2ge2ℓ(1 + |119909|)

4120574

10038161003816100381610038161199101003816100381610038161003816119871le (120574 +

1

2)

119871

|119909|119871

(18)

Also we note that by [12 Proposition 20314] the left Taylorpolynomial 119875119909120601 is of the form

119875119909120601 (119910)

= 120601 (119909) +

119871

sum

ℎ=1

ℎ

sum

119896=1

sum

1le1198941119894119896le119899

1198891198941+sdotsdotsdot+119889

119894119896=ℎ

1205781198941

(119910) 120578119894119896

(119910)

1198961198831198941

sdot sdot sdot 119883119894119896

120601 (119909)

(19)


119895

From these remarks it follows that

int1198632

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

times int1198632

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+ sum

0leℎle119871

10038161003816100381610038161199101003816100381610038161003816ℎ

(1 + |119909|)119873]

times2ℓΔ

(1 +10038161003816100381610038162ℓ1199101003816100381610038161003816)119873119889119910

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873) int

1198632

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+

|119909|119871

(1 + |119909|)119873]

times2ℓΔ

[2ℓ (1 + |119909|)]119873119889119910

=10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ(119873minusΔ)

(1 + |119909|)119873

times [int119866

1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873119889119910

+|119909|

119871

(1 + |119909|)119873int|119909119910minus1|le|119909|2120574

119889119910]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873

times [int119866

1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873119889119910

+|119909|

119871

(1 + |119909|)119873int|119910|le120574(|119909|+|119909|2120574)

119889119910]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873[1 +

|119909|119871+Δ

(1 + |119909|)119873]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873

(20)

where we used that |119910 |119910| le 120574(|119909| + |119909|2120574)| sim |119909|Δ and

119873 minus Δ gt 119871For 119910 isin 1198633 we have |119909119910

minus1| gt |119909|2120574 and hence

int1198633

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

times int1198633

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+ sum

0leℎle119871

10038161003816100381610038161199101003816100381610038161003816ℎ

(1 + |119909|)119873]

times2ℓΔ

(1 +10038161003816100381610038162ℓ1199101003816100381610038161003816)119873119889119910

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2ℓΔ

(1 + |119909|)119873int|119910|ge1

10038161003816100381610038161199101003816100381610038161003816119871

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889119910

le10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873int119866

100381610038161003816100381610038162ℓ11991010038161003816100381610038161003816

119871

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889 (2

ℓ119910)

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873

(21)

where for the last inequality we used [3 Corollary 117] and119873 minus 119871 gt Δ + 1

Combining the above estimates we arrive at

1003816100381610038161003816120601 lowast (1198632ℓ120595) (119909)1003816100381610038161003816 le

10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(119871+1119873)2

minusℓ119871 1

(1 + |119909|)119873

(22)

This is exactly what we need

LetZ(119866) denote the space of Schwartz functions with allmoments vanishingWe then considerZ(119866) as a subspace ofS(119866) including the topology It is shown in [5 Lemma 33]thatZ(119866) is a closed subspace ofS(119866) and the topology dualZ1015840(119866) of Z(119866) can be canonically identified with the factor

space S1015840(119866)P

We now have the following Calderon type reproducingformula


Lemma 3 Suppose L is a sub-Laplacian on 119866 and 120601 isin

S(R+) is a function with compact support vanishing identi-

cally near the origin and satisfying

sum

119895isinZ

120601 (2minus2119895120582) = 1 forall120582 isin R

+ (23)

Then for all 119892 isinZ(119866) it holds that

119892 = lim119896rarrinfin

sum

|119895|le119896

119892 lowast (1198632119895120601) (24)

with convergence in Z(119866) Duality entails that for all 119906 isin

S1015840(119866)P


sum

|119895|le119896

119906 lowast (1198632119895120601) (25)

and the convergence is in S1015840(119866)P

Proof First note that the 2-homogeneity of L implies thatthe distribution kernel of 120601(2minus2119895L) coincides with 1198632119895120601 Let119868 isin N119899 and 119873 isin N be arbitrarily chosen Then take 119871 isin N

such that 119871 ge 119873 + 119889(119868) + 1 Since both 119892 and 120601 are Schwartzfunctions with all moments vanishing it follows by Lemma 2that

(1 + |119909|)119873 10038161003816100381610038161003816119883119868[119892 lowast (1198632119895120601)] (119909)

10038161003816100381610038161003816

= 2119889(119868)119895

(1 + |119909|)119873 10038161003816100381610038161003816119892 lowast [1198632119895 (119883

119868120601)] (119909)

10038161003816100381610038161003816

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus(119871+Δ+2)

(1 + |119909|)119873

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus119873

(1 + |119909|)119873

le 1198622minus|119895|(119871minus119873minus119889(119868))

(26)

where the constant 119862 is a suitable multiple of119892(119871+1119871+Δ+2)119883

119868120601(119871+1119871+Δ+2) This implies that

sum119895isinZ(1 + |119909|)119873|119883

119868[119892 lowast (1198632119895120601)](119909)| converges uniformly

in 119909 for every 119868 isin N119899 and every119873 isin N Consequently thereexists ℎ isin S(119866) such that sum|119895|le119896 119892 lowast (1198632119895120601) converges in thetopology of S(119866) to ℎ as 119896 rarr infin On the other hand by(23) and the spectral theorem (cf [13 Theorem VII2])

119892 = sum

119895isinZ

120601 (2minus2119895

L) 119892 = sum

119895isinZ

119892 lowast (1198632119895120601) (27)

holds in 1198712-norm Therefore ℎ = 119892 which completes theproof

Let A denote the class of all functions 120601 in S(R+)

satisfying

supp120601 sub [2minus2 22]

10038161003816100381610038161003816120601 (120582)

10038161003816100381610038161003816ge 119862 gt 0 for 120582 isin [(3

5)

2

(5

3)

2

]

(28)

Definition 4 Let 120572 isin R 0 lt 119901 lt infin and 0 lt 119902 le infin Let Lbe a sub-Laplacian on 119866 and 120601 isin A We define 120572

119901119902(L 120601) as

the space of all 119891 isin S1015840(119866)P such that

10038171003817100381710038171198911003817100381710038171003817120572119901119902(L120601)

equiv

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

lt infin (29)

with the usual modification for 119902 = infin

We then introduce the Peetre type maximal functionsGiven 119891 isin S1015840

(119866) 120601 isin S(R+) L a sub-Laplacian and

119886 119905 gt 0 we define

119872lowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1003816100381610038161003816119891 lowast (119863119905minus1120601) (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

119872lowastlowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (119863119905minus1120601)] (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(30)

Lemma 5 Suppose L is a sub-Laplacian and 120601 isin A Thenfor every 119886 gt 0 there is a constant 119862 gt 0 such that for all119891 isin S1015840

(119866)P all 119895 isin Z and all 119909 isin 119866

119872lowastlowast

1198862minus119895(119891L 120601) (119909) le 1198622

119895119872

lowast

1198862minus119895(119891L 120601) (119909) (31)

Proof Because of (28) it is possible to find a func-tion isin S(R+

) supported in [2minus2 2

2] such that

sum119895isinZ 120601(2minus2119895120582)(2

minus2119895120582) = 1 for 120582 isin R+ Set 120577(120582) =

sum1

119895=minus1120601(2

minus2119895120582)(2

minus2119895120582) 120582 isin R+ Then 120601(120582) = 120601(120582)120577(120582) for

all 120582 isin R+ Consequently for 119895 isin Z and 1 le 119896 le ]1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601)] (119910)

1003816100381610038161003816

=1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601) lowast (1198632119895120577)] (119910)

1003816100381610038161003816

le int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816[119883119896 (1198632119895120577)] (119911

minus1119910)10038161003816100381610038161003816119889119911

= 2119895(Δ+1)

int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816(119883119896120577) (2

119895(119911

minus1119910))

10038161003816100381610038161003816119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

times int 2119895Δ(1 + 2

119895 10038161003816100381610038161003816119911minus111990910038161003816100381610038161003816)119886

(1 + 2119895 10038161003816100381610038161003816119911minus111991010038161003816100381610038161003816)minus119886minus(Δ+1)

119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(32)

where for the last inequality we used [3 Corollary 117] andthat (1+2119895|119911minus1119909|)119886(1+2119895|119911minus1119910|)minus119886 ≲ (1+2119895|119910minus1119909|)119886 Dividingboth sides of the above estimate by (1 + 2119895|119910minus1119909|)119886 and thentaking the supremum over 119910 isin 119866 and 1 le 119896 le ] we obtainthe desired estimate

Lemma 6 Suppose L is a sub-Laplacian 120601 isin A and 119903 gt 0Then there exists a constant 119862 such that for all 119891 isin S1015840

(119866)Pall 119895 isin Z and all 119909 isin 119866

119872lowast

1198862minus119895(119891L 120601) (119909) le 119862 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

(33)


where 119886 = Δ119903 and 119872 is the Hardy-Littlewood maximaloperator on 119866

Proof Let 119892 isin 119862infin(119866) and 1199100 isin 119866 The stratified mean valuetheorem (cf [3 Theorem 141]) gives that for every 120575 gt 0 andevery 119910 isin 119866 with |119910minus11199100| lt 120575

1003816100381610038161003816119892 (119910) minus 119892 (1199100)1003816100381610038161003816

le 11986210038161003816100381610038161003816119910minus1

011991010038161003816100381610038161003816

sup|119911|le119887|119910

minus1

0119910|

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

le 119862120575 sup|119911|le119887120575

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

(34)

where 119887 is a suitable positive constant Hence we have1003816100381610038161003816119892 (1199100)

1003816100381610038161003816 le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896119892 (119910)1003816100381610038161003816

+ 120575minus119886(int

|119910minus11199100|le120575

1003816100381610038161003816119892 (119910)1003816100381610038161003816119903119889119910)

1119903

(35)

Putting 119892 = 119891lowast(1198632119895120601) dividing both sides by (1+2119895|119910minus1

0119909|)

119886and using Lemma 5 we have1003816100381610038161003816119891 lowast (1198632119895120601) (1199100)

1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601)] (119910)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(1 + 2119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886(int

|119910minus11199100|le120575

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575 sup|119910minus11199100|le119887120575

(1 + 2119895 10038161003816100381610038161003816119910minus1119910010038161003816100381610038161003816)119886

119872lowastlowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

times (int|119910minus1119909|le120574(120575+|119910minus1

0119909|)

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575(1 + 2119895119887120575)

119886

2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886120574119886(120575 +

10038161003816100381610038161003816119910minus1

011990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

le 119862120576(1 + 119887120576)119886119872

lowast

1198862minus119895(119891L 120601) (119909)

+ 120574119886(1 + 120576

minus1)119886

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119909)]

1119903

(36)

wherewehave set120575 = 2minus119895120576 andused that (1+2119895120576minus1|119910minus10119909|)(1+

2119895|119910minus1

0119909|) ≲ 1 + 120576

minus1 Finally taking 120576 sufficiently small (such

that 119862120576(1 + 119887120576)119886 lt 12) and taking the supremum over 1199100 isin119866 we get the desired estimate

Theorem 7 Suppose L(1)L(2) are any two sub-Laplacians

on 119866 and 120601(1) 120601(2) are any two functions in A Then for 120572 isinR 0 lt 119901 lt infin and 0 lt 119902 le infin we have the (quasi-)normequivalence

10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(1) 120601(1))

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(2) 120601(2))

119891 isin S1015840(119866) P (37)

Proof First we note that if 119886 gt Δmin119901 119902 then we have

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(119894) 120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119891 lowast (1198632119895120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(38)

for 119894 = 1 2 where 120601(119894) denotes the distribution kernel of120601(119894)(L(119894)

) Indeed the direction ldquo≳rdquo of (38) is obvious and theother direction follows by Lemma 6 and the Fefferman-Steinvector-valuedmaximal inequality on spaces of homogeneoustype (see eg [14]) Thus to prove (37) it suffices to showthat

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(1) 120601

(1))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(2) 120601

(2))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(39)

To this end let (1) be a function in S(R+) with support

in [2minus2 22] such that sum119895isinZ 120601(1)(2minus2119895120582)

(1)(2minus2119895120582) = 1 for 120582 isin

R+ For 119891 isin S1015840(119866)P by Lemma 3 we have

119891 = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1)) (40)

with convergence in S1015840(119866)P Here 120595(1) is the distribution

kernel of (1)(L(1)) Hence since 120601(2) isin Z(119866) we have the

pointwise representation

119891 lowast (1198632ℓ120601(2)) (119910) = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1))

lowast (1198632ℓ120601(2)) (119910) forall119910 isin 119866

(41)


It follows that10038161003816100381610038161003816119891 lowast (1198632ℓ120601

(2)) (119910)

10038161003816100381610038161003816

le sum

119895isinZ

int10038161003816100381610038161003816119891 lowast (1198632119895120601

(1)) (119911)

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911

minus1119910)10038161003816100381610038161003816119889119911

le sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

times int (1 + 2119895|119911|)

119886 10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911)

10038161003816100381610038161003816119889119911

equiv sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910) 119868119895ℓ

(42)

Since both 120595(1) and 120601

(2) are Schwartz functions with allmoments vanishing we can use Lemma 2 to estimate thatwith 119871 sufficiently large

119868119895ℓ le 119862int (1 + 2119895|119911|)

119886

2minus|119895minusℓ|119871

2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119871+Δ+2)

119889119911

le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119886+Δ+1)

119889119911

le 1198622minus|119895minusℓ|(119871minus119886)

int2(119895andℓ)Δ

(1 + 2119895andℓ|119911|)

minus(Δ+1)

119889119911


(43)

Here the constant 119862 is a suitable multiple of120595

(1)(119871+1119871+Δ+2)120601

(2)(119871+1119871+Δ+2) On the other hand we

observe that

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

le 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

≲ 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

times (1 + 2ℓ 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

max (1 2(119895minusℓ)119886)

(44)

Putting these estimates into (42) multiplying both sides by2ℓ120572 dividing both sides by (1 + 2ℓ|119910minus1119909|)119886 and then takingthe supremum over 119910 isin 119866 we obtain

2ℓ120572119872

lowast

1198862minusℓ(119891L

(2) 120601

(2)) (119909)

le sum

119895isinZ

2minus|119895minusℓ|(119871minus2119886minus|120572|)

2119895120572119872

lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

(45)

In view of [15 Lemma 2] taking 119871 gt 2119886 + |120572| in the aboveinequality yields the direction ldquo≲rdquo of (39) By symmetricity(39) holds and the proof is complete

Remark 8 FromTheorem 7 we see that the space 120572119901119902(L 120601)

is actually independent of the choice of L and 120601 Thus inwhat follows we donrsquot specify the choice ofL and 120601 and write120572

119901119902(119866) instead of 120572

119901119902(L 120601) Henceforth we shall fix any sub-

LaplacianLMoreover for the sake of briefness wewill write119872

lowast

119886119905(119891 120601)(119909) instead of119872lowast

119886119905(119891L 120601)(119909)

Proposition 9 For 120572 isin R 0 lt 119901 lt infin and 0 lt 119902 le infinone has the continuous inclusion maps Z(119866) 997893rarr

120572

119901119902(119866) 997893rarr

S1015840(119866)P

Proof Let 119892 isin Z(119866) and 120601 isin A Choose119873 ge (Δ + 1)119901 and119871 ge 119873 + |120572| + 1 Since both 119892 and 120601 are Schwartz functionswith all moments vanishing it follows by Lemma 2 that

1003816100381610038161003816119892 lowast (1198632119895120601) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus(119871+Δ+2)

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus119873

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(46)

for all 119895 isin Z and all 119909 isin 119866 where the constant 119862 is a suitablemultiple of 120601(119871+1119871+Δ+2) This together with [3 Corollary117] give that

10038171003817100381710038171198921003817100381710038171003817120572119901119902(119866)

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)(sum

119895isinZ

2minus|119895|(119871minus119873minus|120572|)119902

)

1119902

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)

(47)

which implies thatZ(119866) 997893rarr 119904

119901119902(119866) continuously

Now we show the other embedding Let 119891 isin 120572

119901119902(119866)

and take any 120595 isin Z(119866) Let 120601 isin A and then let 120577 isin

S(R+) be a function with support in [2

minus2 2

2] such that

sum119895isinZ 120601(2minus2119895120582)120577(2

minus2119895120582) = 1 for 120582 isin R+ Then by Lemma 3

we have

1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⟨sum

119895isinZ

119891 lowast (1198632119895120601) lowast (1198632119895120577) 120595⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

119895isinZ

10038161003816100381610038161003816⟨119891 lowast (1198632119895120601) 120595 lowast (1198632119895120577)⟩

10038161003816100381610038161003816

le sum

119895isinZ

1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin

10038171003817100381710038171003817120595 lowast (1198632119895120577)

100381710038171003817100381710038171198711

(48)

To proceed we claim that

1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin le 1198622

119895Δ119901 1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871119901

(49)


for all 0 lt 119901 lt infin Assuming the claim for a moment itfollows from (48) that

1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 ≲ (sup

119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901)

times sum

119895isinZ

2119895[(Δ119901)minus120572]10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

(50)

It is easy to see that

sup119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (51)

To estimate the sum in (50) we note that if we choose 119873 ge

Δ + 1 and 119871 ge 119873 + (Δ119901) + |120572| + 1 then similarly to (46) wehave10038161003816100381610038161003816120595 lowast (1198632119895120577) (119909)

10038161003816100381610038161003816le 119862

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(52)

where the constant 119862 is a suitable multiple of 120577(119871+1119871+Δ+2)From this it follows that

sum

119895isinZ

2119895(Δ119901minus120572)10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

≲ sum

119895isinZ

2minus|119895|[119871minus119873minus(Δ119901)minus|120572|]1003817100381710038171003817120595

1003817100381710038171003817(119871+1119871+Δ+2)

≲10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)

(53)

Therefore1003816100381610038161003816⟨119891 120595⟩

1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2) (54)

This implies that 120572119901119902(119866) 997893rarr S1015840

(119866)P continuouslyWe are leftwith showing the claim Indeed if 119903 gt 0 is fixed

then by Lemma 6 we have for all 119909 isin 119866

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816 sim inf

119910isin119861(1199092minus119895)

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119909minus1119910

1003816100381610038161003816)Δ119903

le inf119910isin119861(1199092minus119895)

119872lowast

Δ1199032minus119895(119891 120601) (119910)

≲ inf119910isin119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

1119903

(55)

Taking 119903 lt 119901 in the above estimate and using Hardy-Littlewood maximal inequality we have1003816100381610038161003816119891 lowast (1198632119895120601) (119909)

1003816100381610038161003816119901

≲1

1003816100381610038161003816119861 (119909 2minus119895)1003816100381610038161003816

int119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

119901119903

119889119910

le 2119895Δint1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816119901119889119910

(56)

Since 119909 is arbitrary the claim follows

Since all the necessary tools are developed in the abovearguments the following proposition can be proved in thesame manner as its Euclidean counterpart see for examplethe proof of [16 Theorem 233]

Proposition 10 For 120572 isin R 0 lt 119901 lt infin and 0 lt 119902 le infin120572

119901119902(119866) is a quasi-Banach space

Let us introduce a class of functions We say that 119891 isin

R(119866) if there exists 120601 isin S(R+) whose support is compact

andwhich vanishes identically near the origin and119892 isin S(119866)such that 119891 = 120601(L)119892 ClearlyR(119866) subZ(119866)

Lemma 11 Let 120572 isin R and 0 lt 119901 119902 lt infin ThenR(119866) is densein 120572

119901119902(119866) In particularZ(119866) is dense in 120572

119901119902(119866)

Proof Take any 119906 isin 120572119901119902(119866) and any 120576 gt 0 In the appendix

we show that 120572119901119902(119866) admits smooth atomic decomposition

By the smooth atomic decomposition we see that 119862infin0(119866) cap

120572

119901119902(119866) is dense in 120572

119901119902(119866) for 120572 isin R and 0 lt 119901 119902 lt infinThus

we can find 119892 isin 119862infin0(119866)cap

120572

119901119902(119866) such that 119892minus119906120572

119901119902(119866) lt 1205762

On the other hand the argument in Step 5 of the proof of[16 Theorem 233] shows that there exists a sufficiently large119873 isin N such that 119892 minussum119873

119895=minus119873119892lowast (1198632119895120601)120572

119901119902(119866) lt 1205762 Now we

put

ℎ =

119873

sum

119895=minus119873

120601 (2minus2119895

L) 119892 =

119873

sum

119895=minus119873

119892 lowast (1198632119895120601) (57)

Then ℎ isinR(119866) and we have

ℎ minus 119906120572119901119902(119866) ≲

1003817100381710038171003817ℎ minus 1198921003817100381710038171003817120572119901119902(119866)+1003817100381710038171003817119892 minus 119906

1003817100381710038171003817120572119901119902(119866)

lt 120576 (58)

This proves the claimed statement

We next consider lifting property of 120572119901119902(119866) For 120590 isin R

the powerL120590 is naturally given by

L120590119891 = int

infin

0

120582120590119889119864 (120582) 119891 119891 isin Dom (L

120590) sub 119871

2(119866) (59)

Remark 12 By [17 Theorem 1324] we have R(119866) sub

Dom(L120590) for all 120590 isin R As a consequence Dom(L120590

) cap

120572

119901119902(119866) is dense in 120572

119901119902(119866) for all 120572 120590 isin R and 0 lt 119901 119902 lt infin

We now have the lifting property of 120572119901119902(119866)

Theorem 13 Let 120572 120590 isin R and 0 lt 119901 119902 lt infin

(i) The operator L120590 initially defined on Dom(L120590) cap

120572

119901119902(119866) extends to a continuous operator from

120572

119901119902(119866)

to 120572minus2120590119901119902

(119866)

(ii) Let 119879120590 120572119901119902(119866) rarr 120572minus2120590

119901119902(119866) denote the continuous

extension of L120590 Then 119879120590 is an isomorphism and119879120590119891120572minus2120590

119901119902(119866) is an equivalent quasi-norm of 120572

119901119902(119866)

Proof (i) Let 120601 isin A Set 120577(120582) = 120582120590 and (120582) =

120601(120582)120577(120582) = 120601(120582)120582120590 Clearly is also in A and


22119895120590(2

minus2119895120582) = 120601(2

minus2119895120582)120577(120582) By [17Theorem 1324] we have

120601(2minus2119895L) ∘ 120577(L) sub (120601(2

minus2119895sdot)120577(sdot))(L) and moreover

Dom (120601 (2minus2119895

L) ∘ 120577 (L))

= Dom (120577 (L)) cap Dom ((120601 (2minus2119895sdot) 120577 (sdot)) (L))

= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)

Hence for every 119891 isin Dom(L120590) we have 22119895120590(2minus2119895L)119891 =

120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904

119901119902(119866) By the above remarks

we have 22119895120590[119891 lowast (1198632119895120595)] = (L120590119891) lowast (1198632119895120601) It follows that

1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)

Since Dom(L120590)cap

120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)

the mappingL120590 Dom(L120590

) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to

a continuous operator from 120572

119901119902(119866) to 120572minus2120590

119901119902(119866) We denote

this extension by 119879120590 120572

119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)

(ii) Let us first show that the mapping 119879120590 120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is injective Indeed assume 119891 isin

120572

119901119902(119866) such that

119879120590119891 is the zero element of 120572minus2120590119901119902

(119866) By Remark 12 we canfind a sequence 119891ℓ in Dom(L120590

) cap 120572

119901119902(119866) which converges

in 120572119901119902(119866) to 119891 Then applying (i) to L120590 yields that L120590

119891ℓ

converges in 119904minus2120590

119901119902(119866) to the zero element Since L120590

119891ℓ isin

Dom(Lminus120590) cap

120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)

to the operatorLminus120590 we see that119891ℓ converges in 120572

119901119902(119866) to the

zero elementTherefore119891 is the zero element in 120572119901119902(119866)This

proves that 119879120590 is injectiveNext we show that119879120590

120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective

Indeed given 119891 isin 120572minus2120590

119901119902(119866) we let 119891ℓ be a sequence in


120572minus2120590

119901119902(119866) which converges in 120572minus2120590

119901119902(119866) to 119891

Then from (i) we see thatLminus120590119891ℓ converges in

120572

119901119902(119866) Denote

this limit by 119892 We claim that 119879120590119892 = 119891 Indeed since 119891ℓ =L120590

(Lminus120590119891ℓ) it follows from (i) that 119891ℓ converges to 119879120590119892 in

120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590

119901119902(119866) This proves that 119879120590 is

surjectiveThe above arguments also show that both 119879minus120590 ∘ 119879120590 and

119879120590 ∘ 119879minus120590 are identity operators on 120572119901119902(119866) Furthermore by

an easy density argument we see that (61) holds for all 119891 isin

120572

119901119902(119866) provided thatL120590 in (61) is replaced by 119879120590 Thus 119879120590

120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is an isomorphism and 119879120590119891120572minus2120590

119901119902(119866) is

an equivalent quasi-norm of 120572119901119902(119866)

Our next goal is to show the Lusin and Littlewood-Paleyfunction characterizations of 120572

119901119902(119866) If 120572 isin R 0 lt 119902 lt infin

120582 gt 0 119887 isin R and 119906(119909 119896) is a function on 119866 times Z we define

119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

times(1 + 2119896 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)minus120582119902

2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin

int|119910minus1119909|le2119887minus119896

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)

The following proposition shows that the spaces 120572119901119902(119866) are

characterized by Lusin and Littlewood-Paley functions

Proposition 14 Let 120572 isin R 0 lt 119901 119902 lt infin 119887 isin R and 120582 gtmaxΔ119901 Δ119902 Let 119891 isin S1015840

(119866)P and 120601 isin A Put 119906(119909 119896) =119891 lowast (1198632119896120601)(119909) for 119909 isin 119866 and 119896 isin Z Then one has

10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)

Proof Step 1 Show that if 120582 gt maxΔ119901 Δ119902 then

10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)

Indeed the proof of the first inequality in (64) is essentiallythe same as that of [18 Theorem 23] To see the secondinequality in (64) one only needs to examine the proofs ofTheorems 1 2 and 4 in [19 Chapter 4] and observe thatalthough the function 119906 considered in [19] is defined on thehalf space R119899+1

+= R119899

times (0infin) the arguments there can alsobe adapted to functions 119906 which are defined on 119866 times Z

Step 2Weprove that 119878120572119887119902(119906)119871119901 ≲ 119892

120572

119902(119906)119871119901 First note that

by an argument similar to the proofs of [19 Theorems 1 and2] it follows that 119878120572

119887119902(119906)119871119901 sim 119878

120572

0119902(119906) for every fixed 119887 isin R

Hence in view of (38) it is enough to show that

10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)


But this is a consequence of the following elementary esti-mate

int|119910minus1119909|lt2minus119895

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

le sup119910isin119866

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)

The proof of Proposition 14 is thus complete

Corollary 15 Let 119866 be a stratified Lie group Then 01199012(119866) =

119867119901(119866) with equivalent (quasi-)norms for 0 lt 119901 lt infin Here

119867119901(119866) are Hardy spaces on 119866

Proof In [3 Chapter 7] Folland and Stein proved the char-acterization of Hardy spaces 119867119901

(119866) by continuous versionLusin function for 0 lt 119901 lt infin Note that the argumentsin [3 Chapter 7] are still valid if we replace the continuousversion Lusin function by discrete version one defined abovesee also [20] for a treatment of discrete version Lusin funcionThis fact together with Proposition 14 yield the identificationof 0

1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin

4 Convolution Singular Integral Operators on120572

119901119902(119866)

In this section we study boundedness of convolution singularintegral operators on homogeneous Triebel-Lizorkin spaceson stratified Lie groups Motivated by [21 Section 53 inChapter XIII] we introduce a class of singular convolutionkernels as follows

Definition 16 Let 119903 be a positive integer A kernel of order 119903is a distribution119870 isin S1015840

(119866) with the following properties(i) 119870 coincides with a 119862119903 function 119870(119909) away from the

group identity 0 and enjoys the regularity condition10038161003816100381610038161003816119883119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|

minus119876minus119889(119868) for |119868| le 119903 119909 = 0 (67)

(ii)119870 satisfies the cancellation condition For all normal-ized bump function 120601 and all 119877 gt 0 we have

10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)

where 120601119877(119909) = 120601(119877119909) and 119862 is a constant independent of120601 and 119877 Here by a normalized bump function we mean afunction 120601 supported in 119861(0 1) and satisfying

10038161003816100381610038161003816119883119868120601 (119909)

10038161003816100381610038161003816le 1 forall |119868| le 119873 forall119909 isin 119866 (69)

for some fixed positive integer119873The convolution operator119879with kernel of order 119903 is called

a singular integral operator of order 119903

Remark 17 Using [3 Proposition 129] it is easy to verify that(67) is equivalent to the following condition

10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|


Examples of such kernels include the class of distributionswhich are homogeneous of degree minusΔ (see Folland and Stein[3 p 11] for definition) and agree with 119862infin functions awayfrom 0 Indeed assume119870 isin S1015840

(119866) is such a distribution thenit is easy to verify that119870 satisfies the regularity condition (i) inDefinition 16moreover from [3 Proposition 613]we see that119870 is a principle value distribution such that int

120576lt|119909|lt119871119870(119909)119889119909 =

0 for all 0 lt 120576 lt 119871 lt infin Hence for every normalized bumpfunction 120601 by the homogeneity of 119870 we have

10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)

Using stratified mean value theorem (cf [3 Theorem 141])and (67)ndash(69) it is easy to verify that the last integralconverges absolutely and is bounded by a constant inde-pendent of 120601 and 119877 Hence 119870 satisfies the condition (ii) inDefinition 16

Now we state the main result of this section

Theorem 18 Let 120572 isin R 0 lt 119901 119902 lt infin and let 119903 be a positiveinteger such that 119903 gt Δmin119901 119902 + |120572| + 2 Suppose 119879 isa singular integral operator of order 119903 Then 119879 extends to abounded operator on 120572

119901119902(119866)

If 119870 isin S1015840(119866) and 119905 gt 0 we define 119863119905119870 as the tempered

distribution given by ⟨119863119905119870 120601⟩ = ⟨119870 120601(119905minus1sdot)⟩ (forall120601 isin S(119866))

For the proof of Theorem 18 we will need the followinglemma in which 119887 is the positive constant as in [3 Corollary144]

Lemma 19 Let 119903 be a positive integer Suppose119870 is a kernel oforder 119903 and 120601 is a smooth function supported in 119861(0 141205742119887119903)and having vanishing moments of order 119903minus1Then there existsa constant119862 gt 0 such that for all 119895 isin Z all 119868 isin N119899 with |119868| le 119903and all 119909 isin 119866

max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)

Moreover 120601lowast(1198632119895119870) have vanishing moments of the sameorder as 120601

Proof Recall that the convolution of 120601 isin S(119866) with 119870 isin

S1015840(119866) is defined by 120601 lowast 119870(119909) = ⟨119870 (

119909120601)

sim⟩ where 119909

120601 isthe function given by 119909

120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909

minus1) for any function 119891 119866 rarr C From [3 p 38] we

see that 120601 lowast (1198632119895119870) are 119862infin functions 119895 isin Z We claim that

for every 119909 with |119909| le 12120574 the function 119911 997891rarr (119909120601)

sim(119911)


is a normalized bump function multiplied with a constantindependent of 119909 Indeed using the quasi-triangle inequalitysatisfied by the homogeneous norm it is easy to verify thatthe function 119911 997891rarr (

119909120601)

sim(119911) is supported in 119861(0 1) moreover

since |119909| le 12120574 and since

119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)

where119875119868119869 are polynomials of homogeneous degree119889(119869)minus119889(119868)(see [3 Proposition 129]) we have

10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)

Here 119862119868 is a constant depending on 119868 but not on 119909 Hencethe claim is true The above argument also shows that forevery 119909 with |119909| le 12120574 and for every 119868 isin N119899 [119909(119884119868120601)]sim isa normalized bump function multiplied with a constant 1198621015840

119868

independent of 119909Thus by the condition (ii) in Definition 16there exits a constant 119862 gt 0 such that for all 119895 isin Z all 119868 isin N119899

with |119868| le 119903 and all 119909 with |119909| le 1212057410038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)

From this and [3 Proposition 129] we also get that for all119895 isin Z all 119868 isin N119899 with |119868| le 119903 and all 119909 with |119909| le 1212057410038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)

Let now |119909| gt 12120574 Let 119910 isin supp120601 Let 119868 isin N119899 with|119868| le 119903 and denote by 119909119884119868(119863

2119895119870)

the right Taylor polynomialof119884119868(1198632119895119870) at 119909 of homogeneous degree 119903minus|119868|minus1 (see [3 pp26-27]) Then by the right-invariant version of [3 Corollary144] we have

10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)

Observe that 1198632119895119870 satisfies (67) with the bound 119862 indepen-dent of 119895 isin Z Also note that for 119910 isin supp120601 and |119911| le 119887119903minus|119868||119910|we have 119911119909 isin 119866 0 Thus for all 119869 with 119889(119869) = 119903 minus |119868| and all119911 with |119911| le 119887119903minus|119868||119910| by using (73) and (67) (with119870 replacedby1198632119895119870) we have

10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)

≲ |119911119909|minusΔminus119903+|119868|minus119889(119868)

(78)

Here for the second inequality we also used the observationthat when 119889(119869) = 119903 minus |119868| and |1198691015840| le |119869| we have |1198691015840 + 119868| le|119869 + 119868| le 119889(119869) + |119868| = 119903 minus |119868| + |119868| = 119903 Inserting (78) into (77)we obtain

10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|

|119911119909|minusΔminus119903+|119868|minus119889(119868)

(79)

Notice that for |119909| ge 12120574 119910 isin supp120601 and |119911| le 119887119903minus|119868||119910| wehave |119911119909| sim |119909| Thus by using the vanishing moments of 120601and (79) we have

10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|

int |119911119909|minusΔminus119903+|119868|minus119889(119868)1003816100381610038161003816119910

1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910

le 119862|119909|minusΔminus119903+|119868| minus119889(119868)

int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)

Combining (76) and (80) we see that for all 119895 isin Z all 119868 isin N119899

with |119868| lt 119903 and all 119909 isin 119866

10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)

minusΔminus119903+|119868|minus119889(119868) (81)


From this and (73) we also get that for all 119895 isin Z all 119868 isin N119899

with |119868| lt 119903 and all 119909 isin 11986610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)

le 119862(1 + |119909|)minusΔminus119903+|119868|minus119889(119868)

(82)

Since |119868| le 119889(119868) (81) along with (82) yield (72)It is straightforward to verify that 120601 lowast (1198632119895119870) have

vanishing moments of the same order as 120601 The proof ofLemma 19 is therefore complete

The proof of Theorem 18 also relies on the existenceof smooth functions with compact support and havingarbitrarily high order vanishing moments

Lemma 20 Given any nonnegative integer 119871 and any positivenumber 120575 there exists a function 120577 isin S(R+

)with the followingproperties

(i) |120577(120582)| ge 119862 gt 0 for 120582 isin [2minus2(1198960+1) 2

minus2(1198960minus1)] with 1198960

some (large) positive integer(ii) 120577 is a Schwartz function on 119866 having vanishing

moments of order 119871(iii) supp 120577 sub 119861(0 120575)

Proof From the appendix of [22] we see that there exists120579 isin S(R+

) such that 120579(0) = 1 and 120579 has compact supportNow let us define 120577(120582) = (119905minus2120582)119896120579(119905minus2120582) 120582 isin R+ Here 119905 gt 0and 119896 is a nonnegative integer Then 120577(119909) = 119863119905(L

119896120579)(119909) =

119905Δ(L119896

120579)(119905119909) Hence if we take 119905 119896 sufficiently large then(ii) and (iii) follow immediately Moreover since 120579(0) = 1it is easy to see that (i) is also satisfied provided that 1198960 issufficiently large

We are now ready to proveTheorem 18

Proof of Theorem 1 Choose a function 120577 isin S(R+) which

satisfies conditions (i)ndash(iii) in Lemma 20 with 119871 = 119903 minus 1 and120575 = 14120574

2119887119903 The condition (i) guarantees the existence of a

function isin S(R+) with the following properties

supp sub [2minus2(1198960+1) 2minus2(1198960minus1)]

1003816100381610038161003816 (120582)1003816100381610038161003816 gt 0 for 120582 isin (2minus2(1198960+1) 2minus2(1198960minus1))

sum

119895isinZ

(2minus2119895120582) 120577 (2

minus2119895120582) = 1 forall120582 isin R

+

(83)

Note that (2minus21198960 sdot) isin A For 119891 isinZ(119866) by Lemma 3 we have

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)

with convergence in Z(119866) Let 120601 isin A and let 119870 be theconvolution kernel of the operaotor 119879 Then we have therepresentation

119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ

119891 lowast (1198632119895120595) lowast 1198632119895 [120577 lowast (1198632minus119895119870)] lowast (1198632ℓ120601)

(85)

which holds pointwise and also in the sense of S1015840(119866) Since

(by Lemma 19) 120577 lowast (1198632minus119895119870) satisfies the decay condition (72)(with the bound 119862 independent of 119895 isin Z) and has vanishingmoments of the same order as 120577 from the proof of Lemma 2we see that

10038161003816100381610038161198632119895 [120577 lowast (1198632minus119895119870)] lowast (1198632ℓ120601) (119910)1003816100381610038161003816

le 1198622minus|119895minusℓ|(119903minus2) 2

(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)

This together with (85) gives that1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

times100381610038161003816100381610038161198632119895 [120577 lowast (1198632minus119895119870)] lowast (1198632ℓ120601) (119911

minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ

2minus|119895minusℓ|(119903minus2)

int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ

2minus|119895minusℓ|(119903minus2minus119886)

[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]

times int2(119895andℓ)Δ

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)

where for the last inequality we used that (1+ 2119895andℓ|119911minus1119909|)minus119886 le2|119895minusℓ|119886

(1 + 2119895|119911minus1119909|)

minus119886 By the hypothesis we can choose 119886such that 119886 gt Δmin119901 119902 and 119903 minus 2 minus 119886 minus |120572| gt 0From [3 Corollary 117] we see that the last integral convergesabsolutely Consequently we obtain

2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ

2minus|119895minusℓ|(119903minus2minus119886minus|120572|)

2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)

Hence it follows by [15 Lemma 2] that10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)


This together with (38) imply that 119891 lowast 119870120572119901119902(119866) ≲ 119891120572

119901119902(119866)

for all 119891 isinZ(119866) SinceZ(119866) is dense in 120572119901119902(119866) 119891 997891rarr 119891 lowast119870

extends to an bounded operator on 120572119901119902(119866) This completes

the proof of Theorem 18

Corollary 21 Let 120572 isin R 0 lt 119901 119902 lt infin and let 119896 be anonnegative integer Then

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)

Proof Note that by the Poincare-Birkhoff-Witt theorem (cf[23 I27]) the operators119883119868 form a basis of the algebra of theleft-invariant differential operators on 119866 By this fact and thestratification of 119866 it suffices to show that

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)

To this end we first note that when restricted to Schwartzfunctions 119883119895L

minus12 are convolution operators with distribu-tion kernels homogeneous of degree minusΔ and coincide withsmooth functions in1198660This follows from the fact that theoperatorLminus12 is a convolution operator whose distributionkernel is homogeneous of degree minusΔ+1 and coincides with asmooth function in 119866 0 (see [2 Proposition 317]) Henceby Theorem 18 119883119895L

minus12 extend to bounded operators on120572

119901119902(119866) From this fact and the lifting property (Theorem 7)

we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902

(119866) ≲ 119891120572119901119902(119866) To see the converse we

need to use [2 Lemma 412] which asserts that there existstempered distributions 1198701 119870] homogeneous of degreeminusΔ + 1 and coinciding with smooth functions in 119866 0 suchthat 119891 = sum]

119895=1(119883119895119891) lowast 119870119895 for all 119891 isin S(119866) By this result and

Theorem 7 we have at least for 119891 isinR(119866) (sub Dom(Lminus12))

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)

where 1198771 is the convolution kernel of the operatorLminus12 Asis indicated in [2 p 190]L(119870119895lowast1198771) are distributions homo-geneous of degree minusΔ and coincide with smooth functionsaway from 0 Thus it follows byTheorem 18 that

10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)

Inserting this into (93) we obtain 119891120572119901119902(119866) ≲ sum

]119895=1

119883119895119891120572minus1119901119902

(119866) for all119891 isinR(119866) SinceR(119866) is dense in 120572119901119902(119866)

the latter inequality also holds for all 119891 isin 120572

119901119902(119866) This

completes the proof

Appendix

Smooth Atomic Decomposition of 120572119901119902(119866)

In this appendixwe show that homogeneous Triebel-Lizorkinspaces on stratified Lie groups admit smooth atomic decom-position We follow the proof of [24 Theorem 663] withnecessary modifications

Equipped with Haar measure and the quasi-distancedefined by the homogeneous norm the group 119866 is a space ofhomogeneous type in the sense of Coifman and Weiss [25]On such type of spaces Christ [26] constructed a dyadic gridanalogous to that of the Euclidean space as follows

Lemma A1 Let 119866 be a stratified Lie group There exists acollection 119876119896

120572 119896 isin Z 120572 isin A119896 of open subsets of 119866 where

A119896 is some (possibly finite) index set and constants 120575 isin (0 1)and 1198601 1198602 gt 0 such that

(i) |119866 ⋃120572isinA119896

119876119896

120572| = 0 for each fixed 119896 and 119876119896

120572cap 119876

119896

120573= 0

if 120572 = 120573

(ii) for any 120572 120573 119896 ℓ with ℓ ge 119896 either 119876ℓ

120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0

(iii) for each (119896 120572) and ℓ lt 119896 there exists a unique 120573 suchthat 119876119896

120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896

120572contains some ball 119861(119911119896

120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896

120572can be thought of as a dyadic cube with

diameter roughly 120575119896 and centered at 119911119896120572 We denote byD the

family of all dyadic cubes on 119866 For 119896 isin Z we setD119896 = 119876119896

120572

120572 isin A119896 so thatD = ⋃119896isinZ D119896 For any dyadic cube 119876 isin Dwe denote by 119911119876 the ldquocenterrdquo of119876 andby 119896119876 the unique integer119896 such that 119876 isin D119896

Without loss of generality in what follows we assume 120575 =12 Otherwise we need to replace 2119895 in Definition 4 by 120575minus119895and alsomake some other necessary changes see [27 pp 96ndash98] for more details

Definition A2 Let 119876 be a dyadic cube and let 119871 be anonnegative integer A smooth function 119886119876 on 119866 is called asmooth L-atom for 119876 if it satisfies

(i) 119886119876 is supported in 119861(119911119876 (120574(1 + 1198601)1198602) 2minus119896119876)

(ii) int 119886119876(119909)119875(119909)119889119909 = 0 for all 119875 isin P119871

(iii) |119883119868119886119876(119909)| le |119876|

minus(119889(119868)Δ)minus(12) for all multi-indices 119868with |119868| le 119871 + 1


Definition A3 Let 120572 isin R and 0 lt 119901 119902 lt infin The sequencespace 119891

120572

119901119902consists of all sequences 119904119876119876isinD such that the

function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)

is in 119871119901(119866) For such a sequence 119904 = 119904119876119876 we set

119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)

The smooth atomic decomposition of homogeneousTriebel-Lizorkin spaces on stratified Lie groups can be statedas follows

Theorem A4 Let 120572 isin R 0 lt 119901 119902 lt infin and let 119871 be anonnegative integer satisfying 119871 gt [Δmax(1 1119901 1119902) minus Δ minus120572] + 1 Then there is a constant 119862Δ119901119902120572 such that for everysequence of smooth 119871-atoms 119886119876119876isinD and every sequence ofcomplex scalars 119904119876119876isinD one has

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)

Conversely there is a constant 1198621015840Δ119901119902120572

such that given anydistribution 119891 isin 120572

119901119902(119866) and any 119871 ge 0 there exists a sequence

of smooth 119871-atoms 119886119876119876isinD such that

119891 = sum

119876isinD

119904119876119886119876 (A4)

where the sum converges in S1015840(119866)P and moreover

10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)

Proof Let 119876 isin Dℓ for some ℓ isin Z and let 119886119876 be a smooth119871-atom for119876 We set 1198861015840

119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported

in 119861(0 (120574(1 + 1198601)1198602)2minusℓ) moreover (since 1 + 2ℓ|119910| 1 for

119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)

where 119873 can be chosen to be arbitrarily large Set 11988610158401015840119876(119909) =

2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840

119876 Then the above inequality

can be rewritten as1003816100381610038161003816100381611988311986811988610158401015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)

Using this estimate (73) and that supp 11988610158401015840119876sub 119861(0 (120574(1 +

1198601))1198602) we also deduce that10038161003816100381610038161003816119883119868(119886

10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)

Thus it follows from Lemma 2 that

100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2

minus(119895andℓ)Δ

1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)

where 120601 isin A and 119873 can be taken to be arbitrarily largeConsequently

1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)

Note that if 119903 isin (0 1] and119873 gt Δ119903 then we have

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860

1)2minus(119895andℓ)

( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0

int(21205741198601)21198962minus(119895andℓ)le|119909minus1119911|le(2120574119860

1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903


times int(21205741198601)21198962minus(119895andℓ)le|119909minus1119911|le(2120574119860

1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903

2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860

12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903

times 119872( sum119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903

2minus(119895andℓ)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)

Here we used the fact that for 119911 isin 119876 and |119909minus1119911| ge

(21205741198601)21198962minus(119895andℓ) one has (1 + 2119895andℓ|119911minus1

119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903

which can be easily verified by using Lemma 19 (iv) and thequasi-triangle inequality satisfied by the homogeneous normThe above estimate and (A10) along with the argument in[24 pp 80-81] yield (A3)

Now we show the converse statement of the theoremBy Lemma 20 there exists 120577 isin S(R+

) such that |120577(120582)| ge119862 gt 0 for 120582 isin [2

minus2(1198960+1) 2

minus2(1198960minus1)] for some (large)

positive integer 1198960 and that 120577 is supported in 119861(0 1) and hasvanishing moments of order 119871 Then it is possible to finda function isin S(R+

) with the properties that supp120595 sub

[2minus2(1198960+1) 2

minus2(1198960minus1)] |(120582)| gt 0 for 120582 isin (2

minus2(1198960+1) 2

minus2(1198960minus1))

and sum119895isinZ (2minus2119895120582)120577(2

minus2119895120582) = 1 for 120582 isin R+ Note that

(2minus21198960 sdot) isin A Given 119891 isin

120572

119901119902(119866) it follows from Lemma 3

that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)

with the convergence in S1015840(119866)P Let us decompose 119891 as

119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)

where we have set for 119876 isin D119895

119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)

It is straightforward to verify that 119886119876 is supported in119861(119911119876 (120574(1 + 1198601)1198602) 2

minus119895) and that 119886119876 has vanishing

moments of the same order as 120577 Moreover for 119876 isin D119895 wehave

1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)

for all |119868| le 119871+1 Hence the function 119886119876 a smooth 119871-atom for119876 Choose any 120582 gt Δmin119901 119902 We note that

sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)

We thus obtain using (38)

10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments

The author would like to thank Professor Hitoshi Arai for hispatient guidance and constant support He is also grateful toProfessor Yoshihiro Sawano for his valuable comments

References

[1] Y Han D Muller and D Yang ldquoA theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Caratheodory spacesrdquo Abstract and Applied Analysis vol 2008Article ID 893409 250 pages 2008

[2] G B Folland ldquoSubelliptic estimates and function spaces onnilpotent Lie groupsrdquoArkiv forMatematik vol 13 no 2 pp 161ndash207 1975

[3] G B Folland and E M Stein Hardy Spaces on HomogeneousGroups vol 28 of Mathematical Notes Princeton UniversityPress Princeton NJ USA 1982

[4] K Saka ldquoBesov spaces and Sobolev spaces on a nilpotent LiegrouprdquoTheTohokuMathematical Journal vol 31 no 4 pp 383ndash437 1979

[5] H Fuhr and A Mayeli ldquoHomogeneous Besov spaces onstratified Lie groups and their wavelet characterizationrdquo Journalof Function Spaces andApplications vol 2012 Article ID 52358641 pages 2012

[6] A W Knapp and E M Stein ldquoIntertwining operators forsemisimple groupsrdquo Annals of Mathematics vol 93 pp 489ndash578 1971

[7] A Koranyi and S Vagi ldquoSingular integrals on homogeneousspaces and some problems of classical analysisrdquo Annali dellaScuola Normale Superiore di Pisa vol 25 pp 575ndash648 1971

[8] M Christ ldquo119871119901 bounds for spectral multipliers on nilpotentgroupsrdquoTransactions of the AmericanMathematical Society vol328 no 1 pp 73ndash81 1991

[9] A Hulanicki ldquoA functional calculus for Rockland operators onnilpotent Lie groupsrdquo Studia Mathematica vol 78 no 3 pp253ndash266 1984

[10] D Geller and A Mayeli ldquoContinuous wavelets and frames onstratified Lie groups Irdquo The Journal of Fourier Analysis andApplications vol 12 no 5 pp 543ndash579 2006

[11] M Frazier and B Jawerth ldquoA discrete transform and decom-positions of distribution spacesrdquo Journal of Functional Analysisvol 93 no 1 pp 34ndash170 1990

[12] A Bonfiglioli E Lanconelli and F Uguzzoni Stratified LieGroups and Potential Theory for Their Sub-Laplacians SpringerMonographs in Mathematics Springer Berlin Germany 2007

[13] M Reed and B Simon Methods of Modern MathematicalPhysics I Functional Analysis Academic Press New York NYUSA 2nd edition 1980

[14] L Grafakos L Liu and D Yang ldquoVector-valued singularintegrals and maximal functions on spaces of homogeneoustyperdquo Mathematica Scandinavica vol 104 no 2 pp 296ndash3102009

[15] V S Rychkov ldquoOn a theorem of Bui Paluszynski and Taible-sonrdquo Proceedings of the Steklov Institute of Mathematics vol 227pp 280ndash292 1999

[16] H TriebelTheory of Function Spaces vol 78 ofMonographs inMathematics Birkhauser Basel Switzerland 1983

[17] W Rudin Functional Analysis International Series in Pure andApplied Mathematics McGraw-Hill New York NY USA 2ndedition 1991

[18] L Paivarinta ldquoEquivalent quasinorms and Fourier multipliersin the Triebel spaces 119865119904

119901119902rdquoMathematische Nachrichten vol 106

pp 101ndash108 1982[19] J-O Stromberg and A TorchinskyWeighted Hardy Spaces vol

1381 of LectureNotes inMathematics Springer Berlin Germany1989

[20] M Bownik B Li D Yang and Y Zhou ldquoWeighted anisotropicproduct Hardy spaces and boundedness of sublinear operatorsrdquoMathematische Nachrichten vol 283 no 3 pp 392ndash442 2010

[21] E M SteinHarmonic Analysis Real-Variable Methods Orthog-onality and Oscillatory Integrals vol 43 of PrincetonMathemat-ical Series PrincetonUniversity Press PrincetonNJUSA 1993

[22] L Grafakos and X Li ldquoBilinear operators on homogeneousgroupsrdquo Journal of Operator Theory vol 44 no 1 pp 63ndash902000

[23] N Bourbaki Groupes et Algebres de Lie (Elements de MathFasc 26 et 37) Chapter IndashIII Hermann Paris France 1960 and1972

[24] L GrafakosModern Fourier Analysis vol 250 ofGraduate Textsin Mathematics Springer New York NY USA 2nd edition2009

[25] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977

[26] M Christ ldquoA 119879(119887) theorem with remarks on analytic capacityand the Cauchy integralrdquo Colloquium Mathematicum vol 60-61 no 2 pp 601ndash628 1990

[27] Y S Han and E T Sawyer ldquoLittlewood-Paley theory onspaces of homogeneous type and the classical function spacesrdquoMemoirs of the AmericanMathematical Society vol 110 no 530pp 1ndash126 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


paper the letter 119862 will denote a positive constant which isindependent of the main variables involved but whose valuemay differ from line to line The notation 119886 ≲ 119887 or 119887 ≳ 119886 forsome variable quantities 119886 and 119887means that 119886 le 119862119887 for someconstant 119862 gt 0 119886 sim 119887 stands for 119886 ≲ 119887 ≲ 119886 We agree that theset N of natural numbers contains 0

2 Preliminaries

In this section we briefly review the basic notions concerningstratified Lie groups and their associated sub-Laplacians Formore details we refer the reader to the monograph by Follandand Stein [3] A Lie group 119866 is called a stratified Lie groupif it is connected and simply connected and its Lie algebra gmay be decomposed as a direct sum g = 1198811 oplus sdot sdot sdot oplus 119881119898 with[1198811 119881119896] = 119881119896+1 for 1 le 119896 le 119898 minus 1 and [1198811 119881119898] = 0 Sucha group 119866 is clearly nilpotent and thus it may be identifiedwith g (as a manifold) via the exponential map exp g rarr 119866Examples of stratified Lie groups include Euclidean spacesR119899

and the Heisenberg group H119899The algebra g is equipped with a family of dilations 120575119905

119905 gt 0 which are the algebra automorphisms defined by

120575119905(

119898

sum

119895=1

119883119895) =

119898

sum

119895=1

119905119895119883119895 (119883119895 isin 119881119895) (1)

Under our identification of 119866 with g 120575119905 may also be viewedas a map 119866 rarr 119866 We generally write 119905119909 instead of 120575119905(119909) for119909 isin 119866 We shall denote by

Δ =

119898

sum

119895=1

119895 [dim (119881119895)] (2)

the homogeneous dimension of 119866A homogeneous norm on G is a continuous function

119909 997891rarr |119909| from 119866 to [0infin) smooth away from 0 (the groupidentity) vanishing only at 0 and satisfying |119909minus1| = |119909| and|119905119909| = 119905|119909| for all 119909 isin 119866 and 119905 gt 0 Homogeneous norms on 119866always exist and any two of them are equivalent We assume119866 is provided with a fixed homogeneous norm It satisfies atriangle inequality there exists a constant 120574 ge 1 such that|119909119910| le 120574(|119909|+|119910|) for all 119909 119910 isin 119866 If 119909 isin 119866 and 119903 gt 0we definethe ball of radius 119903 about 119909 by 119861(119909 119903) = 119910 isin 119866 |119910

minus1119909| lt

119903 The Lebesgue measure on g induces a bi-invariant Haarmeasure on 119866 As done in [3] we fix the normalization ofHaar measure by requiring that the measure of 119861(0 1) be 1We shall denote the measure of any measurable 119864 sub 119866 by |119864|Clearly we have |120575119905(119864)| = 119905

Δ|119864| All integrals on 119866 are with

respect to (the normalization of) Haar measure Convolutionis defined by

119891 lowast 119892 (119909) = int119891 (119910) 119892 (119910minus1119909) 119889119910 = int119891 (119909119910

minus1) 119892 (119910) 119889119910

(3)

We consider g as the Lie algebra of all left-invariant vectorfields on119866 and fix a basis1198831 119883119899 of g obtained as a unionof bases of the119881119895 In particular1198831 119883] with ] = dim(1198811)

is a basis of 1198811 We denote by 1198841 119884119899 the correspondingbasis for right-invariant vector fields that is

119884119895119891 (119909) =119889

119889119905119891 (exp (119905119883119895) 119909) |119905=0 (4)

If 119868 = (1198941 119894119899) isin N119899 is a multi-index we set119883119868= 119883

1198941

1sdot sdot sdot 119883

119894119899

119899

and 119884119868 = 11988411989411sdot sdot sdot 119884

119894119899

119899 Moreover we set

|119868| =

119899

sum

119896=1

119894119896 119889 (119868) =

119899

sum

119896=1

119889119896119894119896 (5)


119896

Then 119883119868 (resp 119884119868) is a left-invariant (resp right-invariant) differential operator homogeneous of degree 119889(119868)with respect to the dilations 120575119905 119905 gt 0

A complex-valued function 119875 on119866 is called a polynomialon119866 if 119875 ∘ exp is a polynomial on g Let 1205851 120585119899 be the basisfor the linear forms on g dual to the basis 1198831 119883119899 for gand set 120578119895 = 120585119895 ∘ exp

minus1 From our definition of polynomialson 119866 1205781 120578119899 are generators of the algebra of polynomialson 119866 Thus every polynomial on 119866 can be written uniquelyas

119875 = sum

119868

119886119868120578119868 119886119868 isin C (6)

where all but finitely many of the coefficients vanish and120578119868= 120578

1198941 sdot sdot sdot 120578

119894119899 A polynomial of the type (6) is called of

homogeneous degree 119871 where 119871 isin N if 119889(119868) le 119871 holds forall multi-indices 119868 with 119886119868 = 0 We let P denote the spaceof all polynomials on 119866 and let P119871 denote the space ofpolynomials on 119866 of homogeneous degree 119871 Note thatP119871 isinvariant under left and right translations (see [3 Proposition125]) A function 119891 119866 rarr C is said to have vanishingmoments of order 119871 if

forall119875 isin P119871 int119866

119891 (119909) 119875 (119909) 119889119909 = 0 (7)

with the absolute convergence of the integralThe Schwartz class on 119866 is defined by

S (119866) = 119891 isin 119862infin(119866) 119875

120597|119868|119891

1205971205781198941 sdot sdot sdot 120597120578119894119899isin 119871

infin(119866)

forall119868 isin N119899 forall119875 isin P

(8)

that is 119891 isin S(119866) if and only if 119891 ∘ exp isin S(g) As is indicatedin [3 p 35] S(119866) is a Frechet space and several differentchoices of families of norms induce the same topology onS(119866) In this paper for our purpose we use the family sdot (119871119873) 119871119873 isin N of norms given by

10038171003817100381710038171206011003817100381710038171003817(119871119873) = sup

119889(119868)le119871119909isin119866

(1 + |119909|)119873(10038161003816100381610038161003816119883119868120601 (119909)

10038161003816100381610038161003816+10038161003816100381610038161003816119883119868120601 (119909)

10038161003816100381610038161003816) (9)

Here and in what follows we use the notation convention120601(119909) = 120601(119909

minus1) for any function 120601 119866 rarr C The dual space






]119895=1

1198832



2(119866) L119906 isin 119871



L = int

infin

0

120582 119889119864 (120582) (10)


119898(L) = int

infin

0

119898(120582) 119889119864 (120582) (11)







119898(L) is in S(119866)








119863119905ℎ (119909) = 119905Δℎ (119905119909) (13)



1003816100381610038161003816(1198632119895120601) lowast (1198632ℓ120595) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(119871+1119873)2


(1 + 2119895andℓ |119909|)119873

(14)






1003816100381610038161003816120601 lowast (1198632ℓ120595) (119909)1003816100381610038161003816

le int10038161003816100381610038161003816120601 (119909119910

minus1) minus 119875119909120601 (119910

minus1)100381610038161003816100381610038161003816100381610038161003816(1198632ℓ120595) (119910)

1003816100381610038161003816 119889119910

equiv int1198631

+int1198632

+int1198633

(15)


10038161003816100381610038161003816120601 (119909119910

minus1) minus 119875119909120601 (119910

minus1)10038161003816100381610038161003816

≲10038161003816100381610038161199101003816100381610038161003816119871+1 sup

|119911|le119887119871+1

|119910|119889(119868)=119871+1

10038161003816100381610038161003816(119883

119868120601) (119909119911)

10038161003816100381610038161003816

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873) sup

|119911|le119887119871+1

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + |119909119911|)119873≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + |119909|)119873

(16)



int1198631

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2ℓΔ

(1 + |119909|)119873int1198631

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889119910

=10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ(119871+1)

(1 + |119909|)119873int119866

100381610038161003816100381610038162ℓ11991010038161003816100381610038161003816

119871+1

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889 (2

ℓ119910)

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ(119871+1)

(1 + |119909|)119873

(17)





|119909|2120574) = (120574 + (12))|119909| Thus we have

1 + 2ℓ 1003816100381610038161003816119910

1003816100381610038161003816 ge 2ℓ 1003816100381610038161003816119910

1003816100381610038161003816 ge2ℓ(1 +

10038161003816100381610038161199101003816100381610038161003816)

2ge2ℓ(1 + |119909|)

4120574

10038161003816100381610038161199101003816100381610038161003816119871le (120574 +

1

2)

119871

|119909|119871

(18)


119875119909120601 (119910)

= 120601 (119909) +

119871

sum

ℎ=1

ℎ

sum

119896=1

sum

1le1198941119894119896le119899

1198891198941+sdotsdotsdot+119889

119894119896=ℎ

1205781198941

(119910) 120578119894119896

(119910)

1198961198831198941

sdot sdot sdot 119883119894119896

120601 (119909)

(19)


119895


int1198632

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

times int1198632

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+ sum

0leℎle119871

10038161003816100381610038161199101003816100381610038161003816ℎ

(1 + |119909|)119873]

times2ℓΔ

(1 +10038161003816100381610038162ℓ1199101003816100381610038161003816)119873119889119910

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873) int

1198632

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+

|119909|119871

(1 + |119909|)119873]

times2ℓΔ

[2ℓ (1 + |119909|)]119873119889119910

=10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873

times [int119866

1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873119889119910

+|119909|

119871

(1 + |119909|)119873int|119909119910minus1|le|119909|2120574

119889119910]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873

times [int119866

1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873119889119910

+|119909|

119871

(1 + |119909|)119873int|119910|le120574(|119909|+|119909|2120574)

119889119910]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873[1 +

|119909|119871+Δ

(1 + |119909|)119873]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873

(20)




int1198633

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

times int1198633

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+ sum

0leℎle119871

10038161003816100381610038161199101003816100381610038161003816ℎ

(1 + |119909|)119873]

times2ℓΔ

(1 +10038161003816100381610038162ℓ1199101003816100381610038161003816)119873119889119910

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2ℓΔ

(1 + |119909|)119873int|119910|ge1

10038161003816100381610038161199101003816100381610038161003816119871

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889119910

le10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873int119866

100381610038161003816100381610038162ℓ11991010038161003816100381610038161003816

119871

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889 (2

ℓ119910)

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873

(21)



1003816100381610038161003816120601 lowast (1198632ℓ120595) (119909)1003816100381610038161003816 le

10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(119871+1119873)2

minusℓ119871 1

(1 + |119909|)119873

(22)



space S1015840(119866)P






sum

119895isinZ


+ (23)



sum

|119895|le119896

119892 lowast (1198632119895120601) (24)


S1015840(119866)P


sum

|119895|le119896

119906 lowast (1198632119895120601) (25)




(1 + |119909|)119873 10038161003816100381610038161003816119883119868[119892 lowast (1198632119895120601)] (119909)

10038161003816100381610038161003816

= 2119889(119868)119895

(1 + |119909|)119873 10038161003816100381610038161003816119892 lowast [1198632119895 (119883

119868120601)] (119909)

10038161003816100381610038161003816

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus(119871+Δ+2)

(1 + |119909|)119873

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus119873

(1 + |119909|)119873


(26)



sum119895isinZ(1 + |119909|)119873|119883



119892 = sum

119895isinZ

120601 (2minus2119895

L) 119892 = sum

119895isinZ

119892 lowast (1198632119895120601) (27)



satisfying


10038161003816100381610038161003816120601 (120582)

10038161003816100381610038161003816ge 119862 gt 0 for 120582 isin [(3

5)

2

(5

3)

2

]

(28)


119901119902(L 120601) as


10038171003817100381710038171198911003817100381710038171003817120572119901119902(L120601)

equiv

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

lt infin (29)




119886 119905 gt 0 we define

119872lowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1003816100381610038161003816119891 lowast (119863119905minus1120601) (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

119872lowastlowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (119863119905minus1120601)] (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(30)



119872lowastlowast

1198862minus119895(119891L 120601) (119909) le 1198622

119895119872

lowast

1198862minus119895(119891L 120601) (119909) (31)



2] such that

sum119895isinZ 120601(2minus2119895120582)(2


sum1

119895=minus1120601(2

minus2119895120582)(2



1003816100381610038161003816

=1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601) lowast (1198632119895120577)] (119910)

1003816100381610038161003816

le int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816[119883119896 (1198632119895120577)] (119911

minus1119910)10038161003816100381610038161003816119889119911

= 2119895(Δ+1)

int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816(119883119896120577) (2

119895(119911

minus1119910))

10038161003816100381610038161003816119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

times int 2119895Δ(1 + 2

119895 10038161003816100381610038161003816119911minus111990910038161003816100381610038161003816)119886


119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(32)




119872lowast

1198862minus119895(119891L 120601) (119909) le 119862 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

(33)




1003816100381610038161003816119892 (119910) minus 119892 (1199100)1003816100381610038161003816

le 11986210038161003816100381610038161003816119910minus1

011991010038161003816100381610038161003816

sup|119911|le119887|119910

minus1

0119910|

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

le 119862120575 sup|119911|le119887120575

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

(34)


1003816100381610038161003816 le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896119892 (119910)1003816100381610038161003816

+ 120575minus119886(int

|119910minus11199100|le120575

1003816100381610038161003816119892 (119910)1003816100381610038161003816119903119889119910)

1119903

(35)


0119909|)


1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601)] (119910)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(1 + 2119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886(int

|119910minus11199100|le120575

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575 sup|119910minus11199100|le119887120575

(1 + 2119895 10038161003816100381610038161003816119910minus1119910010038161003816100381610038161003816)119886

119872lowastlowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886


0119909|)

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575(1 + 2119895119887120575)

119886

2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886120574119886(120575 +

10038161003816100381610038161003816119910minus1

011990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

le 119862120576(1 + 119887120576)119886119872

lowast

1198862minus119895(119891L 120601) (119909)

+ 120574119886(1 + 120576

minus1)119886

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119909)]

1119903

(36)


2119895|119910minus1

0119909|) ≲ 1 + 120576





10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(1) 120601(1))

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(2) 120601(2))

119891 isin S1015840(119866) P (37)


100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(119894) 120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119891 lowast (1198632119895120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(38)



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(1) 120601

(1))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(2) 120601

(2))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(39)



(1)(2minus2119895120582) = 1 for 120582 isin


119891 = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1)) (40)




119891 lowast (1198632ℓ120601(2)) (119910) = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1))


(41)



(2)) (119910)

10038161003816100381610038161003816

le sum

119895isinZ

int10038161003816100381610038161003816119891 lowast (1198632119895120601

(1)) (119911)

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911

minus1119910)10038161003816100381610038161003816119889119911

le sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

times int (1 + 2119895|119911|)

119886 10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911)

10038161003816100381610038161003816119889119911

equiv sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910) 119868119895ℓ

(42)

Since both 120595(1) and 120601


119868119895ℓ le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119871+Δ+2)

119889119911

le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119886+Δ+1)

119889119911



(1 + 2119895andℓ|119911|)

minus(Δ+1)

119889119911


(43)


(1)(119871+1119871+Δ+2)120601


observe that

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

le 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

≲ 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

times (1 + 2ℓ 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

max (1 2(119895minusℓ)119886)

(44)


2ℓ120572119872

lowast

1198862minusℓ(119891L

(2) 120601

(2)) (119909)

le sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

(45)




119901119902(119866) instead of 120572



lowast


119886119905(119891L 120601)(119909)


120572

119901119902(119866) 997893rarr

S1015840(119866)P


1003816100381610038161003816119892 lowast (1198632119895120601) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus(119871+Δ+2)

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus119873

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(46)


10038171003817100381710038171198921003817100381710038171003817120572119901119902(119866)

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)(sum

119895isinZ


)

1119902

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)

(47)


119901119902(119866) continuously


119901119902(119866)



minus2 2

2] such that

sum119895isinZ 120601(2minus2119895120582)120577(2


we have

1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⟨sum

119895isinZ

119891 lowast (1198632119895120601) lowast (1198632119895120577) 120595⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

119895isinZ

10038161003816100381610038161003816⟨119891 lowast (1198632119895120601) 120595 lowast (1198632119895120577)⟩

10038161003816100381610038161003816

le sum

119895isinZ

1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin

10038171003817100381710038171003817120595 lowast (1198632119895120577)

100381710038171003817100381710038171198711

(48)


1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin le 1198622

119895Δ119901 1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871119901

(49)



1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 ≲ (sup

119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901)

times sum

119895isinZ

2119895[(Δ119901)minus120572]10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

(50)


sup119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (51)



10038161003816100381610038161003816le 119862

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(52)


sum

119895isinZ

2119895(Δ119901minus120572)10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

≲ sum

119895isinZ


1003817100381710038171003817(119871+1119871+Δ+2)

≲10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)

(53)

Therefore1003816100381610038161003816⟨119891 120595⟩

1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2) (54)




1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816 sim inf

119910isin119861(1199092minus119895)

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119909minus1119910

1003816100381610038161003816)Δ119903

le inf119910isin119861(1199092minus119895)

119872lowast

Δ1199032minus119895(119891 120601) (119910)

≲ inf119910isin119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

1119903

(55)


1003816100381610038161003816119901

≲1

1003816100381610038161003816119861 (119909 2minus119895)1003816100381610038161003816

int119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

119901119903

119889119910

le 2119895Δint1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816119901119889119910

(56)










119901119902(119866)




120572

119901119902(119866) is dense in 120572



120572

119901119902(119866) such that 119892minus119906120572

119901119902(119866) lt 1205762


119895=minus119873119892lowast (1198632119895120601)120572

119901119902(119866) lt 1205762 Now we

put

ℎ =

119873

sum

119895=minus119873

120601 (2minus2119895

L) 119892 =

119873

sum

119895=minus119873

119892 lowast (1198632119895120601) (57)


ℎ minus 119906120572119901119902(119866) ≲

1003817100381710038171003817ℎ minus 1198921003817100381710038171003817120572119901119902(119866)+1003817100381710038171003817119892 minus 119906

1003817100381710038171003817120572119901119902(119866)

lt 120576 (58)




L120590119891 = int

infin

0

120582120590119889119864 (120582) 119891 119891 isin Dom (L

120590) sub 119871

2(119866) (59)



) cap

120572

119901119902(119866) is dense in 120572





120572


120572

119901119902(119866)

to 120572minus2120590119901119902

(119866)

(ii) Let 119879120590 120572119901119902(119866) rarr 120572minus2120590




119901119902(119866)




22119895120590(2

minus2119895120582) = 120601(2


120601(2minus2119895L) ∘ 120577(L) sub (120601(2


Dom (120601 (2minus2119895

L) ∘ 120577 (L))


= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)


120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904



1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)


120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)


) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to


119901119902(119866) to 120572minus2120590

119901119902(119866) We denote


119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)


119901119902(119866) rarr

120572minus2120590


120572

119901119902(119866) such that



) cap 120572

119901119902(119866) which converges


119891ℓ



119891ℓ isin


120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)


119901119902(119866) to the



120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective




120572minus2120590


119901119902(119866) to 119891


120572

119901119902(119866) Denote



120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590





120572


120572

119901119902(119866) rarr

120572minus2120590


119901119902(119866) is





119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902


2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin


(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)





10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)


10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)


+= R119899



120572

119902(119906)119871119901 First note that


119887119902(119906)119871119901 sim 119878

120572



10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)




(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














]119895=1

1198832



2(119866) L119906 isin 119871



L = int

infin

0

120582 119889119864 (120582) (10)


119898(L) = int

infin

0

119898(120582) 119889119864 (120582) (11)







119898(L) is in S(119866)








119863119905ℎ (119909) = 119905Δℎ (119905119909) (13)



1003816100381610038161003816(1198632119895120601) lowast (1198632ℓ120595) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(119871+1119873)2


(1 + 2119895andℓ |119909|)119873

(14)






1003816100381610038161003816120601 lowast (1198632ℓ120595) (119909)1003816100381610038161003816

le int10038161003816100381610038161003816120601 (119909119910

minus1) minus 119875119909120601 (119910

minus1)100381610038161003816100381610038161003816100381610038161003816(1198632ℓ120595) (119910)

1003816100381610038161003816 119889119910

equiv int1198631

+int1198632

+int1198633

(15)


10038161003816100381610038161003816120601 (119909119910

minus1) minus 119875119909120601 (119910

minus1)10038161003816100381610038161003816

≲10038161003816100381610038161199101003816100381610038161003816119871+1 sup

|119911|le119887119871+1

|119910|119889(119868)=119871+1

10038161003816100381610038161003816(119883

119868120601) (119909119911)

10038161003816100381610038161003816

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873) sup

|119911|le119887119871+1

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + |119909119911|)119873≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + |119909|)119873

(16)



int1198631

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2ℓΔ

(1 + |119909|)119873int1198631

10038161003816100381610038161199101003816100381610038161003816119871+1

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889119910

=10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ(119871+1)

(1 + |119909|)119873int119866

100381610038161003816100381610038162ℓ11991010038161003816100381610038161003816

119871+1

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889 (2

ℓ119910)

≲10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ(119871+1)

(1 + |119909|)119873

(17)





|119909|2120574) = (120574 + (12))|119909| Thus we have

1 + 2ℓ 1003816100381610038161003816119910

1003816100381610038161003816 ge 2ℓ 1003816100381610038161003816119910

1003816100381610038161003816 ge2ℓ(1 +

10038161003816100381610038161199101003816100381610038161003816)

2ge2ℓ(1 + |119909|)

4120574

10038161003816100381610038161199101003816100381610038161003816119871le (120574 +

1

2)

119871

|119909|119871

(18)


119875119909120601 (119910)

= 120601 (119909) +

119871

sum

ℎ=1

ℎ

sum

119896=1

sum

1le1198941119894119896le119899

1198891198941+sdotsdotsdot+119889

119894119896=ℎ

1205781198941

(119910) 120578119894119896

(119910)

1198961198831198941

sdot sdot sdot 119883119894119896

120601 (119909)

(19)


119895


int1198632

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

times int1198632

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+ sum

0leℎle119871

10038161003816100381610038161199101003816100381610038161003816ℎ

(1 + |119909|)119873]

times2ℓΔ

(1 +10038161003816100381610038162ℓ1199101003816100381610038161003816)119873119889119910

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873) int

1198632

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+

|119909|119871

(1 + |119909|)119873]

times2ℓΔ

[2ℓ (1 + |119909|)]119873119889119910

=10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873

times [int119866

1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873119889119910

+|119909|

119871

(1 + |119909|)119873int|119909119910minus1|le|119909|2120574

119889119910]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873

times [int119866

1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873119889119910

+|119909|

119871

(1 + |119909|)119873int|119910|le120574(|119909|+|119909|2120574)

119889119910]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873[1 +

|119909|119871+Δ

(1 + |119909|)119873]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873

(20)




int1198633

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

times int1198633

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+ sum

0leℎle119871

10038161003816100381610038161199101003816100381610038161003816ℎ

(1 + |119909|)119873]

times2ℓΔ

(1 +10038161003816100381610038162ℓ1199101003816100381610038161003816)119873119889119910

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2ℓΔ

(1 + |119909|)119873int|119910|ge1

10038161003816100381610038161199101003816100381610038161003816119871

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889119910

le10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873int119866

100381610038161003816100381610038162ℓ11991010038161003816100381610038161003816

119871

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889 (2

ℓ119910)

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873

(21)



1003816100381610038161003816120601 lowast (1198632ℓ120595) (119909)1003816100381610038161003816 le

10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(119871+1119873)2

minusℓ119871 1

(1 + |119909|)119873

(22)



space S1015840(119866)P






sum

119895isinZ


+ (23)



sum

|119895|le119896

119892 lowast (1198632119895120601) (24)


S1015840(119866)P


sum

|119895|le119896

119906 lowast (1198632119895120601) (25)




(1 + |119909|)119873 10038161003816100381610038161003816119883119868[119892 lowast (1198632119895120601)] (119909)

10038161003816100381610038161003816

= 2119889(119868)119895

(1 + |119909|)119873 10038161003816100381610038161003816119892 lowast [1198632119895 (119883

119868120601)] (119909)

10038161003816100381610038161003816

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus(119871+Δ+2)

(1 + |119909|)119873

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus119873

(1 + |119909|)119873


(26)



sum119895isinZ(1 + |119909|)119873|119883



119892 = sum

119895isinZ

120601 (2minus2119895

L) 119892 = sum

119895isinZ

119892 lowast (1198632119895120601) (27)



satisfying


10038161003816100381610038161003816120601 (120582)

10038161003816100381610038161003816ge 119862 gt 0 for 120582 isin [(3

5)

2

(5

3)

2

]

(28)


119901119902(L 120601) as


10038171003817100381710038171198911003817100381710038171003817120572119901119902(L120601)

equiv

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

lt infin (29)




119886 119905 gt 0 we define

119872lowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1003816100381610038161003816119891 lowast (119863119905minus1120601) (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

119872lowastlowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (119863119905minus1120601)] (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(30)



119872lowastlowast

1198862minus119895(119891L 120601) (119909) le 1198622

119895119872

lowast

1198862minus119895(119891L 120601) (119909) (31)



2] such that

sum119895isinZ 120601(2minus2119895120582)(2


sum1

119895=minus1120601(2

minus2119895120582)(2



1003816100381610038161003816

=1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601) lowast (1198632119895120577)] (119910)

1003816100381610038161003816

le int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816[119883119896 (1198632119895120577)] (119911

minus1119910)10038161003816100381610038161003816119889119911

= 2119895(Δ+1)

int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816(119883119896120577) (2

119895(119911

minus1119910))

10038161003816100381610038161003816119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

times int 2119895Δ(1 + 2

119895 10038161003816100381610038161003816119911minus111990910038161003816100381610038161003816)119886


119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(32)




119872lowast

1198862minus119895(119891L 120601) (119909) le 119862 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

(33)




1003816100381610038161003816119892 (119910) minus 119892 (1199100)1003816100381610038161003816

le 11986210038161003816100381610038161003816119910minus1

011991010038161003816100381610038161003816

sup|119911|le119887|119910

minus1

0119910|

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

le 119862120575 sup|119911|le119887120575

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

(34)


1003816100381610038161003816 le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896119892 (119910)1003816100381610038161003816

+ 120575minus119886(int

|119910minus11199100|le120575

1003816100381610038161003816119892 (119910)1003816100381610038161003816119903119889119910)

1119903

(35)


0119909|)


1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601)] (119910)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(1 + 2119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886(int

|119910minus11199100|le120575

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575 sup|119910minus11199100|le119887120575

(1 + 2119895 10038161003816100381610038161003816119910minus1119910010038161003816100381610038161003816)119886

119872lowastlowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886


0119909|)

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575(1 + 2119895119887120575)

119886

2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886120574119886(120575 +

10038161003816100381610038161003816119910minus1

011990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

le 119862120576(1 + 119887120576)119886119872

lowast

1198862minus119895(119891L 120601) (119909)

+ 120574119886(1 + 120576

minus1)119886

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119909)]

1119903

(36)


2119895|119910minus1

0119909|) ≲ 1 + 120576





10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(1) 120601(1))

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(2) 120601(2))

119891 isin S1015840(119866) P (37)


100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(119894) 120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119891 lowast (1198632119895120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(38)



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(1) 120601

(1))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(2) 120601

(2))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(39)



(1)(2minus2119895120582) = 1 for 120582 isin


119891 = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1)) (40)




119891 lowast (1198632ℓ120601(2)) (119910) = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1))


(41)



(2)) (119910)

10038161003816100381610038161003816

le sum

119895isinZ

int10038161003816100381610038161003816119891 lowast (1198632119895120601

(1)) (119911)

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911

minus1119910)10038161003816100381610038161003816119889119911

le sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

times int (1 + 2119895|119911|)

119886 10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911)

10038161003816100381610038161003816119889119911

equiv sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910) 119868119895ℓ

(42)

Since both 120595(1) and 120601


119868119895ℓ le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119871+Δ+2)

119889119911

le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119886+Δ+1)

119889119911



(1 + 2119895andℓ|119911|)

minus(Δ+1)

119889119911


(43)


(1)(119871+1119871+Δ+2)120601


observe that

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

le 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

≲ 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

times (1 + 2ℓ 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

max (1 2(119895minusℓ)119886)

(44)


2ℓ120572119872

lowast

1198862minusℓ(119891L

(2) 120601

(2)) (119909)

le sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

(45)




119901119902(119866) instead of 120572



lowast


119886119905(119891L 120601)(119909)


120572

119901119902(119866) 997893rarr

S1015840(119866)P


1003816100381610038161003816119892 lowast (1198632119895120601) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus(119871+Δ+2)

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus119873

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(46)


10038171003817100381710038171198921003817100381710038171003817120572119901119902(119866)

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)(sum

119895isinZ


)

1119902

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)

(47)


119901119902(119866) continuously


119901119902(119866)



minus2 2

2] such that

sum119895isinZ 120601(2minus2119895120582)120577(2


we have

1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⟨sum

119895isinZ

119891 lowast (1198632119895120601) lowast (1198632119895120577) 120595⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

119895isinZ

10038161003816100381610038161003816⟨119891 lowast (1198632119895120601) 120595 lowast (1198632119895120577)⟩

10038161003816100381610038161003816

le sum

119895isinZ

1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin

10038171003817100381710038171003817120595 lowast (1198632119895120577)

100381710038171003817100381710038171198711

(48)


1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin le 1198622

119895Δ119901 1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871119901

(49)



1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 ≲ (sup

119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901)

times sum

119895isinZ

2119895[(Δ119901)minus120572]10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

(50)


sup119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (51)



10038161003816100381610038161003816le 119862

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(52)


sum

119895isinZ

2119895(Δ119901minus120572)10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

≲ sum

119895isinZ


1003817100381710038171003817(119871+1119871+Δ+2)

≲10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)

(53)

Therefore1003816100381610038161003816⟨119891 120595⟩

1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2) (54)




1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816 sim inf

119910isin119861(1199092minus119895)

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119909minus1119910

1003816100381610038161003816)Δ119903

le inf119910isin119861(1199092minus119895)

119872lowast

Δ1199032minus119895(119891 120601) (119910)

≲ inf119910isin119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

1119903

(55)


1003816100381610038161003816119901

≲1

1003816100381610038161003816119861 (119909 2minus119895)1003816100381610038161003816

int119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

119901119903

119889119910

le 2119895Δint1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816119901119889119910

(56)










119901119902(119866)




120572

119901119902(119866) is dense in 120572



120572

119901119902(119866) such that 119892minus119906120572

119901119902(119866) lt 1205762


119895=minus119873119892lowast (1198632119895120601)120572

119901119902(119866) lt 1205762 Now we

put

ℎ =

119873

sum

119895=minus119873

120601 (2minus2119895

L) 119892 =

119873

sum

119895=minus119873

119892 lowast (1198632119895120601) (57)


ℎ minus 119906120572119901119902(119866) ≲

1003817100381710038171003817ℎ minus 1198921003817100381710038171003817120572119901119902(119866)+1003817100381710038171003817119892 minus 119906

1003817100381710038171003817120572119901119902(119866)

lt 120576 (58)




L120590119891 = int

infin

0

120582120590119889119864 (120582) 119891 119891 isin Dom (L

120590) sub 119871

2(119866) (59)



) cap

120572

119901119902(119866) is dense in 120572





120572


120572

119901119902(119866)

to 120572minus2120590119901119902

(119866)

(ii) Let 119879120590 120572119901119902(119866) rarr 120572minus2120590




119901119902(119866)




22119895120590(2

minus2119895120582) = 120601(2


120601(2minus2119895L) ∘ 120577(L) sub (120601(2


Dom (120601 (2minus2119895

L) ∘ 120577 (L))


= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)


120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904



1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)


120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)


) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to


119901119902(119866) to 120572minus2120590

119901119902(119866) We denote


119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)


119901119902(119866) rarr

120572minus2120590


120572

119901119902(119866) such that



) cap 120572

119901119902(119866) which converges


119891ℓ



119891ℓ isin


120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)


119901119902(119866) to the



120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective




120572minus2120590


119901119902(119866) to 119891


120572

119901119902(119866) Denote



120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590





120572


120572

119901119902(119866) rarr

120572minus2120590


119901119902(119866) is





119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902


2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin


(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)





10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)


10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)


+= R119899



120572

119902(119906)119871119901 First note that


119887119902(119906)119871119901 sim 119878

120572



10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)




(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












|119909|2120574) = (120574 + (12))|119909| Thus we have

1 + 2ℓ 1003816100381610038161003816119910

1003816100381610038161003816 ge 2ℓ 1003816100381610038161003816119910

1003816100381610038161003816 ge2ℓ(1 +

10038161003816100381610038161199101003816100381610038161003816)

2ge2ℓ(1 + |119909|)

4120574

10038161003816100381610038161199101003816100381610038161003816119871le (120574 +

1

2)

119871

|119909|119871

(18)


119875119909120601 (119910)

= 120601 (119909) +

119871

sum

ℎ=1

ℎ

sum

119896=1

sum

1le1198941119894119896le119899

1198891198941+sdotsdotsdot+119889

119894119896=ℎ

1205781198941

(119910) 120578119894119896

(119910)

1198961198831198941

sdot sdot sdot 119883119894119896

120601 (119909)

(19)


119895


int1198632

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

times int1198632

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+ sum

0leℎle119871

10038161003816100381610038161199101003816100381610038161003816ℎ

(1 + |119909|)119873]

times2ℓΔ

(1 +10038161003816100381610038162ℓ1199101003816100381610038161003816)119873119889119910

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873) int

1198632

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+

|119909|119871

(1 + |119909|)119873]

times2ℓΔ

[2ℓ (1 + |119909|)]119873119889119910

=10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873

times [int119866

1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873119889119910

+|119909|

119871

(1 + |119909|)119873int|119909119910minus1|le|119909|2120574

119889119910]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873

times [int119866

1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873119889119910

+|119909|

119871

(1 + |119909|)119873int|119910|le120574(|119909|+|119909|2120574)

119889119910]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)


(1 + |119909|)119873[1 +

|119909|119871+Δ

(1 + |119909|)119873]

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873

(20)




int1198633

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

times int1198633

[1

(1 +1003816100381610038161003816119909119910

minus11003816100381610038161003816)119873+ sum

0leℎle119871

10038161003816100381610038161199101003816100381610038161003816ℎ

(1 + |119909|)119873]

times2ℓΔ

(1 +10038161003816100381610038162ℓ1199101003816100381610038161003816)119873119889119910

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2ℓΔ

(1 + |119909|)119873int|119910|ge1

10038161003816100381610038161199101003816100381610038161003816119871

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889119910

le10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873int119866

100381610038161003816100381610038162ℓ11991010038161003816100381610038161003816

119871

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873119889 (2

ℓ119910)

≲10038171003817100381710038171206011003817100381710038171003817(119871119873)

10038171003817100381710038171205951003817100381710038171003817(0119873)

2minusℓ119871

(1 + |119909|)119873

(21)



1003816100381610038161003816120601 lowast (1198632ℓ120595) (119909)1003816100381610038161003816 le

10038171003817100381710038171206011003817100381710038171003817(119871+1119873)

10038171003817100381710038171205951003817100381710038171003817(119871+1119873)2

minusℓ119871 1

(1 + |119909|)119873

(22)



space S1015840(119866)P






sum

119895isinZ


+ (23)



sum

|119895|le119896

119892 lowast (1198632119895120601) (24)


S1015840(119866)P


sum

|119895|le119896

119906 lowast (1198632119895120601) (25)




(1 + |119909|)119873 10038161003816100381610038161003816119883119868[119892 lowast (1198632119895120601)] (119909)

10038161003816100381610038161003816

= 2119889(119868)119895

(1 + |119909|)119873 10038161003816100381610038161003816119892 lowast [1198632119895 (119883

119868120601)] (119909)

10038161003816100381610038161003816

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus(119871+Δ+2)

(1 + |119909|)119873

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus119873

(1 + |119909|)119873


(26)



sum119895isinZ(1 + |119909|)119873|119883



119892 = sum

119895isinZ

120601 (2minus2119895

L) 119892 = sum

119895isinZ

119892 lowast (1198632119895120601) (27)



satisfying


10038161003816100381610038161003816120601 (120582)

10038161003816100381610038161003816ge 119862 gt 0 for 120582 isin [(3

5)

2

(5

3)

2

]

(28)


119901119902(L 120601) as


10038171003817100381710038171198911003817100381710038171003817120572119901119902(L120601)

equiv

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

lt infin (29)




119886 119905 gt 0 we define

119872lowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1003816100381610038161003816119891 lowast (119863119905minus1120601) (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

119872lowastlowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (119863119905minus1120601)] (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(30)



119872lowastlowast

1198862minus119895(119891L 120601) (119909) le 1198622

119895119872

lowast

1198862minus119895(119891L 120601) (119909) (31)



2] such that

sum119895isinZ 120601(2minus2119895120582)(2


sum1

119895=minus1120601(2

minus2119895120582)(2



1003816100381610038161003816

=1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601) lowast (1198632119895120577)] (119910)

1003816100381610038161003816

le int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816[119883119896 (1198632119895120577)] (119911

minus1119910)10038161003816100381610038161003816119889119911

= 2119895(Δ+1)

int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816(119883119896120577) (2

119895(119911

minus1119910))

10038161003816100381610038161003816119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

times int 2119895Δ(1 + 2

119895 10038161003816100381610038161003816119911minus111990910038161003816100381610038161003816)119886


119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(32)




119872lowast

1198862minus119895(119891L 120601) (119909) le 119862 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

(33)




1003816100381610038161003816119892 (119910) minus 119892 (1199100)1003816100381610038161003816

le 11986210038161003816100381610038161003816119910minus1

011991010038161003816100381610038161003816

sup|119911|le119887|119910

minus1

0119910|

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

le 119862120575 sup|119911|le119887120575

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

(34)


1003816100381610038161003816 le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896119892 (119910)1003816100381610038161003816

+ 120575minus119886(int

|119910minus11199100|le120575

1003816100381610038161003816119892 (119910)1003816100381610038161003816119903119889119910)

1119903

(35)


0119909|)


1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601)] (119910)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(1 + 2119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886(int

|119910minus11199100|le120575

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575 sup|119910minus11199100|le119887120575

(1 + 2119895 10038161003816100381610038161003816119910minus1119910010038161003816100381610038161003816)119886

119872lowastlowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886


0119909|)

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575(1 + 2119895119887120575)

119886

2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886120574119886(120575 +

10038161003816100381610038161003816119910minus1

011990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

le 119862120576(1 + 119887120576)119886119872

lowast

1198862minus119895(119891L 120601) (119909)

+ 120574119886(1 + 120576

minus1)119886

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119909)]

1119903

(36)


2119895|119910minus1

0119909|) ≲ 1 + 120576





10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(1) 120601(1))

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(2) 120601(2))

119891 isin S1015840(119866) P (37)


100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(119894) 120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119891 lowast (1198632119895120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(38)



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(1) 120601

(1))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(2) 120601

(2))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(39)



(1)(2minus2119895120582) = 1 for 120582 isin


119891 = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1)) (40)




119891 lowast (1198632ℓ120601(2)) (119910) = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1))


(41)



(2)) (119910)

10038161003816100381610038161003816

le sum

119895isinZ

int10038161003816100381610038161003816119891 lowast (1198632119895120601

(1)) (119911)

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911

minus1119910)10038161003816100381610038161003816119889119911

le sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

times int (1 + 2119895|119911|)

119886 10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911)

10038161003816100381610038161003816119889119911

equiv sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910) 119868119895ℓ

(42)

Since both 120595(1) and 120601


119868119895ℓ le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119871+Δ+2)

119889119911

le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119886+Δ+1)

119889119911



(1 + 2119895andℓ|119911|)

minus(Δ+1)

119889119911


(43)


(1)(119871+1119871+Δ+2)120601


observe that

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

le 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

≲ 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

times (1 + 2ℓ 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

max (1 2(119895minusℓ)119886)

(44)


2ℓ120572119872

lowast

1198862minusℓ(119891L

(2) 120601

(2)) (119909)

le sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

(45)




119901119902(119866) instead of 120572



lowast


119886119905(119891L 120601)(119909)


120572

119901119902(119866) 997893rarr

S1015840(119866)P


1003816100381610038161003816119892 lowast (1198632119895120601) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus(119871+Δ+2)

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus119873

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(46)


10038171003817100381710038171198921003817100381710038171003817120572119901119902(119866)

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)(sum

119895isinZ


)

1119902

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)

(47)


119901119902(119866) continuously


119901119902(119866)



minus2 2

2] such that

sum119895isinZ 120601(2minus2119895120582)120577(2


we have

1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⟨sum

119895isinZ

119891 lowast (1198632119895120601) lowast (1198632119895120577) 120595⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

119895isinZ

10038161003816100381610038161003816⟨119891 lowast (1198632119895120601) 120595 lowast (1198632119895120577)⟩

10038161003816100381610038161003816

le sum

119895isinZ

1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin

10038171003817100381710038171003817120595 lowast (1198632119895120577)

100381710038171003817100381710038171198711

(48)


1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin le 1198622

119895Δ119901 1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871119901

(49)



1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 ≲ (sup

119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901)

times sum

119895isinZ

2119895[(Δ119901)minus120572]10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

(50)


sup119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (51)



10038161003816100381610038161003816le 119862

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(52)


sum

119895isinZ

2119895(Δ119901minus120572)10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

≲ sum

119895isinZ


1003817100381710038171003817(119871+1119871+Δ+2)

≲10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)

(53)

Therefore1003816100381610038161003816⟨119891 120595⟩

1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2) (54)




1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816 sim inf

119910isin119861(1199092minus119895)

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119909minus1119910

1003816100381610038161003816)Δ119903

le inf119910isin119861(1199092minus119895)

119872lowast

Δ1199032minus119895(119891 120601) (119910)

≲ inf119910isin119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

1119903

(55)


1003816100381610038161003816119901

≲1

1003816100381610038161003816119861 (119909 2minus119895)1003816100381610038161003816

int119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

119901119903

119889119910

le 2119895Δint1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816119901119889119910

(56)










119901119902(119866)




120572

119901119902(119866) is dense in 120572



120572

119901119902(119866) such that 119892minus119906120572

119901119902(119866) lt 1205762


119895=minus119873119892lowast (1198632119895120601)120572

119901119902(119866) lt 1205762 Now we

put

ℎ =

119873

sum

119895=minus119873

120601 (2minus2119895

L) 119892 =

119873

sum

119895=minus119873

119892 lowast (1198632119895120601) (57)


ℎ minus 119906120572119901119902(119866) ≲

1003817100381710038171003817ℎ minus 1198921003817100381710038171003817120572119901119902(119866)+1003817100381710038171003817119892 minus 119906

1003817100381710038171003817120572119901119902(119866)

lt 120576 (58)




L120590119891 = int

infin

0

120582120590119889119864 (120582) 119891 119891 isin Dom (L

120590) sub 119871

2(119866) (59)



) cap

120572

119901119902(119866) is dense in 120572





120572


120572

119901119902(119866)

to 120572minus2120590119901119902

(119866)

(ii) Let 119879120590 120572119901119902(119866) rarr 120572minus2120590




119901119902(119866)




22119895120590(2

minus2119895120582) = 120601(2


120601(2minus2119895L) ∘ 120577(L) sub (120601(2


Dom (120601 (2minus2119895

L) ∘ 120577 (L))


= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)


120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904



1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)


120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)


) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to


119901119902(119866) to 120572minus2120590

119901119902(119866) We denote


119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)


119901119902(119866) rarr

120572minus2120590


120572

119901119902(119866) such that



) cap 120572

119901119902(119866) which converges


119891ℓ



119891ℓ isin


120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)


119901119902(119866) to the



120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective




120572minus2120590


119901119902(119866) to 119891


120572

119901119902(119866) Denote



120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590





120572


120572

119901119902(119866) rarr

120572minus2120590


119901119902(119866) is





119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902


2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin


(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)





10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)


10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)


+= R119899



120572

119902(119906)119871119901 First note that


119887119902(119906)119871119901 sim 119878

120572



10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)




(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













sum

119895isinZ


+ (23)



sum

|119895|le119896

119892 lowast (1198632119895120601) (24)


S1015840(119866)P


sum

|119895|le119896

119906 lowast (1198632119895120601) (25)




(1 + |119909|)119873 10038161003816100381610038161003816119883119868[119892 lowast (1198632119895120601)] (119909)

10038161003816100381610038161003816

= 2119889(119868)119895

(1 + |119909|)119873 10038161003816100381610038161003816119892 lowast [1198632119895 (119883

119868120601)] (119909)

10038161003816100381610038161003816

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus(119871+Δ+2)

(1 + |119909|)119873

le 1198622119889(119868)119895

2minus|119895|119871

2(119895and0)Δ

(1 + 2(119895and0)

|119909|)minus119873

(1 + |119909|)119873


(26)



sum119895isinZ(1 + |119909|)119873|119883



119892 = sum

119895isinZ

120601 (2minus2119895

L) 119892 = sum

119895isinZ

119892 lowast (1198632119895120601) (27)



satisfying


10038161003816100381610038161003816120601 (120582)

10038161003816100381610038161003816ge 119862 gt 0 for 120582 isin [(3

5)

2

(5

3)

2

]

(28)


119901119902(L 120601) as


10038171003817100381710038171198911003817100381710038171003817120572119901119902(L120601)

equiv

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

lt infin (29)




119886 119905 gt 0 we define

119872lowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1003816100381610038161003816119891 lowast (119863119905minus1120601) (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

119872lowastlowast

119886119905(119891L 120601) (119909) = sup

119910isin119866

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (119863119905minus1120601)] (119909)1003816100381610038161003816

(1 + 119905minus11003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(30)



119872lowastlowast

1198862minus119895(119891L 120601) (119909) le 1198622

119895119872

lowast

1198862minus119895(119891L 120601) (119909) (31)



2] such that

sum119895isinZ 120601(2minus2119895120582)(2


sum1

119895=minus1120601(2

minus2119895120582)(2



1003816100381610038161003816

=1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601) lowast (1198632119895120577)] (119910)

1003816100381610038161003816

le int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816[119883119896 (1198632119895120577)] (119911

minus1119910)10038161003816100381610038161003816119889119911

= 2119895(Δ+1)

int1003816100381610038161003816119891 lowast (1198632119895120601) (119911)

100381610038161003816100381610038161003816100381610038161003816(119883119896120577) (2

119895(119911

minus1119910))

10038161003816100381610038161003816119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

times int 2119895Δ(1 + 2

119895 10038161003816100381610038161003816119911minus111990910038161003816100381610038161003816)119886


119889119911

≲ 2119895119872

lowast

1198862minus119895(119891L 120601) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(32)




119872lowast

1198862minus119895(119891L 120601) (119909) le 119862 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

(33)




1003816100381610038161003816119892 (119910) minus 119892 (1199100)1003816100381610038161003816

le 11986210038161003816100381610038161003816119910minus1

011991010038161003816100381610038161003816

sup|119911|le119887|119910

minus1

0119910|

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

le 119862120575 sup|119911|le119887120575

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

(34)


1003816100381610038161003816 le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896119892 (119910)1003816100381610038161003816

+ 120575minus119886(int

|119910minus11199100|le120575

1003816100381610038161003816119892 (119910)1003816100381610038161003816119903119889119910)

1119903

(35)


0119909|)


1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601)] (119910)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(1 + 2119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886(int

|119910minus11199100|le120575

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575 sup|119910minus11199100|le119887120575

(1 + 2119895 10038161003816100381610038161003816119910minus1119910010038161003816100381610038161003816)119886

119872lowastlowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886


0119909|)

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575(1 + 2119895119887120575)

119886

2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886120574119886(120575 +

10038161003816100381610038161003816119910minus1

011990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

le 119862120576(1 + 119887120576)119886119872

lowast

1198862minus119895(119891L 120601) (119909)

+ 120574119886(1 + 120576

minus1)119886

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119909)]

1119903

(36)


2119895|119910minus1

0119909|) ≲ 1 + 120576





10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(1) 120601(1))

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(2) 120601(2))

119891 isin S1015840(119866) P (37)


100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(119894) 120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119891 lowast (1198632119895120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(38)



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(1) 120601

(1))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(2) 120601

(2))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(39)



(1)(2minus2119895120582) = 1 for 120582 isin


119891 = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1)) (40)




119891 lowast (1198632ℓ120601(2)) (119910) = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1))


(41)



(2)) (119910)

10038161003816100381610038161003816

le sum

119895isinZ

int10038161003816100381610038161003816119891 lowast (1198632119895120601

(1)) (119911)

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911

minus1119910)10038161003816100381610038161003816119889119911

le sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

times int (1 + 2119895|119911|)

119886 10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911)

10038161003816100381610038161003816119889119911

equiv sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910) 119868119895ℓ

(42)

Since both 120595(1) and 120601


119868119895ℓ le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119871+Δ+2)

119889119911

le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119886+Δ+1)

119889119911



(1 + 2119895andℓ|119911|)

minus(Δ+1)

119889119911


(43)


(1)(119871+1119871+Δ+2)120601


observe that

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

le 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

≲ 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

times (1 + 2ℓ 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

max (1 2(119895minusℓ)119886)

(44)


2ℓ120572119872

lowast

1198862minusℓ(119891L

(2) 120601

(2)) (119909)

le sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

(45)




119901119902(119866) instead of 120572



lowast


119886119905(119891L 120601)(119909)


120572

119901119902(119866) 997893rarr

S1015840(119866)P


1003816100381610038161003816119892 lowast (1198632119895120601) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus(119871+Δ+2)

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus119873

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(46)


10038171003817100381710038171198921003817100381710038171003817120572119901119902(119866)

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)(sum

119895isinZ


)

1119902

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)

(47)


119901119902(119866) continuously


119901119902(119866)



minus2 2

2] such that

sum119895isinZ 120601(2minus2119895120582)120577(2


we have

1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⟨sum

119895isinZ

119891 lowast (1198632119895120601) lowast (1198632119895120577) 120595⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

119895isinZ

10038161003816100381610038161003816⟨119891 lowast (1198632119895120601) 120595 lowast (1198632119895120577)⟩

10038161003816100381610038161003816

le sum

119895isinZ

1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin

10038171003817100381710038171003817120595 lowast (1198632119895120577)

100381710038171003817100381710038171198711

(48)


1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin le 1198622

119895Δ119901 1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871119901

(49)



1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 ≲ (sup

119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901)

times sum

119895isinZ

2119895[(Δ119901)minus120572]10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

(50)


sup119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (51)



10038161003816100381610038161003816le 119862

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(52)


sum

119895isinZ

2119895(Δ119901minus120572)10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

≲ sum

119895isinZ


1003817100381710038171003817(119871+1119871+Δ+2)

≲10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)

(53)

Therefore1003816100381610038161003816⟨119891 120595⟩

1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2) (54)




1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816 sim inf

119910isin119861(1199092minus119895)

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119909minus1119910

1003816100381610038161003816)Δ119903

le inf119910isin119861(1199092minus119895)

119872lowast

Δ1199032minus119895(119891 120601) (119910)

≲ inf119910isin119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

1119903

(55)


1003816100381610038161003816119901

≲1

1003816100381610038161003816119861 (119909 2minus119895)1003816100381610038161003816

int119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

119901119903

119889119910

le 2119895Δint1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816119901119889119910

(56)










119901119902(119866)




120572

119901119902(119866) is dense in 120572



120572

119901119902(119866) such that 119892minus119906120572

119901119902(119866) lt 1205762


119895=minus119873119892lowast (1198632119895120601)120572

119901119902(119866) lt 1205762 Now we

put

ℎ =

119873

sum

119895=minus119873

120601 (2minus2119895

L) 119892 =

119873

sum

119895=minus119873

119892 lowast (1198632119895120601) (57)


ℎ minus 119906120572119901119902(119866) ≲

1003817100381710038171003817ℎ minus 1198921003817100381710038171003817120572119901119902(119866)+1003817100381710038171003817119892 minus 119906

1003817100381710038171003817120572119901119902(119866)

lt 120576 (58)




L120590119891 = int

infin

0

120582120590119889119864 (120582) 119891 119891 isin Dom (L

120590) sub 119871

2(119866) (59)



) cap

120572

119901119902(119866) is dense in 120572





120572


120572

119901119902(119866)

to 120572minus2120590119901119902

(119866)

(ii) Let 119879120590 120572119901119902(119866) rarr 120572minus2120590




119901119902(119866)




22119895120590(2

minus2119895120582) = 120601(2


120601(2minus2119895L) ∘ 120577(L) sub (120601(2


Dom (120601 (2minus2119895

L) ∘ 120577 (L))


= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)


120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904



1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)


120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)


) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to


119901119902(119866) to 120572minus2120590

119901119902(119866) We denote


119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)


119901119902(119866) rarr

120572minus2120590


120572

119901119902(119866) such that



) cap 120572

119901119902(119866) which converges


119891ℓ



119891ℓ isin


120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)


119901119902(119866) to the



120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective




120572minus2120590


119901119902(119866) to 119891


120572

119901119902(119866) Denote



120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590





120572


120572

119901119902(119866) rarr

120572minus2120590


119901119902(119866) is





119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902


2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin


(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)





10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)


10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)


+= R119899



120572

119902(119906)119871119901 First note that


119887119902(119906)119871119901 sim 119878

120572



10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)




(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1003816100381610038161003816119892 (119910) minus 119892 (1199100)1003816100381610038161003816

le 11986210038161003816100381610038161003816119910minus1

011991010038161003816100381610038161003816

sup|119911|le119887|119910

minus1

0119910|

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

le 119862120575 sup|119911|le119887120575

1le119896le]

1003816100381610038161003816(119883119896119892) (1199100119911)1003816100381610038161003816

(34)


1003816100381610038161003816 le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896119892 (119910)1003816100381610038161003816

+ 120575minus119886(int

|119910minus11199100|le120575

1003816100381610038161003816119892 (119910)1003816100381610038161003816119903119889119910)

1119903

(35)


0119909|)


1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

le 119862120575 sup|119910minus11199100|le119887120575

1le119896le]

1003816100381610038161003816119883119896 [119891 lowast (1198632119895120601)] (119910)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)119886

(1 + 2119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886(int

|119910minus11199100|le120575

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575 sup|119910minus11199100|le119887120575

(1 + 2119895 10038161003816100381610038161003816119910minus1119910010038161003816100381610038161003816)119886

119872lowastlowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886


0119909|)

1003816100381610038161003816119891 lowast (1198632119895120601) (119910)1003816100381610038161003816119903119889119910)

1119903

le 119862120575(1 + 2119895119887120575)

119886

2119895119872

lowast

1198862minus119895(119891L 120601) (119909)

+120575minus119886120574119886(120575 +

10038161003816100381610038161003816119910minus1

011990910038161003816100381610038161003816)119886

(1 + 21198951003816100381610038161003816119910minus101199091003816100381610038161003816)119886 [119872(

1003816100381610038161003816119891 lowast (1198632119895120601)1003816100381610038161003816119903) (119909)]

1119903

le 119862120576(1 + 119887120576)119886119872

lowast

1198862minus119895(119891L 120601) (119909)

+ 120574119886(1 + 120576

minus1)119886

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119909)]

1119903

(36)


2119895|119910minus1

0119909|) ≲ 1 + 120576





10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(1) 120601(1))

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(L(2) 120601(2))

119891 isin S1015840(119866) P (37)


100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(119894) 120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119891 lowast (1198632119895120601

(119894))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(38)



100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(1) 120601

(1))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891L

(2) 120601

(2))10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(39)



(1)(2minus2119895120582) = 1 for 120582 isin


119891 = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1)) (40)




119891 lowast (1198632ℓ120601(2)) (119910) = sum

119895isinZ

119891 lowast (1198632119895120601(1)) lowast (1198632119895120595

(1))


(41)



(2)) (119910)

10038161003816100381610038161003816

le sum

119895isinZ

int10038161003816100381610038161003816119891 lowast (1198632119895120601

(1)) (119911)

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911

minus1119910)10038161003816100381610038161003816119889119911

le sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

times int (1 + 2119895|119911|)

119886 10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911)

10038161003816100381610038161003816119889119911

equiv sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910) 119868119895ℓ

(42)

Since both 120595(1) and 120601


119868119895ℓ le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119871+Δ+2)

119889119911

le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119886+Δ+1)

119889119911



(1 + 2119895andℓ|119911|)

minus(Δ+1)

119889119911


(43)


(1)(119871+1119871+Δ+2)120601


observe that

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

le 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

≲ 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

times (1 + 2ℓ 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

max (1 2(119895minusℓ)119886)

(44)


2ℓ120572119872

lowast

1198862minusℓ(119891L

(2) 120601

(2)) (119909)

le sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

(45)




119901119902(119866) instead of 120572



lowast


119886119905(119891L 120601)(119909)


120572

119901119902(119866) 997893rarr

S1015840(119866)P


1003816100381610038161003816119892 lowast (1198632119895120601) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus(119871+Δ+2)

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus119873

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(46)


10038171003817100381710038171198921003817100381710038171003817120572119901119902(119866)

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)(sum

119895isinZ


)

1119902

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)

(47)


119901119902(119866) continuously


119901119902(119866)



minus2 2

2] such that

sum119895isinZ 120601(2minus2119895120582)120577(2


we have

1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⟨sum

119895isinZ

119891 lowast (1198632119895120601) lowast (1198632119895120577) 120595⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

119895isinZ

10038161003816100381610038161003816⟨119891 lowast (1198632119895120601) 120595 lowast (1198632119895120577)⟩

10038161003816100381610038161003816

le sum

119895isinZ

1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin

10038171003817100381710038171003817120595 lowast (1198632119895120577)

100381710038171003817100381710038171198711

(48)


1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin le 1198622

119895Δ119901 1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871119901

(49)



1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 ≲ (sup

119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901)

times sum

119895isinZ

2119895[(Δ119901)minus120572]10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

(50)


sup119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (51)



10038161003816100381610038161003816le 119862

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(52)


sum

119895isinZ

2119895(Δ119901minus120572)10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

≲ sum

119895isinZ


1003817100381710038171003817(119871+1119871+Δ+2)

≲10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)

(53)

Therefore1003816100381610038161003816⟨119891 120595⟩

1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2) (54)




1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816 sim inf

119910isin119861(1199092minus119895)

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119909minus1119910

1003816100381610038161003816)Δ119903

le inf119910isin119861(1199092minus119895)

119872lowast

Δ1199032minus119895(119891 120601) (119910)

≲ inf119910isin119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

1119903

(55)


1003816100381610038161003816119901

≲1

1003816100381610038161003816119861 (119909 2minus119895)1003816100381610038161003816

int119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

119901119903

119889119910

le 2119895Δint1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816119901119889119910

(56)










119901119902(119866)




120572

119901119902(119866) is dense in 120572



120572

119901119902(119866) such that 119892minus119906120572

119901119902(119866) lt 1205762


119895=minus119873119892lowast (1198632119895120601)120572

119901119902(119866) lt 1205762 Now we

put

ℎ =

119873

sum

119895=minus119873

120601 (2minus2119895

L) 119892 =

119873

sum

119895=minus119873

119892 lowast (1198632119895120601) (57)


ℎ minus 119906120572119901119902(119866) ≲

1003817100381710038171003817ℎ minus 1198921003817100381710038171003817120572119901119902(119866)+1003817100381710038171003817119892 minus 119906

1003817100381710038171003817120572119901119902(119866)

lt 120576 (58)




L120590119891 = int

infin

0

120582120590119889119864 (120582) 119891 119891 isin Dom (L

120590) sub 119871

2(119866) (59)



) cap

120572

119901119902(119866) is dense in 120572





120572


120572

119901119902(119866)

to 120572minus2120590119901119902

(119866)

(ii) Let 119879120590 120572119901119902(119866) rarr 120572minus2120590




119901119902(119866)




22119895120590(2

minus2119895120582) = 120601(2


120601(2minus2119895L) ∘ 120577(L) sub (120601(2


Dom (120601 (2minus2119895

L) ∘ 120577 (L))


= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)


120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904



1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)


120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)


) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to


119901119902(119866) to 120572minus2120590

119901119902(119866) We denote


119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)


119901119902(119866) rarr

120572minus2120590


120572

119901119902(119866) such that



) cap 120572

119901119902(119866) which converges


119891ℓ



119891ℓ isin


120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)


119901119902(119866) to the



120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective




120572minus2120590


119901119902(119866) to 119891


120572

119901119902(119866) Denote



120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590





120572


120572

119901119902(119866) rarr

120572minus2120590


119901119902(119866) is





119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902


2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin


(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)





10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)


10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)


+= R119899



120572

119902(119906)119871119901 First note that


119887119902(119906)119871119901 sim 119878

120572



10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)




(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(2)) (119910)

10038161003816100381610038161003816

le sum

119895isinZ

int10038161003816100381610038161003816119891 lowast (1198632119895120601

(1)) (119911)

10038161003816100381610038161003816

times10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911

minus1119910)10038161003816100381610038161003816119889119911

le sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

times int (1 + 2119895|119911|)

119886 10038161003816100381610038161003816(1198632119895120595

(1)) lowast (1198632ℓ120601

(2)) (119911)

10038161003816100381610038161003816119889119911

equiv sum

119895isinZ

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910) 119868119895ℓ

(42)

Since both 120595(1) and 120601


119868119895ℓ le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119871+Δ+2)

119889119911

le 119862int (1 + 2119895|119911|)

119886


2(119895andℓ)Δ

times (1 + 2119895andℓ|119911|)

minus(119886+Δ+1)

119889119911



(1 + 2119895andℓ|119911|)

minus(Δ+1)

119889119911


(43)


(1)(119871+1119871+Δ+2)120601


observe that

119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119910)

le 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909) (1 + 2

119895 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

≲ 119872lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

times (1 + 2ℓ 10038161003816100381610038161003816119910minus111990910038161003816100381610038161003816)119886

max (1 2(119895minusℓ)119886)

(44)


2ℓ120572119872

lowast

1198862minusℓ(119891L

(2) 120601

(2)) (119909)

le sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891L

(1) 120601

(1)) (119909)

(45)




119901119902(119866) instead of 120572



lowast


119886119905(119891L 120601)(119909)


120572

119901119902(119866) 997893rarr

S1015840(119866)P


1003816100381610038161003816119892 lowast (1198632119895120601) (119909)1003816100381610038161003816

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus(119871+Δ+2)

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|1198712(119895and0)Δ

(1 + 2119895and0

|119909|)minus119873

le 11986210038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(46)


10038171003817100381710038171198921003817100381710038171003817120572119901119902(119866)

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)(sum

119895isinZ


)

1119902

≲10038171003817100381710038171198921003817100381710038171003817(119871+1119871+Δ+2)

(47)


119901119902(119866) continuously


119901119902(119866)



minus2 2

2] such that

sum119895isinZ 120601(2minus2119895120582)120577(2


we have

1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

⟨sum

119895isinZ

119891 lowast (1198632119895120601) lowast (1198632119895120577) 120595⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sum

119895isinZ

10038161003816100381610038161003816⟨119891 lowast (1198632119895120601) 120595 lowast (1198632119895120577)⟩

10038161003816100381610038161003816

le sum

119895isinZ

1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin

10038171003817100381710038171003817120595 lowast (1198632119895120577)

100381710038171003817100381710038171198711

(48)


1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871infin le 1198622

119895Δ119901 1003817100381710038171003817119891 lowast (1198632119895120601)1003817100381710038171003817119871119901

(49)



1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 ≲ (sup

119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901)

times sum

119895isinZ

2119895[(Δ119901)minus120572]10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

(50)


sup119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (51)



10038161003816100381610038161003816le 119862

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(52)


sum

119895isinZ

2119895(Δ119901minus120572)10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

≲ sum

119895isinZ


1003817100381710038171003817(119871+1119871+Δ+2)

≲10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)

(53)

Therefore1003816100381610038161003816⟨119891 120595⟩

1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2) (54)




1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816 sim inf

119910isin119861(1199092minus119895)

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119909minus1119910

1003816100381610038161003816)Δ119903

le inf119910isin119861(1199092minus119895)

119872lowast

Δ1199032minus119895(119891 120601) (119910)

≲ inf119910isin119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

1119903

(55)


1003816100381610038161003816119901

≲1

1003816100381610038161003816119861 (119909 2minus119895)1003816100381610038161003816

int119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

119901119903

119889119910

le 2119895Δint1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816119901119889119910

(56)










119901119902(119866)




120572

119901119902(119866) is dense in 120572



120572

119901119902(119866) such that 119892minus119906120572

119901119902(119866) lt 1205762


119895=minus119873119892lowast (1198632119895120601)120572

119901119902(119866) lt 1205762 Now we

put

ℎ =

119873

sum

119895=minus119873

120601 (2minus2119895

L) 119892 =

119873

sum

119895=minus119873

119892 lowast (1198632119895120601) (57)


ℎ minus 119906120572119901119902(119866) ≲

1003817100381710038171003817ℎ minus 1198921003817100381710038171003817120572119901119902(119866)+1003817100381710038171003817119892 minus 119906

1003817100381710038171003817120572119901119902(119866)

lt 120576 (58)




L120590119891 = int

infin

0

120582120590119889119864 (120582) 119891 119891 isin Dom (L

120590) sub 119871

2(119866) (59)



) cap

120572

119901119902(119866) is dense in 120572





120572


120572

119901119902(119866)

to 120572minus2120590119901119902

(119866)

(ii) Let 119879120590 120572119901119902(119866) rarr 120572minus2120590




119901119902(119866)




22119895120590(2

minus2119895120582) = 120601(2


120601(2minus2119895L) ∘ 120577(L) sub (120601(2


Dom (120601 (2minus2119895

L) ∘ 120577 (L))


= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)


120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904



1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)


120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)


) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to


119901119902(119866) to 120572minus2120590

119901119902(119866) We denote


119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)


119901119902(119866) rarr

120572minus2120590


120572

119901119902(119866) such that



) cap 120572

119901119902(119866) which converges


119891ℓ



119891ℓ isin


120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)


119901119902(119866) to the



120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective




120572minus2120590


119901119902(119866) to 119891


120572

119901119902(119866) Denote



120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590





120572


120572

119901119902(119866) rarr

120572minus2120590


119901119902(119866) is





119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902


2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin


(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)





10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)


10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)


+= R119899



120572

119902(119906)119871119901 First note that


119887119902(119906)119871119901 sim 119878

120572



10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)




(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003816100381610038161003816⟨119891 120595⟩1003816100381610038161003816 ≲ (sup

119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901)

times sum

119895isinZ

2119895[(Δ119901)minus120572]10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

(50)


sup119895isinZ

21198951205721003817100381710038171003817119891 lowast (1198632119895120601)

1003817100381710038171003817119871119901 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (51)



10038161003816100381610038161003816le 119862

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)2

minus|119895|(119871minus119873)(1 + |119909|)

minus119873

(52)


sum

119895isinZ

2119895(Δ119901minus120572)10038171003817100381710038171003817

120595 lowast (1198632119895120577)100381710038171003817100381710038171198711

≲ sum

119895isinZ


1003817100381710038171003817(119871+1119871+Δ+2)

≲10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2)

(53)

Therefore1003816100381610038161003816⟨119891 120595⟩

1003816100381610038161003816 ≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

10038171003817100381710038171205951003817100381710038171003817(119871+1119871+Δ+2) (54)




1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816 sim inf

119910isin119861(1199092minus119895)

1003816100381610038161003816119891 lowast (1198632119895120601) (119909)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119909minus1119910

1003816100381610038161003816)Δ119903

le inf119910isin119861(1199092minus119895)

119872lowast

Δ1199032minus119895(119891 120601) (119910)

≲ inf119910isin119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

1119903

(55)


1003816100381610038161003816119901

≲1

1003816100381610038161003816119861 (119909 2minus119895)1003816100381610038161003816

int119861(1199092minus119895)

[119872(1003816100381610038161003816119891 lowast (1198632119895120601)

1003816100381610038161003816119903) (119910)]

119901119903

119889119910

le 2119895Δint1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816119901119889119910

(56)










119901119902(119866)




120572

119901119902(119866) is dense in 120572



120572

119901119902(119866) such that 119892minus119906120572

119901119902(119866) lt 1205762


119895=minus119873119892lowast (1198632119895120601)120572

119901119902(119866) lt 1205762 Now we

put

ℎ =

119873

sum

119895=minus119873

120601 (2minus2119895

L) 119892 =

119873

sum

119895=minus119873

119892 lowast (1198632119895120601) (57)


ℎ minus 119906120572119901119902(119866) ≲

1003817100381710038171003817ℎ minus 1198921003817100381710038171003817120572119901119902(119866)+1003817100381710038171003817119892 minus 119906

1003817100381710038171003817120572119901119902(119866)

lt 120576 (58)




L120590119891 = int

infin

0

120582120590119889119864 (120582) 119891 119891 isin Dom (L

120590) sub 119871

2(119866) (59)



) cap

120572

119901119902(119866) is dense in 120572





120572


120572

119901119902(119866)

to 120572minus2120590119901119902

(119866)

(ii) Let 119879120590 120572119901119902(119866) rarr 120572minus2120590




119901119902(119866)




22119895120590(2

minus2119895120582) = 120601(2


120601(2minus2119895L) ∘ 120577(L) sub (120601(2


Dom (120601 (2minus2119895

L) ∘ 120577 (L))


= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)


120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904



1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)


120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)


) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to


119901119902(119866) to 120572minus2120590

119901119902(119866) We denote


119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)


119901119902(119866) rarr

120572minus2120590


120572

119901119902(119866) such that



) cap 120572

119901119902(119866) which converges


119891ℓ



119891ℓ isin


120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)


119901119902(119866) to the



120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective




120572minus2120590


119901119902(119866) to 119891


120572

119901119902(119866) Denote



120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590





120572


120572

119901119902(119866) rarr

120572minus2120590


119901119902(119866) is





119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902


2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin


(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)





10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)


10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)


+= R119899



120572

119902(119906)119871119901 First note that


119887119902(119906)119871119901 sim 119878

120572



10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)




(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










22119895120590(2

minus2119895120582) = 120601(2


120601(2minus2119895L) ∘ 120577(L) sub (120601(2


Dom (120601 (2minus2119895

L) ∘ 120577 (L))


= Dom (L120590) cap 119871

2(119866) = Dom (L

120590)

(60)


120601(2minus2119895L)L120590

119891Now let 119891 isin Dom(L120590

) cap 119904



1003817100381710038171003817L1205901198911003817100381710038171003817120572minus2120590119901119902

(119866)

sim

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895(120572minus2120590) 1003816100381610038161003816(L

120590119891) lowast (1198632119895120601)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595)

1003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(61)


120572

119901119902(119866) is dense in 120572

119901119902(119866) (see Remark 12)


) cap 120572

119901119902(119866) 997891rarr

120572

119901119902(119866) extends to


119901119902(119866) to 120572minus2120590

119901119902(119866) We denote


119901119902(119866) 997891rarr

120572minus2120590

119901119902(119866)


119901119902(119866) rarr

120572minus2120590


120572

119901119902(119866) such that



) cap 120572

119901119902(119866) which converges


119891ℓ



119891ℓ isin


120572minus2120590

119901119902(119866) and 119891ℓ = Lminus120590

(L120590119891ℓ) applying (i)


119901119902(119866) to the



120572

119901119902(119866) rarr

120572minus2120590

119901119902(119866) is surjective




120572minus2120590


119901119902(119866) to 119891


120572

119901119902(119866) Denote



120572

119901119902(119866) Hence 119879120590119892 = 119891 in 120572minus2120590





120572


120572

119901119902(119866) rarr

120572minus2120590


119901119902(119866) is





119892120572

119902(119906) (119909) = (

infin

sum

119896=minusinfin

(2119896120572|119906 (119909 119896)|)

119902

)

1119902

119866120572

120582119902(119906) (119909) = [

infin

sum

119896=minusinfin

int119866

(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902


2119896Δ119889119910]

1119902

119878120572

119887119902(119906) (119909)

= (

infin

sum

119896=minusinfin


(2119896120572 1003816100381610038161003816119906 (119910 119896)

1003816100381610038161003816)119902

2(119896minus119887)Δ

119889119910)

1119902

(62)





10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901

sim10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901 (63)


10038171003817100381710038171003817119892120572

119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119866120572

120582119902(119906)10038171003817100381710038171003817119871119901

≲10038171003817100381710038171003817119878120572

119887119902(119906)10038171003817100381710038171003817119871119901 (64)


+= R119899



120572

119902(119906)119871119901 First note that


119887119902(119906)119871119901 sim 119878

120572



10038171003817100381710038171003817119878120572

0119902(119906)10038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1205822minus119895(119891 120601)

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(65)




(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

2119895Δ119889119910

≲ sup119910isin119861(1199092minus119895)

(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902


(2119895120572 1003816100381610038161003816119891 lowast (1198632119895120601) (119910)

1003816100381610038161003816)119902

(1 + 21198951003816100381610038161003816119910minus1119909

1003816100381610038161003816)120582119902

= [2119895120572119872

lowast

1205822minus119895(119891 120601) (119909)]

119902

(66)







1199012(119866) with119867119901

(119866) for 0 lt 119901 lt infin


119901119902(119866)





10038161003816100381610038161003816le 119862119868|119909|



10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816le 119862 (68)


10038161003816100381610038161003816119883119868120601 (119909)





10038161003816100381610038161003816119884119868119870 (119909)

10038161003816100381610038161003816le 119862119868|119909|




120576lt|119909|lt119871119870(119909)119889119909 =


10038161003816100381610038161003816⟨119870 120601

119877⟩10038161003816100381610038161003816=1003816100381610038161003816⟨119870 120601⟩

1003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816lim120576rarr0

int120576lt|119909|lt2

119870 (119909) [120601 (119909) minus 120601 (0)] 119889119909

10038161003816100381610038161003816100381610038161003816

le int|119909|lt2

|119870 (119909)|1003816100381610038161003816120601 (119909) minus 120601 (0)

1003816100381610038161003816 119889119909

(71)




119901119902(119866)





max 10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

1003816100381610038161003816100381610038161003816100381610038161003816119884119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

le 119862(1 + |119909|)minusΔminus119903

(72)




119909120601)

sim⟩ where 119909


120601(119911) = 120601(119909119911) and as before 119891(119909) =119891(119909




sim(119911)



119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909120601)



119884119868= sum

|119869|le|119868|

119889(119869)ge119889(119868)

119875119868119869119883119869

(73)


10038161003816100381610038161003816119883119868[(119909120601)

sim] (119911)

10038161003816100381610038161003816

=10038161003816100381610038161003816119884119868(119909120601) (119911

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816119883119869(119909120601) (119911

minus1)10038161003816100381610038161003816

= sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119875119868119869 (119911

minus1)10038161003816100381610038161003816

10038161003816100381610038161003816(119883

119869120601) (119909119911

minus1)10038161003816100381610038161003816le 119862119868

(74)


119868



10038161003816100381610038161003816

=10038161003816100381610038161003816(119884

119868120601) lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=10038161003816100381610038161003816⟨1198632119895119870 [

119909(119884

119868120601)]

sim

⟩10038161003816100381610038161003816

=10038161003816100381610038161003816⟨119870 [

119909(119884

119868120601)]

sim

(2minus119895sdot)⟩10038161003816100381610038161003816le 119862

(75)


10038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

1003816100381610038161003816100381610038161198841198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816le 119862 (76)


2119895119870)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868|

|119910|119889(119869)=119903minus|119868|

10038161003816100381610038161003816119884119869119883119868(1198632119869119870) (119911119909)

10038161003816100381610038161003816 (77)


10038161003816100381610038161003816119884119869119883119868(1198632119895119870) (119911119909)

10038161003816100381610038161003816

le sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

10038161003816100381610038161198751198691198691015840 (119911119909)1003816100381610038161003816

1003816100381610038161003816100381610038161198831198691015840+119868(1198632119895119870) (119911119909)

100381610038161003816100381610038161003816

≲ sum

|1198691015840

|le|119869|

119889(1198691015840

)ge119889(119869)

|119911119909|119889(1198691015840)minus119889(119869)

|119911119909|minusΔminus119889(119869

1015840+119868)


(78)


10038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816

le 11986210038161003816100381610038161199101003816100381610038161003816119903minus|119868| sup

|119911|le119887119903minus|119868||119910|


(79)


10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816

=1003816100381610038161003816100381610038161003816int 120601 (119910)119883

119868(1198632119895119870) (119910

minus1119909) 119889119910

1003816100381610038161003816100381610038161003816

le int1003816100381610038161003816120601 (119910)

100381610038161003816100381610038161003816100381610038161003816119883119868(1198632119895119870) (119910

minus1119909) minus 119909119883119868(119863

2119895119870)

(119910minus1)10038161003816100381610038161003816119889119910

le 119862 sup|119911|le119887119903minus|119868||119910|


1003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


int10038161003816100381610038161199101003816100381610038161003816119903minus|119868| 1003816100381610038161003816120601 (119910)

1003816100381610038161003816 119889119910


(80)



10038161003816100381610038161003816119883119868[120601 lowast (1198632119895119870)] (119909)

10038161003816100381610038161003816le 119862(1 + |119909|)





10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












10038161003816100381610038161003816

le sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

10038161003816100381610038161198751198681198681015840 (119909)1003816100381610038161003816 119883

1198681015840

[120601 lowast (1198632119895119870)] (119909)100381610038161003816100381610038161003816

le 119862 sum

|1198681015840

|le|119868|

119889(1198681015840

)ge119889(119868)

|119909|119889(1198681015840)minus119889(119868)

(1 + |119909|)minusΔminus119903+|119868

1015840|minus119889(1198681015840)


(82)












119896120579)(119909) =

119905Δ(L119896









sum

119895isinZ

(2minus2119895120582) 120577 (2


+

(83)


119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (84)


119891 lowast 119870 lowast (1198632ℓ120601)

= sum

119895isinZ


= sum

119895isinZ


(85)





(119895andℓ)Δ

(1 + 2119895andℓ10038161003816100381610038161199101003816100381610038161003816)119903+Δ

(86)


1003816100381610038161003816

le sum

119895isinZ

int1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816


minus1119909)10038161003816100381610038161003816119889119911

≲ sum

119895isinZ


int2(119895andℓ)Δ 1003816100381610038161003816119891 lowast (1198632119895120595) (119911)

1003816100381610038161003816

(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δ

119889119911

≲ sum

119895isinZ


[sup119911isin119866

1003816100381610038161003816119891 lowast (1198632119895120595) (119911)1003816100381610038161003816

(1 + 21198951003816100381610038161003816119911minus1119909

1003816100381610038161003816)119886 ]


(1 + 2119895andℓ1003816100381610038161003816119911minus1119909

1003816100381610038161003816)119903+Δminus119886

119889119911

(87)


(1 + 2119895|119911minus1119909|)


2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816

≲ sum

119895isinZ


2119895120572119872

lowast

1198862minus119895(119891 ) (119909)

(88)


(sum

ℓisinZ

(2ℓ120572 1003816100381610038161003816119891 lowast 119870 lowast (1198632ℓ120601) (119909)

1003816100381610038161003816)119902

)

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

(2119895120572 10038161003816100381610038161003816119872

lowast

1198862minus119895(119891 )

10038161003816100381610038161003816)119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(89)



119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901119902(119866)





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim sum

119889(119868)=119896

1003817100381710038171003817100381711988311986811989110038171003817100381710038171003817120572minus119896119901119902

(119866) (90)


10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

sim

]

sum

119895=1

10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (91)





we deduce that10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866)=10038171003817100381710038171003817(119883119895L

minus12)L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)

≲10038171003817100381710038171003817L

1211989110038171003817100381710038171003817120572minus1119901119902

(119866)sim10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(92)

Hencesum]119895=1

119883119895119891120572minus1119901119902





10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

=10038171003817100381710038171003817L (L

minus12119891)10038171003817100381710038171003817120572minus1119901119902

(119866)

=1003817100381710038171003817L (119891 lowast 1198771)

1003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

L(

]

sum

119895=1

(119883119895119891) lowast 119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

]

sum

119895=1

(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817100381710038171003817100381710038171003817120572minus1119901119902

(119866)

(93)


10038171003817100381710038171003817(119883119895119891) lowastL (119870119895 lowast 1198771)

10038171003817100381710038171003817120572minus1119901119902

(119866)≲10038171003817100381710038171003817119883119895119891

10038171003817100381710038171003817120572minus1119901119902

(119866) (94)


]119895=1

119883119895119891120572minus1119901119902



119901119902(119866) This

completes the proof

Appendix







(i) |119866 ⋃120572isinA119896

119876119896


120572cap 119876

119896

120573= 0

if 120572 = 120573


120573sub 119876

119896

120572or 119876ℓ

120573cap

119876119896

120572= 0


120572sub 119876

ℓ

120573

(iv) diam(119876119896

120572) le 1198601120575

119896 where diam(119876119896

120572) = sup|119909minus1119910|

119909 119910 isin 119876119896

120572

(v) each119876119896


120572 1198602120575

119896) where 119911119896

120572isin 119866

The set 119876119896




120572






(iii) |119883119868119886119876(119909)| le |119876|




120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572


function

119892120572119902(119904119876119876

) = ( sum

119876isinD

(|119876|minus(120572Δ) minus(12) 1003816100381610038161003816119904119876

1003816100381610038161003816 120594119876)119902

)

1119902

(A1)


119904 119891120572119901119902

=1003817100381710038171003817119892

120572119902(119904)1003817100381710038171003817119871119901 (A2)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum

119876isinD

119904119876119886119876

10038171003817100381710038171003817100381710038171003817100381710038171003817120572119901119902(119866)

le 119862Δ11990111990212057210038171003817100381710038171003817119904119876119876isinD

10038171003817100381710038171003817 119891120572119901119902

(A3)





119891 = sum

119876isinD

119904119876119886119876 (A4)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

le 1198621015840

Δ119901119902120572

10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866) (A5)


119876(119909) = 119886119876(119911119876119909) Then 1198861015840

119876is supported


119910 isin supp 1198861015840119876)

100381610038161003816100381610038161198831198681198861015840

119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 2ℓ119889(119868)+ℓΔ

(1 + 2ℓ10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A6)


2minusℓΔ1198861015840

119876(2minusℓ119909) that is 1198861015840

119876= 1198632ℓ119886

10158401015840



119876(119910)

10038161003816100381610038161003816≲ 2

minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1 (A7)



10158401015840

119876)sim

(119910)10038161003816100381610038161003816=10038161003816100381610038161003816(119884

11986811988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ sum

|119869|le|119868|

119889(119869)ge119889(119868)

10038161003816100381610038161003816119910minus110038161003816100381610038161003816

119889(119869)minus119889(119868) 10038161003816100381610038161003816(119883

11986911988610158401015840

119876) (119910

minus1)10038161003816100381610038161003816

≲ 2minusℓΔ2 1

(1 +10038161003816100381610038161199101003816100381610038161003816)119873 forall |119868| le 119871 + 1

(A8)


100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119909)

10038161003816100381610038161003816=10038161003816100381610038161003816(1198632ℓ119886

10158401015840

119876) lowast (1198632119895120601) (119909)

10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873(A9)


1003816100381610038161003816119886119876 lowast (1198632119895120601) (119909)1003816100381610038161003816

=100381610038161003816100381610038161198861015840

119876lowast (1198632119895120601) (119911

minus1

119876119909)10038161003816100381610038161003816

≲ 2minusℓΔ2

2minus|119895minusℓ|119871 2


1 + (2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

(A10)


sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873

le ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)

1119903

= (int119866

sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903|119876|

minus1120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903119889119911)

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

+

infin

sum

119896=0


1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911)

(1 + 2119895andℓ10038161003816100381610038161003816119911minus111987611990910038161003816100381610038161003816)119873119903)119889119911

1119903

≲ 2ℓΔ119903

int|119909minus1119911|le(2120574119860


( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903



1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1)2119896+12minus(119895andℓ)

times( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

≲ 2ℓΔ119903


1003816100381610038161003816119861 (119909 212057411986012minus(119895andℓ))

1003816100381610038161003816

int119861(1199092120574119860


times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903 2

(119896+1)Δ2minus(119895andℓ)Δ

1003816100381610038161003816119861 (119909 212057411986012119896+12minus(119895andℓ))

1003816100381610038161003816

times int119861(1199092120574119860

12119896+12minus(119895andℓ))

times ( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876 (119911))119889119911

1119903

le 2ℓΔ119903


+

infin

sum

119896=0

(1 + 11986012119896)minus119873119903

2(119896+1)Δ


1119903


10038161003816100381610038161199041198761003816100381610038161003816119903120594119876) (119909)

1119903

≲ 2ℓΔ119903


119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

= 2max(ℓminus1198950)Δ119903

119872( sum

119876isinDℓ

10038161003816100381610038161199041198761003816100381610038161003816119903120594119876)(119909)

1119903

(A11)



119876119909|)

119873119903ge (1 + 11986012

minus119896)119873119903




minus2(1198960+1) 2




[2minus2(1198960+1) 2


minus2(1198960+1) 2





120572


that

119891 = sum

119895isinZ

119891 lowast (1198632119895120595) lowast (1198632119895120577) (A12)


119891 = sum

119895isinZ

sum

119876isinD119895

int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

= sum

119895isinZ

sum

119876isinD119895

119904119876119886119876

(A13)


119904119876 equiv |119876|12sup

119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 sup|119868|le119871+1

10038171003817100381710038171003817119883119868120577100381710038171003817100381710038171198711

119886119876 (119909) equiv1

119904119876int119876

119891 lowast (1198632119895120595) (119910) (1198632119895120577) (119910minus1119909) 119889119910

(A14)




1003816100381610038161003816100381611988311986811988611987610038161003816100381610038161003816le2Δ1198952119889(119868)119895

119904119876int119876

10038161003816100381610038161003816119891 lowast (1198632119895120595) (119910) (119883

119868120577) (2

119895(119910

minus1119909))

10038161003816100381610038161003816119889119910

le2119889(119868)119895

119904119876

10038171003817100381710038171003817(119883

119868120577)100381710038171003817100381710038171198711

sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816

le |119876|minus(119889(119868)Δ)minus(12)

(A15)


sum

119876isinD119895

(|119876|minus(120572Δ)minus(12)

119904119876120594119876 (119909))119902

sim sum

119876isinD119895

(2119895120572sup119910isin119876

1003816100381610038161003816119891 lowast (1198632119895120595) (119910)1003816100381610038161003816 120594119876 (119909))

119902

≲ sup119910isin119861(11990911986012

minus119895)

[2119895120572 1003816100381610038161003816119891 lowast (1198632119895120595) (119910)

1003816100381610038161003816]119902

≲ [2119895120572119872

lowast

1205822minus119895(119891 ) (119909)]

119902

(A16)


10038171003817100381710038171003817119904119876119876

10038171003817100381710038171003817 119891120572119901119902

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

100381610038161003816100381610038162119895120572119872

lowast

1205822minus119895(119891 )

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817120572119901119902(119866)

(A17)

This proves (A5)


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article homogeneous triebel-lizorkin spaces on...

Documents