research article some function spaces via orthonormal...

Research ArticleSome Function Spaces via Orthonormal Bases on Spaces ofHomogeneous Type

Chuang Chen1 Ji Li2 and Fanghui Liao3

1Department of Mathematics Sichuan University Chengdu 610064 China2Department of Mathematics Macquarie University NSW 2109 Australia3Department of Mathematics China University of Mining amp Technology Beijing Beijing 100083 China

Correspondence should be addressed to Fanghui Liao liaofanghui1028163com

Received 20 June 2014 Accepted 26 August 2014 Published 16 December 2014

Academic Editor Daoyi Xu

Copyright copy 2014 Chuang Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Let (119883 119889 120583) be a space of homogeneous type in the sense of Coifman and Weiss where the quasi-metric dmay have no regularityand the measure 120583 satisfies only the doubling property Adapting the recently developed randomized dyadic structures of X andapplying orthonormal bases of 1198712

(119883) constructed recently by Auscher and Hytonen we develop the Besov and Triebel-Lizorkinspaces on such a general setting In this paper we establish the wavelet characterizations and provide the dualities for these spacesThe results in this paper extend earlier related results with additional assumptions on the quasi-metric d and the measure 120583 to thefull generality of the theory of these function spaces

1 Introduction

The aim of this paper is to introduce the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the senseof Coifman and Weiss without any additional assumptionon the quasi-metric and the doubling measure The maintool used in this paper is the remarkable orthonormal basisconstructed recently by Auscher and Hytonen [1] It is wellknown that function spaces play an important role in bothclassical and modern analysis ordinary and partial differen-tial equations and approximation theory and so forth Sincethe seventies of last century classical theory of the Besovspaces [2] as well as the Triebel-Lizorkin spaces [3 4] hasbeen developed rapidly from Euclidean spaces to spaces ofhomogeneous type in the sense of Coifman and Weiss withsome additional assumptions on the quasi-metric and thedoubling measure See for example [5 6] and referencestherein

Let us recall briefly spaces of homogeneous type intro-duced by Coifman and Weiss [7] A function 119889 119883 times 119883 rarr

[0infin) is called a quasi-metric on a set 119883 if 119889 satisfies thefollowing (1)119889(119909 119910) = 119889(119910 119909) for all119909 119910 isin 119883 (2)119889(119909 119910) = 0

if and only if 119909 = 119910 and (3) the quasi-triangle inequalityholds there is a constant 119860

0ge 1 such that

119889 (119909 119910) le 1198600[119889 (119909 119911) + 119889 (119911 119910)] (1)

for all 119909 119910 and 119911 isin 119883 In addition the quasi-metric ball119861(119909 119903) with center 119909 isin 119883 and radius 119903 gt 0 is definedby 119861(119909 119903) = 119910 isin 119883 119889(119909 119910) lt 119903 We say that anonzero measure 120583 satisfies the doubling condition if thereis a constant 119862

120583gt 0 such that for all 119909 isin 119883 and all 119903 gt 0

120583 (119861 (119909 2119903)) le 119862120583120583 (119861 (119909 119903)) lt infin (2)

We point out that the doubling condition (2) implies thatthere exist positive constants Q (the upper dimension of 120583)and 119862 such that for all 119909 isin 119883 and 120582 ge 1

120583 (119861 (119909 120582119903)) le 119862120582Q120583 (119861 (119909 119903)) (3)

A space (119883 119889 120583) is said to be the space of homogeneoustype in the sense of Coifman and Weiss [7] if 119889 is a quasi-metric on 119883 and 120583 satisfies the doubling condition Suchspaces have many applications in the theory of singular

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 265378 13 pageshttpdxdoiorg1011552014265378

2 Abstract and Applied Analysis

integrals and function spaces [8] Unfortunately for someapplications additional assumptions were imposed on thesegeneral spaces due to the facts that the original quasi-metricmay have no (Holder) regularity and quasi-metric balls evenBorel sets may not be open

A recent work on the Besov spaces and the Triebel-Lizorkin spaces on spaces (119883 119889 120583) of homogeneous type inthe sense of Coifman and Weiss was developed by Han etal [6] More precisely in [6] in order to apply Coifmanrsquosconstruction for the approximation to the identity they needthat the quasi-metric 119889 satisfies the Holder regularity andthe doubling measure 120583 is required to satisfy the reverseddoubling condition that is there are constants 120581 isin (0Q] and119888 isin (0 1] such that

119888120582120581120583 (119861 (119909 119903)) le 120583 (119861 (119909 120582119903)) (4)

for all 119909 isin 119883 0 lt 119903 lt sup119909119910isin119883

119889(119909 119910)2 and 1 le 120582 lt

sup119909119910isin119883

119889(119909 119910)2119903 This assumption ensures that

sum

119896isinZ120575119896ge119903

1

120583 (119861 (119909 120575119896))le

119862

120583 (119861 (119909 119903)) (5)

which is the key to develop the theory of the Besov andTriebel-Lizorkin spaces on spaces of homogeneous type Themain tool used in [6] is the so-called frame developed byHan [9] (see also [10]) because the Fourier transform ismissing in general spaces of homogeneous type and waveletswere not available at that time However things seem to bechanged after the quite recent work of Auscher and Hytonen[1] where remarkable orthonormal bases (wavelets) wereconstructed by using spline functions with the original quasi-metric and the doubling measure These orthonormal basesopen the door for developing function spaces on these generalsettings See the very recent work in [11] for the theoryof product 119867119901 CMO119901 VMO and duality on such generalspaces Motivated by the work in [11] in this paper wedevelop the theory of the Besov and Triebel-Lizorkin spaceson spaces of homogeneous type in the sense of Coifman andWeiss without any additional assumption on the quasi-metricand the doubling measure Therefore results in this paperextend earlier related results with additional assumptions onthe quasi-metric 119889 and the measure 120583 to the full generality ofthe theory of the Besov and Triebel-Lizorkin spaces

This paper is organized as follows In Section 2 we givesome notations and preliminaries including a useful almostorthogonality estimate (see Lemma 7) Wavelet character-izations and dualities of the inhomogeneous Besov andTriebel-Lizorkin spaces are given in Section 3 We introducehomogeneous Besov and Triebel-Lizorkin spaces and discussrelationships between homogeneous and inhomogeneousBesov and Triebel-Lizorkin spaces in Section 4

2 Preliminaries and Notations

Throughout this paper (119883 119889 120583) denotes a space of homoge-neous type with quasi-triangle constant 119860

0and 120583(119883) = infin

By 119881119903(119909) we denote the measure of 119861(119909 119903) the ball centered

at 119909 with radius 119903 gt 0 and by 119881(119909 119910) we denote the measure

of 119861(119909 119889(119909 119910)) the ball centered at 119909with radius 119889(119909 119910) gt 0In addition we use the notation 119886 ≲ 119887 to mean that there isa constant 119862 gt 0 such that 119886 le 119862119887 and the notation 119886 sim 119887 tomean that 119886 ≲ 119887 ≲ 119886 The implicit constants 119862 are meant tobe independent of other relevant quantities

Wefirst recall the orthonormal basis of1198712(119883) constructedby Auscher and Hytonen [1] Let 119904119896

120572be spline functions

constructed in [1 Section 3] and let 119881119896be the closed linear

span in 1198712(119883) of 119904119896120572120572isinX119896 By [1Theorem 51] we observe that

119881119896sub 119881

119896+1 ⋂

119896isinZ

119881119896= 0 ⋃

119896isinZ

119881119896= 119871

2(119883) (6)

Further on by writing 119882119896as the orthogonal (in 1198712(119883))

complement of119881119896in119881

119896+1 Auscher and Hytonen pointed out

the following result

Theorem 1 (Theorems 61 and 71 [1]) Let 119886 = (1 +

2log2119860

0)minus1 There exists an orthonormal basis 120601119896

120572120572isinX119896 of 119881119896

satisfying the following exponential decay

10038161003816100381610038161003816120601119896

120572(119909)

10038161003816100381610038161003816le

119862

radic119881120575119896 (119909119896

120572)

exp minus] (120575minus119896119889 (119909 119909119896

120572))

119886

(7)

and the Holder regularity of order 120578

10038161003816100381610038161003816120601119896

120572(119909) minus 120601

119896

120572(119910)

10038161003816100381610038161003816le

119862

radic119881120575119896 (119909119896

120572)

(119889 (119909 119910)

120575119896)

120578

times exp minus] (120575minus119896119889 (119909 119909119896

120572))

119886

(8)

whenever 119889(119909 119910) le 120575119896 for some 120578 isin (0 1] Here 120575 gt 0 is a fixedparameter small enough say 120575 lt (11000)119860minus10

0 and 119862 ] are

two positive constants independent of 119896 120572 119909 and 119909119896

120572 Clearly

120601119896

120572120572isinX119896119896isinZ is an orthonormal basis of 1198712(119883)Also there exists an orthonormal basis 120595119896

120572120572isinY119896 of 119882119896


10038161003816100381610038161003816120595

119896

120572(119909)

10038161003816100381610038161003816le

119862

radic119881120575119896 (119910119896

120572)

exp minus] (120575minus119896119889 (119909 119910119896

120572))

119886

(9)

the Holder regularity of order 120578

10038161003816100381610038161003816120595

119896

120572(119909) minus 120595

119896

120572(119910)

10038161003816100381610038161003816le

119862

radic119881120575119896 (119910119896

120572)

(119889 (119909 119910)

120575119896)

120578


120572))

119886

(10)

whenever 119889(119909 119910) le 120575119896 for some 120578 isin (0 1] and the cancellationproperty

int119883

120595119896

120572(119909) 119889120583 (119909) = 0 119896 isin Z 120572 isin Y

119896 (11)

where 120575 is given above and 119862 ] are two positive constantsindependent of 119896 120572 119909 and 119910119896

120572 Clearly 120595119896

120572120572isinY119896119896isinZ is an

orthonormal basis of 1198712(119883) In what follows we also refer tothe functions 120595119896

120572as wavelets

Abstract and Applied Analysis 3

In order to introduce the (inhomogeneous) Besov andTriebel-Lizorkin spaces we need the following concepts oftest functions and distributions Refer to [12] for details andalso see [11]

Definition 2 For fixed 1199090isin 119883 119903 gt 0 120573 isin (0 120578] where 120578

is given in Theorem 1 and 120574 gt 0 A function 119891 is said to bea test function of type (119909

0 119903 120573 120574) centered at 119909

0with width

119903 gt 0 if 119891 satisfies the following decay and Holder regularityproperties

(i) |119891(119909)| le 119862(1(119881119903(119909

0) + 119881(119909 119909

0)))(119903(119903 + 119889(119909 119909

0)))120574

for all 119909 isin 119883

(ii) |119891(119909)minus119891(119910)| le 119862(119889(119909 119910)(119903+119889(119909 1199090)))

120573(1(119881

119903(119909

0)+

119881(119909 1199090)))(119903(119903 + 119889(119909 119909

0)))

120574 for all 119909 119910 isin 119883 such that119889(119909 119910) le (12119860

0)(119903 + 119889(119909 119909

0))

If 119891 is a test function of type (1199090 119903 120573 120574) we write 119891 isin

119866(1199090 119903 120573 120574) The norm of 119891 on 119866(119909

0 119903 120573 120574) is defined by

10038171003817100381710038171198911003817100381710038171003817119866(1199090 119903120573120574)

= inf 119862 gt 0 (i) and (ii) hold (12)

We denote119866(120573 120574) = 119866(1199090 1 120573 120574) It is easy to check that

119866(1199091 119903 120573 120574) = 119866(120573 120574) with equivalent norms for any fixed

1199091isin 119883 and 119903 gt 0 Furthermore it is also easy to see that

119866(120573 120574) is a Banach space with respect to the norm on119866(120573 120574)For given 120598 isin (0 120578] let

∘

119866(120573 120574) be the completion of the space119866(120598 120598) in 119866(120573 120574) with 0 lt 120573 120574 le 120598 Obviously

∘

119866(120598 120598) =

119866(120598 120598) In addition we say that119891 isin∘

119866(120573 120574) if119891 isin 119866(120573 120574) andthere is a sequence 119891

119899 sub 119866(120598 120598) such that 119891

119899minus119891

119866(120573120574)rarr 0

as 119899 rarr infin For given 119891 isin∘

119866(120573 120574) we define 119891 ∘119866(120573120574)

=

119891119866(120573120574)

Obviously∘

119866(120573 120574) is a Banach space with respect to

the norm sdot ∘119866(120573120574)

As usual we denote by (∘

119866(120573 120574))1015840

the dual

space of∘

119866(120573 120574) consisting of all continuous linear functionalson∘

119866(120573 120574)Han et al [11] showed that the wavelets constructed by

Auscher and Hytonen [1] are test functions defined above

Lemma3 (see [11]Theorem33) Let120595119896

120572be given inTheorem 1

with the Holder regularity of 120578 Then 120595119896

120572radic119881

120575119896(119909119896

120572) isin

119866(119909119896

120572 120575119896 120578 120598) for each 120598 gt 0

By analogous arguments we can obtain the next result

Proposition 4 Let 120601119896

120572be given in Theorem 1 with Holder

regularity of 120578 Then 120601119896

120572radic119881

120575119896(119909119896

120572) isin 119866(119909

119896

120572 120575

119896 120578 120598) for each

120598 gt 0

Let 119875119896and119876

119896be orthogonal projections onto 119881

119896and119882

119896

respectively We next give some estimates on the kernels ofoperators 119875

119896and 119876

119896

Lemma 5 The kernel 119875119896(119909 119910) is symmetric in 119909 and 119910 and

for any 120598 gt 0 119875119896(119909 119910) satisfies that

(i) for all 119909 119910 isin 119883

1003816100381610038161003816119875119896 (119909 119910)1003816100381610038161003816 le 119862

1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(13)

(ii) for 119889(119910 1199101015840) le (121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816119875119896(119909 119910) minus 119875

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

120598

(14)

(iii) for 119889(119909 1199091015840) le (12119860

0)(120575

119896+ 119889(119909 119910)) and 119889(119910 1199101015840

) le

(121198600)(120575119896 + 119889(119909 1199101015840))

10038161003816100381610038161003816[119875

119896(119909 119910) minus 119875

119896(119909

1015840 119910)] minus [119875

119896(119909 119910

1015840) minus 119875

119896(119909

1015840 119910

1015840)]10038161003816100381610038161003816

le 119862(119889 (119909 1199091015840)

120575119896 + 119889 (119909 119910))

120578

(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(15)

Proof Note that for any 119909 1199090isin 119883 and 119903 gt 0 with 119903 le

119889(119909 1199090) from the doubling property on the measure 120583 it

follows that

119881 (119909 1199090) le 119862(

119889 (119909 1199090)

119903)

Q

119881119903(119909

0) (16)

1

119881119903(119909

0)le 119862

1

119881 (119909 1199090) + 119881

119903(119909

0)(119889 (119909 119909

0)

119903)

Q

(17)

Also note that the kernel119875119896(119909 119910) is symmetric in 119909 and119910 due

to [1 Lemma 101] Thus it suffices to prove (13) for the casesthat 120575119896 gt 119889(119909 119910) and 120575119896 le 119889(119909 119910) Again by [1 Lemma 101]we have

1003816100381610038161003816119875119896 (119909 119910)1003816100381610038161003816 le 119862

1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

(18)

for all Γ gt 0 where the constant 119862 is independent of 119896 If120575119896 gt 119889(119909 119910) then 119881(119909 119910) lt 119881

120575119896(119909) and hence

1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

le 1198621

119881 (119909 119910) + 119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

(19)


On the other hand if 120575119896 lt 119889(119909 119910) then (17) implies that

1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

le 1198621

119881 (119909 119910) + 119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

ΓminusQ

(20)

By letting 120598 = Γ minusQ we obtain (13) immediatelyThe proof of (14) is based on the estimates

10038161003816100381610038161003816119875119896(119909 119910) minus 119875

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896)

120578

times(exp minus] (120575minus119896119889 (119909 119910))

119904

radic119881120575119896 (119909) 119881

120575119896 (119910)

+exp minus] (120575minus119896119889 (119909 1199101015840))

119904

radic119881120575119896 (119909) 119881

120575119896 (1199101015840)

)

(21)

given in [1 Lemma 101] Indeed since 119889(119910 1199101015840) le

(121198600)(120575119896+119889(119909 119910)) implies that 120575119896+119889(119909 1199101015840) ge (12119860

0)(120575119896+

119889(119909 119910)) and that 119881120575119896(119909) + 119881(119909 1199101015840) sim 119881

120575119896(119909) + 119881(119909 119910) it

follows from (17) and (21) that

119868119868 le 119862(119889 (119910 1199101015840)

120575119896)

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

Γ

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

Γminus120578

(22)

Applying 120598 = Γminus120578 yields (14) immediatelyThe proof of (15)is similar to that of (13) and (14) and we omit it here

Lemma 6 (see [11] Lemma 36) The kernel 119876119896(119909 119910) is

symmetric in 119909 and 119910 and for any 120598 gt 0 satisfies that

(i) for all 119909 119910 isin 119883

1003816100381610038161003816119876119896(119909 119910)

1003816100381610038161003816 le 1198621

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(23)

(ii) for 119889(119910 1199101015840) le (121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816119876

119896(119909 119910) minus 119876

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(24)

(iii) for 119889(119909 1199091015840) le (121198600)(120575119896 + 119889(119909 119910)) and 119889(119910 1199101015840) le

(121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816[119876

119896(119909 119910) minus 119876

119896(119909

1015840 119910)] minus [119876

119896(119909 119910

1015840) minus 119876

119896(119909

1015840 119910

1015840)]10038161003816100381610038161003816

le 119862(119889 (119909 1199091015840)

120575119896 + 119889 (119909 119910))

120578

(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(25)

Let 120593119897

120573119897isinZ120573isinY119897 be another orthonormal basis of 1198712(119883)

given by the algorithm in Theorem 1 and let 119864119897be the

projection onto 119882119897with respect to the basis 120593119897

120573119897isinZ120573isinY119897

Finally we give the almost orthogonality estimate of thekernel of operator 119876

119896119864

119897to end this section The proof is

similar to that in [6 Lemma 31] and we omit it here

Lemma 7 Let 120598 gt 0 and let 120578 gt 0 be the order of the Holderregularity given in (8) There exists a constant 119862 = 119862

120598gt 0

dependent only on 120598 such that for all 120590 isin (0 120578]

1003816100381610038161003816119876119896119864

119897(119909 119910)

1003816100381610038161003816 le 119862120575|119896minus119897|120590 1

119881120575(119896and119897) (119909) + 119881 (119909 119910)

times (120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

(26)

(ii) for all 120591 isin (0 1)10038161003816100381610038161003816119876

119896119864

119897(119909 119910) minus 119876

119896119864

119897(119909 119910

1015840)10038161003816100381610038161003816

le 119862120575|119896minus119897|120590120591

(119889 (119910 1199101015840)

120575(119896and119897) + 119889 (119909 119910))

120578(1minus120591)

times1

119881120575(119896and119897) (119909) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

(27)

whenever 119889(119910 1199101015840) lt (1(21198600)2)119889(119909 119910) The same estimate

holds with 119909 and 119910 interchanged

3 Inhomogeneous Besov and Triebel-LizorkinSpaces on Spaces of Homogeneous Type

In this section we introduce the inhomogeneous Besov andTriebel-Lizorkin spaces on spaces of homogeneous type andgive wavelet characterizations of corresponding spaces


We would like to point out that the wavelet expansion(Theorem 9) is the key to develop the theory of the Besovand Triebel-Lizorkin spaces As defined previously by 119875

119896and

119876119896we denote the orthogonal projections onto 119881

119896and 119882

119896

respectively These two projections provide us with regularLittlewood-Paley decompositions for spaces of homogeneoustype

Theorem 8 (Theorem 102 [1]) Let 119891 isin 1198712(119883) Then

119891 = sum119896isinZ

119876119896(119891) = sum

119896isinZ

1198762

119896(119891) (28)

which gives homogeneous Littlewood-Paley decompositionwhile

119891 = 119875119897(119891) +sum

119896ge119897

119876119896(119891) = 119875

2

119897(119891) +sum

119896ge119897

1198762

119896(119891) (29)

provides inhomogeneous Littlewood-Paley decomposition

Furthermore following the proof of Theorem 34 in [11]we can show that such wavelet expansions also hold in thespace of distributions

Theorem 9 Let 0 lt 120573 120574 lt 120578 Then wavelet expansion (29)holds in

∘

119866(1205731015840 1205741015840) with 1205731015840 isin (0 120573) and 1205741015840 isin (0 120574) and henceholds in (

∘

119866(1205731015840 1205741015840))1015840

with 1205731015840 isin (120573 120578) and 1205741015840 isin (120574 120578) as well

Proof Since 119866(120573 120574) sub 1198712(119883) and (29) holds in the senseof 1198712(119883) (29) holds for such test function 119891 in the (ae)pointwise sense (for some subsequence with respect to theconvergence)

By analogous arguments given in [11 Theorem 34]we can show the convergence of sum

119896ge1119876

119896(119891) and 119875

1(119891) in

119866(1205731015840 1205741015840) by using the size and smooth conditions of kernels119876

119896(119909 119910) and 119875

1(119909 119910) respectively Therefore

119891 = 1198751(119891) + sum

119896ge1

119876119896(119891) (30)

in the sense of119866(1205731015840 1205741015840)The second equality of (29) holds forthe same reason

We now introduce the inhomogeneous Besov andTriebel-Lizorkin spaces on spaces of homogeneous type

Definition 10 Let 1 lt 119901 119902 le infin and |119904| lt 120578 Theinhomogeneous Besov space 119861119904119902

119901 (119883) is defined by

119861119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

lt infin (31)

where10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(32)

For 1 lt 119901 lt infin and 1 lt 119902 le infin the inhomogeneousTriebel-Lizorkin space 119865119904119902

119901 (119883) is defined by

119865119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

lt infin (33)

where

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(34)

Remark 11 Observe that10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

sim 10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

119891 isin 119861119904119902

119901

(35)

as well as

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

1198751(119891)

119902

+ sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

119891 isin 119865119904119902

119901

(36)

The next result shows that the Besov and Triebel-Lizorkinnorms given in Definition 10 are both independent of choiceof the orthonormal basis 120595119896

120572120572isinY119896119896isinZ of 1198712(119883) and hence

they are independent of the choice of operators 1198751and 119876

119896

with 119896 ge 1Thus the Besov and Triebel-Lizorkin spaces givenabove are both well defined To see this let 120601119896

120572120572isinY119896119896isinZ be

another orthonormal basis of 1198712(119883) given inTheorem 1 andlet 119864

1and 119863

119896be projections onto 119881

1and 119882

119896 respectively

with respect to the basis 120601119896

120572120572isinY119896119896isinZ

Proposition 12 Let |119904| lt 120578 and let 119891 isin (∘

119866(120573 120574))1015840 For all

1 lt 119901 119902 le infin then

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 + sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 + sum

119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(37)

and for all 1 lt 119901 lt infin and 1 lt 119902 le infin

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(38)


Proof Clearly the equivalence of 1198751(119891)

119901and 119864

1(119891)

119901is a

direct consequence of the size condition (13) and the waveletexpansion (29) We next show that

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

≲ sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(39)

Indeed by the wavelet expansion (29) and almost orthogonalestimate (26) we have

1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum119897ge1

int119883

1003816100381610038161003816119876119896119863

119897(119909 119910)

10038161003816100381610038161003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

int119883

1

119881120575(119896and119897) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

times1003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

119872(119863119897(119891)) (119909)

(40)

where 119872 denotes the Hardy-Littlewood maximal operatorThus

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891) (119909)1003817100381710038171003817119901)

119902

1119902

le 119862sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(41)

due to the fact that |119904| lt 120578 This proves (39) and hence weobtain (37) symmetrically

We now prove (38) Clearly by symmetry it suffices toshow that

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

≲

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(42)

To this end from (40) it follows that

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891) (119909)1003816100381610038161003816)

119902

le 119862sum119896ge1

(sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119896ge1

sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119897ge1

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

(43)

due to the fact that sum119897ge1

120575(119897minus119896)119904120575|119896minus119897|120578 lt infin andsum

119896ge1120575(119897minus119896)119904120575|119896minus119897|120578 lt infin for |119904| lt 120578 Thus applying the

Fefferman-Stein vector-valued maximal inequality weobtain (42) immediately The proof is complete

Let 120595119896

120572120572isinY119896 be an orthonormal basis of 119882

119896and

let 1206011

120572120572isinX1 be an orthonormal basis of 119881

1as shown

in Theorem 1 Applying the wavelet expansion given inTheorem 9

119891 = sum

120572isinX1

⟨1206011

120572 119891⟩ 120601

1

120572+ sum

119896ge1

sum

120572isinY119896

⟨120595119896

120572 119891⟩120595

119896

120572 (44)

we now give the wavelet characterizations of the inhomoge-neous Besov and Triebel-Lizorkin spaces

Theorem 13 Let |119904| lt 120578 and let 1206011

120572 120595119896

120572be that given in

Theorem 1 with 119896 ge 1 For 1 lt 119901 119902 le infin then10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(45)

and for 1 lt 119901 lt infin 1 lt 119901 le infin then10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

( sum

120572isinX119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

100381610038161003816100381610038161205941198761120572)

119902

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572

]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(46)

Proof Observe that by analogous arguments in the proof of(26) we have

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

119881120575(1199091

120572) + 119881 (119909 1199091

120572)(

120575

120575 + 119889 (119909 1199091

120572))

120598


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119895

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le119862120575

|119895minus119896|120578

119881120575(119895and119896) (119909

119895

120572) + 119881 (119909 119909119895

120572)(

120575(119895and119896)

120575(119895and119896) + 119889 (119909 119909119895

120572))

120598

(47)

In order to prove (45) by Remark 11 it suffices to show that

10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(48)

By the wavelet expansion (44) we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816

le sum

120572isinX1

120583 (1198761

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(1199091

120572) + 119881 (119909 1199091

120572)

times (120575

120575 + 119889 (119909 1199091

120572))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

= 119862 sum

120572isinX1

int119883

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572(119911) 119889119911

le 119862119872( sum

120572isinX1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572)(119909)

(49)

so that

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 le 119862[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

1119901

(50)

By analogous arguments we also have

1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum

120572isinY119896

120583 (119876119896

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119896

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)(119909)

(51)

and hence

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

120594119876119896120572

]

]

119902119901

(52)


This implies that

10038171003817100381710038171198751

(119891)1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(53)

On the other hand by using the wavelet expansion andthe size condition we have

sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

le sum

120572isinX1

120583 (1198761

120572)

times (int119883

10038161003816100381610038161198751 (119891) (119910)1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910)

119901

le 11986210038171003817100381710038171198751(119891)

1003817100381710038171003817119901

119901

(54)

as well as

sum

120572isinY119894

(10038161003816100381610038161003816⟨120595

119895

120572 119891⟩

10038161003816100381610038161003816120583 (119876

119895

120572)1119901minus12

)119901

le sum

120572isinY119894

120583 (119876119895

120572)(sum

119896ge1

int119883

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int119883

120595119895

120572(119909)

radic120583 (119876119895

120572)

119876119896(119909 119910)119889120583(119909)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

le 119862int119883

(sum119896ge1

120575|119895minus119896|120578

int119883

1

119881120575(119895and119896)(119911) + 119881(119910 119911)

(120575(119895and119896)

120575(119895and119896) + 119889 (119910 119911))

120598

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

119889120583 (119911)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

120575|119895minus119896|120578

119872(119876119896(119891))

1003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

le 119862(sum119896ge1

120575|119895minus119896|120578119901 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119901

(55)

and hence by using the Holder inequality we have furtherthat

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119895ge1

120575minus119895119904119902 [

[

sum

120572isinY119895

(120583 (119876119895

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119895

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

le 119862

10038171003817100381710038171198751(119891)1003817100381710038171003817119902

119901

+sum119895ge1

(sum119896ge1

120575(119896minus119895)119904

120575|119895minus119896|120578

120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(56)

This completes the proof of (48)Analogously we can also obtain (46) by using the

Fefferman-Stein vector-valued maximal inequality

As an application of wavelet characterizations we nowturn to discuss the dual spaces of the inhomogeneous BesovandTriebel-Lizorkin spaces For conveniencewe rewrite (44)as

119891 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119891⟩ 120593

119896

120572 (57)

where Z0 = X1 1205930

120572= 1206011

120572 Z119896 = Y119896 and 120593119896

120572= 120595119896

120572for 119896 ge

1 The following concepts of spaces of sequences were firstintroduced by Frazier and Jawerth [13]


Definition 14 Let 120582 = 120582119896

120572 119896 ge 0 120572 isin Z119896 and let |119904| lt 120578

and 1 lt 119901 119902 le infin The space 119887119904119902119901 of sequences is defined by119887119904119902

119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)

and for 1 lt 119901 lt infin the space 119891119904119902

119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)

Remark 15 It is easy to see that 119891 isin 119861119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)

The next result shows that 1198712 is dense in 119861119904119902

119901 as well as119865

119904119902

119901

Proposition 16 Let 0 lt 120573 120574 lt 120578 and |119904| lt 120578 Then 1198712 isdense in both 119861119904119902

119901 for 1 lt 119901 119902 le infin and 119865119904119902

119901 for 1 lt 119901 lt infin

and 1 lt 119902 le infin

Proof Let |119904| lt 1205781015840 lt 120578 Since 119866(1205781015840 1205781015840) sub 1198712 to show theconclusion it suffices to show that 119866(1205781015840 1205781015840) is dense in 119861119904119902

119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902

119901 For any fixed119872 write

119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)

By the estimates (26) and (27) we can show that 119891119872

isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902

119901 Further on we can also show

that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)

This is a direct consequence of thewavelet expansion (29) andestimate (26) Indeed

1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)

where 119863119897is defined as the same as 119864

119896for 119897 ge 0 By wavelet

expansion (29) we then have

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)

This implies that 119891119872

rarr 119891 in 119861119904119902

119901 as 119872 rarr infin and hence1198712 is dense in 119861119904119902

119901 By analogous arguments we can also showthat 1198712 is dense in 119865119904119902

119901

We are ready to give the dualities of the Besov andTriebel-Lizorkin spaces

Theorem 17 Let 1 lt 119901 119902 lt infin and |119904| lt 120578 and let 1119901 +

11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840

1199011015840 More precisely

given119892 isin 119861minus1199041199021015840

1199011015840 then m

119892(119891) = ⟨119891 119892⟩ defines a linear functional

on 119861119904119902

119901 cap 119866(1205781015840 1205781015840) with |119904| lt 1205781015840 lt 120578 such that10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)

and this linear functional can be extended to 119861119904119902

119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840

Conversely if m is a linear functional on 119861119904119902

119901 then thereexists a unique 119892 isin 119861minus119904119902

1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)

defines a linear functional on 119861119904119902

119901 cap119866(1205781015840 1205781015840) with |119904| lt 1205781015840 lt 120578and m is the extension of m

119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m

Remark 18 Theorem 17 also holds with 119861119904119902

119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840

1199011015840 respectively

Proof of Theorem 17

(i) Suppose that 119892 isin 119861minus1199041199021015840

1199011015840 and 119891 isin 119861

119904119902

119901 cap 119866(1205781015840 1205781015840) ByRemark 15 we observe that

119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)

is an element of 119861minus1199041199021015840

1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)

By the Holder inequality and Minkowski inequality we have10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)

and hence m119892is a linear functional defined on 119861119904119902

119901 cap119866(1205781015840 1205781015840)Clearly m

119892has bounded extension to 119861119904119902

119901 due to the densityof 119861119904119902

119901 cap 119866(1205781015840 1205781015840) in 119861119904119902

119901 This proves that 119861minus1199041199021015840

1199011015840 sub (119861

119904119902

119901 )lowast


(ii) Suppose that m isin (119861119904119902

119901 )lowast We need to prove that there

exists a 119892 isin 119861minus1199041199021015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)

and define a linear functional m on 119878 = ⟨120593119896

120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)

From (71) it follows that10038161003816100381610038161003816m (⟨120593119896

120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)

By the Hahn-Banach theorem m can be extended to 119887119904119902119901 asa continuous linear functional In addition by argumentsanalogous to that in ([13 Remark 511]) we can show that(119887

119904119902

119901 )lowast= 119887

minus1199041199021015840

1199011015840 and hence there exists a unique sequence

119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)

Thus for each 119891 isin 119861119904119902

119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896

120572119892119896 By the orthogonality of 120593119896

120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902

119901 The linear functional m canbe extended to 119861119904119902

119901 Thus m119892(119891) = ⟨119892 119891⟩ for all 119891 isin 119861

119904119902

119901

The next result gives elementary embedding properties ofBesov and Triebel-Lizorkin spaces which can be proved by anapproach analogous to that of Proposition 2 in [14 Section232] and we omit the details here

Proposition 19 (i) Let 1 lt 1199020le 119902

1lt infin 1 lt 119901 lt infin

and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902

1lt infin and |119904 + 120579| lt 120578 Then

119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)

(iii) Let 1 lt 119901 lt infin 1 lt 119902 le infin and |119904| lt 120578 Then

119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)

4 Homogeneous Besov and Triebel-LizorkinSpaces

In this section we develop the theory of the homogeneousBesov and Triebel-Lizorkin spaces and discuss relationshipsbetween homogeneous and inhomogeneous versions

In order to introduce the homogeneous Besov andTriebel-Lizorkin spaces we recall again the spaces of testfunctions and distributions on spaces of homogeneous typegiven in [11] Let 119866(119909

0 119903 120573 120574) be given in Definition 2 By

1198660(120573 120574) we denote the collection of all test functions in

119866(120573 120574) by int119883119891(119909)119889120583(119909) = 0 For 0 lt 120573 120574 lt 120578 let 1198660(120573 120574) be

the completion of the space 1198660(120578 120578) in the norm of 119866(120573 120574)

For 119891 isin 1198660(120573 120574) we define 1198911198660(120573120574)

= 119891119866(120573120574)

The distribution space (1198660(120573 120574))1015840 is defined to be the set

of all linear functionals m from1198660(120573 120574) toCwith the propertythat there exists 119862 gt 0 such that for all 119891 isin 1198660(120573 120574)

1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)

The homogeneous Besov and Triebel-Lizorkin spaces aredefined as follows

Definition 20 Let |119904| lt 120578 and 0 lt 120573 120574 lt 120578 For 1 lt 119901 119902 lt infinthe homogeneous Besov space 119904119902

119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)

and for 1 lt 119901 lt infin 1 lt 119902 le infin the homogeneous Triebel-Lizorkin space 119904119902

119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)

The authors in [11] showed that thewavelet expansion (28)holds in 1198660(120573 120574) as well as (1198660(120573 120574))1015840 with 0 lt 120573 120574 lt 120579 Thenext result shows that Definition 20 is independent of choiceof 119876

119896for 119896 isin Z

Proposition 21 Let 119891 isin (1198660(120573 120574))1015840 and |119904| lt 120578 For 1 lt

119901 119902 le infin then

sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)

and for 1 lt 119901 lt infin and 1 lt 119902 le infin then

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)

We next give some relationships between homogeneousand inhomogeneous spaces

Proposition 22 Let 0 lt 120573 120574 lt 120578 and 0 lt 119904 lt 120578 For 1 lt


119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)

and for 1 lt 119901 lt infin 1 lt 119902 le infin then

119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)

We only give the proof of Proposition 22 Others can beproved by arguments analogous to that of inhomogeneousversions

Proof We first show that 119861119904119902

119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902

119901 Bythe wavelet expansion (29) and the Minkowski inequality wehave 119891 isin 119871

119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)

Applying the wavelet expansion (29) and the size conditionof 119876

1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901

Conversely we show that 119904119902

119901cap 119871119901 sub 119861

119904119902

119901 Indeed let 119891 isin

119904119902

119901cap 119871119901 By the size condition of 119875

1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891

119901 This implies that 119891

119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901

)By analogous arguments we can also obtain the second

desired equality for the inhomogeneous Triebel-Lizorkinspaces

We now introduce the homogeneous Besov and Triebel-Lizorkin spaces by wavelet coefficients

Definition 23 Let 1 lt 119901 119902 le infin and |119904| lt 120578 Thehomogeneous Besov space 119904119902

119901(2)is defined by

119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)

For 1 lt 119901 lt infin the homogeneous Triebel-Lizorkin space

119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)

We are ready to give the wavelet characterizations of thehomogeneous Besov and Triebel-Lizorkin spaces


Theorem 24 Let 0 lt 119904 lt 120578 and 1 lt 119902 le infin Then10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 le infin

10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)

Now we recall the homogenous spaces of sequences 119904119902119901

and 119904119902

119901 refer to [13] for details Let 120582 = 120582119896

120572 119896 isin Z 120572 isin

Y119896 For |119904| lt 120578 and 1 lt 119901 119902 le infin the space of sequence 119904119902119901

is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)

Remark 25 Suppose that 120593119896

120572be the same as in Theorem 1 for

119896 isin Z usingTheorem 24 and [11 Corollary 35] then we have119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901

Proposition 26 Let 0 lt 119904 lt 1205781015840 lt 120578 Then 119866(1205781015840 1205781015840) is densein both 119904119902

119901for 1 lt 119901 and 119902 le infin and 119904119902

119901for 1 lt 119901 lt infin and


Remark 27 As the proof of Proposition 16 we can also obtainthat (28) converges in both the norm of 119904119902

119901for 1 lt 119901 119902 le infin

and the norm of 119904119902

119901for 1 lt 119901 lt infin 1 lt 119902 le infin

The homogeneous version of Theorem 17 is given asfollows

Theorem 28 Let 1 lt 119901 119902 lt infin and |119904| lt 120578 and let 1119901 +11199011015840 = 1119902 + 11199021015840 = 1 Then (119904119902

119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902

119901cap 1198660(1205781015840 1205781015840) with |119904| lt 1205781015840 lt 120578 such that

10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there

exists a unique 119892 isin minus1199041199021015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578

and m is the extension of m119892with 119892

minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840

replaced by 119904119902

119901and minus119904119902

1015840


Finally we give the following embedding properties inhomogeneous Besov and Triebel-Lizorkin spaces to end thissection


1le infin 1 lt 119901 lt infin

and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)

(ii) Let 1 lt 119901 lt infin 1 lt 119902 le infin and |119904| lt 120578 Then

119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Chuang Chen was supported by the NNSF (Grant no11201319) of China and the Science Foundation for YouthScholars of Sichuan University (Grant no 2012SCU11048)and Fanghui Liao was supported by NNSF (Grant no11171345) of China the Doctoral Fund of Ministry of Edu-cation of China (Grant no 20120023110003) and ChinaScholarship Council

References

[1] P Auscher and T Hytonen ldquoOrthonormal bases of regularwavelets in spaces of homogeneous typerdquo Applied and Compu-tational Harmonic Analysis vol 34 no 2 pp 266ndash296 2013

[2] O V Besov ldquoOn some families of functional spaces imbeddingand extension theoremsrdquo Doklady Akademii Nauk SSSR vol126 pp 1163ndash1165 1959

[3] H Triebel ldquoSpaces of distributions of Besov type on Euclidean119899-space Duality interpolationrdquoArkiv for Matematik vol 11 pp13ndash64 1973

[4] P I Lizorkin ldquoProperties of functions in the spaces Λ119903

119901 120579rdquo

Trudy Matematicheskogo Instituta imeni V A Steklova vol 131pp 158ndash181 1974

[5] H Triebel Theory of Function Spaces II Birkhauser BaselSwitzerland 1992


[6] 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

[7] R R Coifman and G Weiss Analyse Harmonique Non-commutative sur Certains Espaces Homogeneous vol 242 ofLecture Notes in Mathematics Springer Berlin Germany 1971

[8] 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

[9] Y S Han ldquoCalderon-type reproducing formula and the 119879119887theoremrdquo Revista Matematica Iberoamericana vol 10 no 1 pp51ndash91 1994

[10] D Deng and Y Han Harmonic Analysis on Spaces of Homo-geneous Type Lecture Notes in Mathematics Springer BerlinGermany 2009

[11] Y Han J Li and L Ward ldquoProduct H119901 CMO119901 VMO andduality via orthonormal bases on spaces of homogeneous typerdquopreprint

[12] Y Han and E Sawyer ldquoLittlewood-Paley theory on spaces ofhomogeneous type and the classical function spacesrdquo Memoirsof the American Mathematical Society vol 110 pp 1ndash126 1994

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

[14] H TriebelTheory of Function Spaces Birkhauser Basel Switz-erland 1983

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


integrals and function spaces [8] Unfortunately for someapplications additional assumptions were imposed on thesegeneral spaces due to the facts that the original quasi-metricmay have no (Holder) regularity and quasi-metric balls evenBorel sets may not be open

A recent work on the Besov spaces and the Triebel-Lizorkin spaces on spaces (119883 119889 120583) of homogeneous type inthe sense of Coifman and Weiss was developed by Han etal [6] More precisely in [6] in order to apply Coifmanrsquosconstruction for the approximation to the identity they needthat the quasi-metric 119889 satisfies the Holder regularity andthe doubling measure 120583 is required to satisfy the reverseddoubling condition that is there are constants 120581 isin (0Q] and119888 isin (0 1] such that

119888120582120581120583 (119861 (119909 119903)) le 120583 (119861 (119909 120582119903)) (4)

for all 119909 isin 119883 0 lt 119903 lt sup119909119910isin119883

119889(119909 119910)2 and 1 le 120582 lt

sup119909119910isin119883

119889(119909 119910)2119903 This assumption ensures that

sum

119896isinZ120575119896ge119903

1

120583 (119861 (119909 120575119896))le

119862

120583 (119861 (119909 119903)) (5)

which is the key to develop the theory of the Besov andTriebel-Lizorkin spaces on spaces of homogeneous type Themain tool used in [6] is the so-called frame developed byHan [9] (see also [10]) because the Fourier transform ismissing in general spaces of homogeneous type and waveletswere not available at that time However things seem to bechanged after the quite recent work of Auscher and Hytonen[1] where remarkable orthonormal bases (wavelets) wereconstructed by using spline functions with the original quasi-metric and the doubling measure These orthonormal basesopen the door for developing function spaces on these generalsettings See the very recent work in [11] for the theoryof product 119867119901 CMO119901 VMO and duality on such generalspaces Motivated by the work in [11] in this paper wedevelop the theory of the Besov and Triebel-Lizorkin spaceson spaces of homogeneous type in the sense of Coifman andWeiss without any additional assumption on the quasi-metricand the doubling measure Therefore results in this paperextend earlier related results with additional assumptions onthe quasi-metric 119889 and the measure 120583 to the full generality ofthe theory of the Besov and Triebel-Lizorkin spaces

This paper is organized as follows In Section 2 we givesome notations and preliminaries including a useful almostorthogonality estimate (see Lemma 7) Wavelet character-izations and dualities of the inhomogeneous Besov andTriebel-Lizorkin spaces are given in Section 3 We introducehomogeneous Besov and Triebel-Lizorkin spaces and discussrelationships between homogeneous and inhomogeneousBesov and Triebel-Lizorkin spaces in Section 4

2 Preliminaries and Notations

Throughout this paper (119883 119889 120583) denotes a space of homoge-neous type with quasi-triangle constant 119860

0and 120583(119883) = infin

By 119881119903(119909) we denote the measure of 119861(119909 119903) the ball centered

at 119909 with radius 119903 gt 0 and by 119881(119909 119910) we denote the measure

of 119861(119909 119889(119909 119910)) the ball centered at 119909with radius 119889(119909 119910) gt 0In addition we use the notation 119886 ≲ 119887 to mean that there isa constant 119862 gt 0 such that 119886 le 119862119887 and the notation 119886 sim 119887 tomean that 119886 ≲ 119887 ≲ 119886 The implicit constants 119862 are meant tobe independent of other relevant quantities

Wefirst recall the orthonormal basis of1198712(119883) constructedby Auscher and Hytonen [1] Let 119904119896

120572be spline functions

constructed in [1 Section 3] and let 119881119896be the closed linear

span in 1198712(119883) of 119904119896120572120572isinX119896 By [1Theorem 51] we observe that

119881119896sub 119881

119896+1 ⋂

119896isinZ

119881119896= 0 ⋃

119896isinZ

119881119896= 119871

2(119883) (6)

Further on by writing 119882119896as the orthogonal (in 1198712(119883))

complement of119881119896in119881

119896+1 Auscher and Hytonen pointed out

the following result

Theorem 1 (Theorems 61 and 71 [1]) Let 119886 = (1 +

2log2119860

0)minus1 There exists an orthonormal basis 120601119896

120572120572isinX119896 of 119881119896


10038161003816100381610038161003816120601119896

120572(119909)

10038161003816100381610038161003816le

119862

radic119881120575119896 (119909119896

120572)

exp minus] (120575minus119896119889 (119909 119909119896

120572))

119886

(7)

and the Holder regularity of order 120578

10038161003816100381610038161003816120601119896

120572(119909) minus 120601

119896

120572(119910)

10038161003816100381610038161003816le

119862

radic119881120575119896 (119909119896

120572)

(119889 (119909 119910)

120575119896)

120578


120572))

119886

(8)

whenever 119889(119909 119910) le 120575119896 for some 120578 isin (0 1] Here 120575 gt 0 is a fixedparameter small enough say 120575 lt (11000)119860minus10

0 and 119862 ] are

two positive constants independent of 119896 120572 119909 and 119909119896

120572 Clearly

120601119896

120572120572isinX119896119896isinZ is an orthonormal basis of 1198712(119883)Also there exists an orthonormal basis 120595119896

120572120572isinY119896 of 119882119896


10038161003816100381610038161003816120595

119896

120572(119909)

10038161003816100381610038161003816le

119862

radic119881120575119896 (119910119896

120572)

exp minus] (120575minus119896119889 (119909 119910119896

120572))

119886

(9)

the Holder regularity of order 120578

10038161003816100381610038161003816120595

119896

120572(119909) minus 120595

119896

120572(119910)

10038161003816100381610038161003816le

119862

radic119881120575119896 (119910119896

120572)

(119889 (119909 119910)

120575119896)

120578


120572))

119886

(10)

whenever 119889(119909 119910) le 120575119896 for some 120578 isin (0 1] and the cancellationproperty

int119883

120595119896

120572(119909) 119889120583 (119909) = 0 119896 isin Z 120572 isin Y

119896 (11)

where 120575 is given above and 119862 ] are two positive constantsindependent of 119896 120572 119909 and 119910119896

120572 Clearly 120595119896

120572120572isinY119896119896isinZ is an

orthonormal basis of 1198712(119883) In what follows we also refer tothe functions 120595119896

120572as wavelets





0 119903 120573 120574) centered at 119909

0with width


(i) |119891(119909)| le 119862(1(119881119903(119909

0) + 119881(119909 119909

0)))(119903(119903 + 119889(119909 119909

0)))120574

for all 119909 isin 119883

(ii) |119891(119909)minus119891(119910)| le 119862(119889(119909 119910)(119903+119889(119909 1199090)))

120573(1(119881

119903(119909

0)+

119881(119909 1199090)))(119903(119903 + 119889(119909 119909

0)))


0)(119903 + 119889(119909 119909

0))


119866(1199090 119903 120573 120574) The norm of 119891 on 119866(119909

0 119903 120573 120574) is defined by

10038171003817100381710038171198911003817100381710038171003817119866(1199090 119903120573120574)






∘


∘

119866(120598 120598) =



119899 sub 119866(120598 120598) such that 119891

119899minus119891

119866(120573120574)rarr 0


119866(120573 120574) we define 119891 ∘119866(120573120574)

=

119891119866(120573120574)

Obviously∘


the norm sdot ∘119866(120573120574)


119866(120573 120574))1015840

the dual

space of∘







120572radic119881

120575119896(119909119896

120572) isin

119866(119909119896

120572 120575119896 120578 120598) for each 120598 gt 0





120572radic119881

120575119896(119909119896

120572) isin 119866(119909

119896

120572 120575

119896 120578 120598) for each

120598 gt 0

Let 119875119896and119876


119896and119882

119896


119896and 119876

119896



(i) for all 119909 119910 isin 119883

1003816100381610038161003816119875119896 (119909 119910)1003816100381610038161003816 le 119862

1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(13)

(ii) for 119889(119910 1199101015840) le (121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816119875119896(119909 119910) minus 119875

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

120598

(14)

(iii) for 119889(119909 1199091015840) le (12119860

0)(120575

119896+ 119889(119909 119910)) and 119889(119910 1199101015840

) le

(121198600)(120575119896 + 119889(119909 1199101015840))

10038161003816100381610038161003816[119875

119896(119909 119910) minus 119875

119896(119909

1015840 119910)] minus [119875

119896(119909 119910

1015840) minus 119875

119896(119909

1015840 119910

1015840)]10038161003816100381610038161003816

le 119862(119889 (119909 1199091015840)

120575119896 + 119889 (119909 119910))

120578

(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(15)



follows that

119881 (119909 1199090) le 119862(

119889 (119909 1199090)

119903)

Q

119881119903(119909

0) (16)

1

119881119903(119909

0)le 119862

1

119881 (119909 1199090) + 119881

119903(119909

0)(119889 (119909 119909

0)

119903)

Q

(17)



1003816100381610038161003816119875119896 (119909 119910)1003816100381610038161003816 le 119862

1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

(18)


120575119896(119909) and hence

1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

le 1198621

119881 (119909 119910) + 119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

(19)



1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

le 1198621

119881 (119909 119910) + 119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

ΓminusQ

(20)


10038161003816100381610038161003816119875119896(119909 119910) minus 119875

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896)

120578


119904

radic119881120575119896 (119909) 119881

120575119896 (119910)

+exp minus] (120575minus119896119889 (119909 1199101015840))

119904

radic119881120575119896 (119909) 119881

120575119896 (1199101015840)

)

(21)


(121198600)(120575119896+119889(119909 119910)) implies that 120575119896+119889(119909 1199101015840) ge (12119860

0)(120575119896+

119889(119909 119910)) and that 119881120575119896(119909) + 119881(119909 1199101015840) sim 119881

120575119896(119909) + 119881(119909 119910) it


119868119868 le 119862(119889 (119910 1199101015840)

120575119896)

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

Γ

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

Γminus120578

(22)




(i) for all 119909 119910 isin 119883

1003816100381610038161003816119876119896(119909 119910)

1003816100381610038161003816 le 1198621

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(23)

(ii) for 119889(119910 1199101015840) le (121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816119876

119896(119909 119910) minus 119876

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(24)

(iii) for 119889(119909 1199091015840) le (121198600)(120575119896 + 119889(119909 119910)) and 119889(119910 1199101015840) le

(121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816[119876

119896(119909 119910) minus 119876

119896(119909

1015840 119910)] minus [119876

119896(119909 119910

1015840) minus 119876

119896(119909

1015840 119910

1015840)]10038161003816100381610038161003816

le 119862(119889 (119909 1199091015840)

120575119896 + 119889 (119909 119910))

120578

(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(25)

Let 120593119897




120573119897isinZ120573isinY119897


119896119864




120598gt 0


1003816100381610038161003816119876119896119864

119897(119909 119910)

1003816100381610038161003816 le 119862120575|119896minus119897|120590 1

119881120575(119896and119897) (119909) + 119881 (119909 119910)

times (120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

(26)

(ii) for all 120591 isin (0 1)10038161003816100381610038161003816119876

119896119864

119897(119909 119910) minus 119876

119896119864

119897(119909 119910

1015840)10038161003816100381610038161003816

le 119862120575|119896minus119897|120590120591

(119889 (119910 1199101015840)

120575(119896and119897) + 119889 (119909 119910))

120578(1minus120591)

times1

119881120575(119896and119897) (119909) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

(27)







119896and


119896and 119882

119896



119891 = sum119896isinZ

119876119896(119891) = sum

119896isinZ

1198762

119896(119891) (28)


119891 = 119875119897(119891) +sum

119896ge119897

119876119896(119891) = 119875

2

119897(119891) +sum

119896ge119897

1198762

119896(119891) (29)




∘


∘

119866(1205731015840 1205741015840))1015840




119896ge1119876

119896(119891) and 119875

1(119891) in


119896(119909 119910) and 119875


119891 = 1198751(119891) + sum

119896ge1

119876119896(119891) (30)




119901 (119883) is defined by

119861119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

lt infin (31)

where10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(32)


119901 (119883) is defined by

119865119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

lt infin (33)

where

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(34)


sim 10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

119891 isin 119861119904119902

119901

(35)

as well as

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

1198751(119891)

119902

+ sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

119891 isin 119865119904119902

119901

(36)




119896


120572120572isinY119896119896isinZ be


1and 119863


1and 119882

119896 respectively


120572120572isinY119896119896isinZ


119866(120573 120574))1015840 For all


10038171003817100381710038171198751(119891)1003817100381710038171003817119901 + sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 + sum

119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(37)


10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(38)



119901and 119864

1(119891)

119901is a


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

≲ sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(39)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum119897ge1

int119883

1003816100381610038161003816119876119896119863

119897(119909 119910)

10038161003816100381610038161003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

int119883

1

119881120575(119896and119897) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

times1003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

119872(119863119897(119891)) (119909)

(40)


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891) (119909)1003817100381710038171003817119901)

119902

1119902

le 119862sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(41)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

≲

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(42)


sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891) (119909)1003816100381610038161003816)

119902

le 119862sum119896ge1

(sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119896ge1

sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119897ge1

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

(43)





Let 120595119896


119896and

let 1206011


1as shown


119891 = sum

120572isinX1

⟨1206011

120572 119891⟩ 120601

1

120572+ sum

119896ge1

sum

120572isinY119896

⟨120595119896

120572 119891⟩120595

119896

120572 (44)



120572 120595119896



sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(45)


sim

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

( sum

120572isinX119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

100381610038161003816100381610038161205941198761120572)

119902

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572

]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(46)


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

119881120575(1199091

120572) + 119881 (119909 1199091

120572)(

120575

120575 + 119889 (119909 1199091

120572))

120598


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119895

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le119862120575

|119895minus119896|120578

119881120575(119895and119896) (119909

119895

120572) + 119881 (119909 119909119895

120572)(

120575(119895and119896)

120575(119895and119896) + 119889 (119909 119909119895

120572))

120598

(47)


10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(48)


10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816

le sum

120572isinX1

120583 (1198761

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(1199091

120572) + 119881 (119909 1199091

120572)

times (120575

120575 + 119889 (119909 1199091

120572))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

= 119862 sum

120572isinX1

int119883

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572(119911) 119889119911

le 119862119872( sum

120572isinX1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572)(119909)

(49)

so that

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 le 119862[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

1119901

(50)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum

120572isinY119896

120583 (119876119896

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119896

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)(119909)

(51)

and hence

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

120594119876119896120572

]

]

119902119901

(52)


This implies that

10038171003817100381710038171198751

(119891)1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(53)


sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

le sum

120572isinX1

120583 (1198761

120572)

times (int119883

10038161003816100381610038161198751 (119891) (119910)1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910)

119901

le 11986210038171003817100381710038171198751(119891)

1003817100381710038171003817119901

119901

(54)

as well as

sum

120572isinY119894

(10038161003816100381610038161003816⟨120595

119895

120572 119891⟩

10038161003816100381610038161003816120583 (119876

119895

120572)1119901minus12

)119901

le sum

120572isinY119894

120583 (119876119895

120572)(sum

119896ge1

int119883

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int119883

120595119895

120572(119909)

radic120583 (119876119895

120572)

119876119896(119909 119910)119889120583(119909)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

le 119862int119883

(sum119896ge1

120575|119895minus119896|120578

int119883

1

119881120575(119895and119896)(119911) + 119881(119910 119911)

(120575(119895and119896)

120575(119895and119896) + 119889 (119910 119911))

120598

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

119889120583 (119911)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

120575|119895minus119896|120578

119872(119876119896(119891))

1003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

le 119862(sum119896ge1

120575|119895minus119896|120578119901 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119901

(55)


[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119895ge1

120575minus119895119904119902 [

[

sum

120572isinY119895

(120583 (119876119895

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119895

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

le 119862

10038171003817100381710038171198751(119891)1003817100381710038171003817119902

119901

+sum119895ge1

(sum119896ge1

120575(119896minus119895)119904

120575|119895minus119896|120578

120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(56)




119891 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119891⟩ 120593

119896

120572 (57)

where Z0 = X1 1205930

120572= 1206011

120572 Z119896 = Y119896 and 120593119896

120572= 120595119896

120572for 119896 ge



Definition 14 Let 120582 = 120582119896



119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)


119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)


119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)


119901 as well as119865

119904119902

119901



119901 for 1 lt 119901 lt infin



119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902


119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)


isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902


that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)


1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)




1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)


rarr 119891 in 119861119904119902



119901



11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840


given119892 isin 119861minus1199041199021015840

1199011015840 then m


on 119861119904119902


10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)


119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840



1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)



119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m


119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840


Proof of Theorem 17


1199011015840 and 119891 isin 119861

119904119902


119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)


1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)


10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)


119901 cap119866(1205781015840 1205781015840)Clearly m



119901 cap 119866(1205781015840 1205781015840) in 119861119904119902


1199011015840 sub (119861

119904119902

119901 )lowast





1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0 119903 120573 120574) centered at 119909

0with width


(i) |119891(119909)| le 119862(1(119881119903(119909

0) + 119881(119909 119909

0)))(119903(119903 + 119889(119909 119909

0)))120574

for all 119909 isin 119883

(ii) |119891(119909)minus119891(119910)| le 119862(119889(119909 119910)(119903+119889(119909 1199090)))

120573(1(119881

119903(119909

0)+

119881(119909 1199090)))(119903(119903 + 119889(119909 119909

0)))


0)(119903 + 119889(119909 119909

0))


119866(1199090 119903 120573 120574) The norm of 119891 on 119866(119909

0 119903 120573 120574) is defined by

10038171003817100381710038171198911003817100381710038171003817119866(1199090 119903120573120574)






∘


∘

119866(120598 120598) =



119899 sub 119866(120598 120598) such that 119891

119899minus119891

119866(120573120574)rarr 0


119866(120573 120574) we define 119891 ∘119866(120573120574)

=

119891119866(120573120574)

Obviously∘


the norm sdot ∘119866(120573120574)


119866(120573 120574))1015840

the dual

space of∘







120572radic119881

120575119896(119909119896

120572) isin

119866(119909119896

120572 120575119896 120578 120598) for each 120598 gt 0





120572radic119881

120575119896(119909119896

120572) isin 119866(119909

119896

120572 120575

119896 120578 120598) for each

120598 gt 0

Let 119875119896and119876


119896and119882

119896


119896and 119876

119896



(i) for all 119909 119910 isin 119883

1003816100381610038161003816119875119896 (119909 119910)1003816100381610038161003816 le 119862

1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(13)

(ii) for 119889(119910 1199101015840) le (121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816119875119896(119909 119910) minus 119875

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

120598

(14)

(iii) for 119889(119909 1199091015840) le (12119860

0)(120575

119896+ 119889(119909 119910)) and 119889(119910 1199101015840

) le

(121198600)(120575119896 + 119889(119909 1199101015840))

10038161003816100381610038161003816[119875

119896(119909 119910) minus 119875

119896(119909

1015840 119910)] minus [119875

119896(119909 119910

1015840) minus 119875

119896(119909

1015840 119910

1015840)]10038161003816100381610038161003816

le 119862(119889 (119909 1199091015840)

120575119896 + 119889 (119909 119910))

120578

(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(15)



follows that

119881 (119909 1199090) le 119862(

119889 (119909 1199090)

119903)

Q

119881119903(119909

0) (16)

1

119881119903(119909

0)le 119862

1

119881 (119909 1199090) + 119881

119903(119909

0)(119889 (119909 119909

0)

119903)

Q

(17)



1003816100381610038161003816119875119896 (119909 119910)1003816100381610038161003816 le 119862

1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

(18)


120575119896(119909) and hence

1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

le 1198621

119881 (119909 119910) + 119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

(19)



1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

le 1198621

119881 (119909 119910) + 119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

ΓminusQ

(20)


10038161003816100381610038161003816119875119896(119909 119910) minus 119875

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896)

120578


119904

radic119881120575119896 (119909) 119881

120575119896 (119910)

+exp minus] (120575minus119896119889 (119909 1199101015840))

119904

radic119881120575119896 (119909) 119881

120575119896 (1199101015840)

)

(21)


(121198600)(120575119896+119889(119909 119910)) implies that 120575119896+119889(119909 1199101015840) ge (12119860

0)(120575119896+

119889(119909 119910)) and that 119881120575119896(119909) + 119881(119909 1199101015840) sim 119881

120575119896(119909) + 119881(119909 119910) it


119868119868 le 119862(119889 (119910 1199101015840)

120575119896)

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

Γ

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

Γminus120578

(22)




(i) for all 119909 119910 isin 119883

1003816100381610038161003816119876119896(119909 119910)

1003816100381610038161003816 le 1198621

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(23)

(ii) for 119889(119910 1199101015840) le (121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816119876

119896(119909 119910) minus 119876

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(24)

(iii) for 119889(119909 1199091015840) le (121198600)(120575119896 + 119889(119909 119910)) and 119889(119910 1199101015840) le

(121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816[119876

119896(119909 119910) minus 119876

119896(119909

1015840 119910)] minus [119876

119896(119909 119910

1015840) minus 119876

119896(119909

1015840 119910

1015840)]10038161003816100381610038161003816

le 119862(119889 (119909 1199091015840)

120575119896 + 119889 (119909 119910))

120578

(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(25)

Let 120593119897




120573119897isinZ120573isinY119897


119896119864




120598gt 0


1003816100381610038161003816119876119896119864

119897(119909 119910)

1003816100381610038161003816 le 119862120575|119896minus119897|120590 1

119881120575(119896and119897) (119909) + 119881 (119909 119910)

times (120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

(26)

(ii) for all 120591 isin (0 1)10038161003816100381610038161003816119876

119896119864

119897(119909 119910) minus 119876

119896119864

119897(119909 119910

1015840)10038161003816100381610038161003816

le 119862120575|119896minus119897|120590120591

(119889 (119910 1199101015840)

120575(119896and119897) + 119889 (119909 119910))

120578(1minus120591)

times1

119881120575(119896and119897) (119909) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

(27)







119896and


119896and 119882

119896



119891 = sum119896isinZ

119876119896(119891) = sum

119896isinZ

1198762

119896(119891) (28)


119891 = 119875119897(119891) +sum

119896ge119897

119876119896(119891) = 119875

2

119897(119891) +sum

119896ge119897

1198762

119896(119891) (29)




∘


∘

119866(1205731015840 1205741015840))1015840




119896ge1119876

119896(119891) and 119875

1(119891) in


119896(119909 119910) and 119875


119891 = 1198751(119891) + sum

119896ge1

119876119896(119891) (30)




119901 (119883) is defined by

119861119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

lt infin (31)

where10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(32)


119901 (119883) is defined by

119865119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

lt infin (33)

where

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(34)


sim 10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

119891 isin 119861119904119902

119901

(35)

as well as

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

1198751(119891)

119902

+ sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

119891 isin 119865119904119902

119901

(36)




119896


120572120572isinY119896119896isinZ be


1and 119863


1and 119882

119896 respectively


120572120572isinY119896119896isinZ


119866(120573 120574))1015840 For all


10038171003817100381710038171198751(119891)1003817100381710038171003817119901 + sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 + sum

119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(37)


10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(38)



119901and 119864

1(119891)

119901is a


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

≲ sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(39)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum119897ge1

int119883

1003816100381610038161003816119876119896119863

119897(119909 119910)

10038161003816100381610038161003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

int119883

1

119881120575(119896and119897) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

times1003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

119872(119863119897(119891)) (119909)

(40)


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891) (119909)1003817100381710038171003817119901)

119902

1119902

le 119862sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(41)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

≲

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(42)


sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891) (119909)1003816100381610038161003816)

119902

le 119862sum119896ge1

(sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119896ge1

sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119897ge1

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

(43)





Let 120595119896


119896and

let 1206011


1as shown


119891 = sum

120572isinX1

⟨1206011

120572 119891⟩ 120601

1

120572+ sum

119896ge1

sum

120572isinY119896

⟨120595119896

120572 119891⟩120595

119896

120572 (44)



120572 120595119896



sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(45)


sim

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

( sum

120572isinX119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

100381610038161003816100381610038161205941198761120572)

119902

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572

]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(46)


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

119881120575(1199091

120572) + 119881 (119909 1199091

120572)(

120575

120575 + 119889 (119909 1199091

120572))

120598


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119895

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le119862120575

|119895minus119896|120578

119881120575(119895and119896) (119909

119895

120572) + 119881 (119909 119909119895

120572)(

120575(119895and119896)

120575(119895and119896) + 119889 (119909 119909119895

120572))

120598

(47)


10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(48)


10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816

le sum

120572isinX1

120583 (1198761

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(1199091

120572) + 119881 (119909 1199091

120572)

times (120575

120575 + 119889 (119909 1199091

120572))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

= 119862 sum

120572isinX1

int119883

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572(119911) 119889119911

le 119862119872( sum

120572isinX1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572)(119909)

(49)

so that

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 le 119862[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

1119901

(50)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum

120572isinY119896

120583 (119876119896

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119896

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)(119909)

(51)

and hence

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

120594119876119896120572

]

]

119902119901

(52)


This implies that

10038171003817100381710038171198751

(119891)1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(53)


sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

le sum

120572isinX1

120583 (1198761

120572)

times (int119883

10038161003816100381610038161198751 (119891) (119910)1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910)

119901

le 11986210038171003817100381710038171198751(119891)

1003817100381710038171003817119901

119901

(54)

as well as

sum

120572isinY119894

(10038161003816100381610038161003816⟨120595

119895

120572 119891⟩

10038161003816100381610038161003816120583 (119876

119895

120572)1119901minus12

)119901

le sum

120572isinY119894

120583 (119876119895

120572)(sum

119896ge1

int119883

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int119883

120595119895

120572(119909)

radic120583 (119876119895

120572)

119876119896(119909 119910)119889120583(119909)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

le 119862int119883

(sum119896ge1

120575|119895minus119896|120578

int119883

1

119881120575(119895and119896)(119911) + 119881(119910 119911)

(120575(119895and119896)

120575(119895and119896) + 119889 (119910 119911))

120598

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

119889120583 (119911)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

120575|119895minus119896|120578

119872(119876119896(119891))

1003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

le 119862(sum119896ge1

120575|119895minus119896|120578119901 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119901

(55)


[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119895ge1

120575minus119895119904119902 [

[

sum

120572isinY119895

(120583 (119876119895

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119895

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

le 119862

10038171003817100381710038171198751(119891)1003817100381710038171003817119902

119901

+sum119895ge1

(sum119896ge1

120575(119896minus119895)119904

120575|119895minus119896|120578

120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(56)




119891 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119891⟩ 120593

119896

120572 (57)

where Z0 = X1 1205930

120572= 1206011

120572 Z119896 = Y119896 and 120593119896

120572= 120595119896

120572for 119896 ge



Definition 14 Let 120582 = 120582119896



119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)


119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)


119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)


119901 as well as119865

119904119902

119901



119901 for 1 lt 119901 lt infin



119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902


119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)


isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902


that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)


1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)




1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)


rarr 119891 in 119861119904119902



119901



11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840


given119892 isin 119861minus1199041199021015840

1199011015840 then m


on 119861119904119902


10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)


119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840



1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)



119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m


119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840


Proof of Theorem 17


1199011015840 and 119891 isin 119861

119904119902


119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)


1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)


10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)


119901 cap119866(1205781015840 1205781015840)Clearly m



119901 cap 119866(1205781015840 1205781015840) in 119861119904119902


1199011015840 sub (119861

119904119902

119901 )lowast





1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

Γ

le 1198621

119881 (119909 119910) + 119881120575119896 (119909)

(120575119896

120575119896 + 119889 (119909 119910))

ΓminusQ

(20)


10038161003816100381610038161003816119875119896(119909 119910) minus 119875

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896)

120578


119904

radic119881120575119896 (119909) 119881

120575119896 (119910)

+exp minus] (120575minus119896119889 (119909 1199101015840))

119904

radic119881120575119896 (119909) 119881

120575119896 (1199101015840)

)

(21)


(121198600)(120575119896+119889(119909 119910)) implies that 120575119896+119889(119909 1199101015840) ge (12119860

0)(120575119896+

119889(119909 119910)) and that 119881120575119896(119909) + 119881(119909 1199101015840) sim 119881

120575119896(119909) + 119881(119909 119910) it


119868119868 le 119862(119889 (119910 1199101015840)

120575119896)

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

Γ

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

1

119881120575119896 (119909) + 119881 (119909 119910)

times (120575119896

120575119896 + 119889 (119909 119910))

Γminus120578

(22)




(i) for all 119909 119910 isin 119883

1003816100381610038161003816119876119896(119909 119910)

1003816100381610038161003816 le 1198621

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(23)

(ii) for 119889(119910 1199101015840) le (121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816119876

119896(119909 119910) minus 119876

119896(119909 119910

1015840)10038161003816100381610038161003816

le 119862(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(24)

(iii) for 119889(119909 1199091015840) le (121198600)(120575119896 + 119889(119909 119910)) and 119889(119910 1199101015840) le

(121198600)(120575119896 + 119889(119909 119910))

10038161003816100381610038161003816[119876

119896(119909 119910) minus 119876

119896(119909

1015840 119910)] minus [119876

119896(119909 119910

1015840) minus 119876

119896(119909

1015840 119910

1015840)]10038161003816100381610038161003816

le 119862(119889 (119909 1199091015840)

120575119896 + 119889 (119909 119910))

120578

(119889 (119910 1199101015840)

120575119896 + 119889 (119909 119910))

120578

times1

119881120575119896 (119909) + 119881 (119909 119910)

(120575119896

120575119896 + 119889 (119909 119910))

120598

(25)

Let 120593119897




120573119897isinZ120573isinY119897


119896119864




120598gt 0


1003816100381610038161003816119876119896119864

119897(119909 119910)

1003816100381610038161003816 le 119862120575|119896minus119897|120590 1

119881120575(119896and119897) (119909) + 119881 (119909 119910)

times (120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

(26)

(ii) for all 120591 isin (0 1)10038161003816100381610038161003816119876

119896119864

119897(119909 119910) minus 119876

119896119864

119897(119909 119910

1015840)10038161003816100381610038161003816

le 119862120575|119896minus119897|120590120591

(119889 (119910 1199101015840)

120575(119896and119897) + 119889 (119909 119910))

120578(1minus120591)

times1

119881120575(119896and119897) (119909) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

(27)







119896and


119896and 119882

119896



119891 = sum119896isinZ

119876119896(119891) = sum

119896isinZ

1198762

119896(119891) (28)


119891 = 119875119897(119891) +sum

119896ge119897

119876119896(119891) = 119875

2

119897(119891) +sum

119896ge119897

1198762

119896(119891) (29)




∘


∘

119866(1205731015840 1205741015840))1015840




119896ge1119876

119896(119891) and 119875

1(119891) in


119896(119909 119910) and 119875


119891 = 1198751(119891) + sum

119896ge1

119876119896(119891) (30)




119901 (119883) is defined by

119861119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

lt infin (31)

where10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(32)


119901 (119883) is defined by

119865119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

lt infin (33)

where

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(34)


sim 10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

119891 isin 119861119904119902

119901

(35)

as well as

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

1198751(119891)

119902

+ sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

119891 isin 119865119904119902

119901

(36)




119896


120572120572isinY119896119896isinZ be


1and 119863


1and 119882

119896 respectively


120572120572isinY119896119896isinZ


119866(120573 120574))1015840 For all


10038171003817100381710038171198751(119891)1003817100381710038171003817119901 + sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 + sum

119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(37)


10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(38)



119901and 119864

1(119891)

119901is a


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

≲ sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(39)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum119897ge1

int119883

1003816100381610038161003816119876119896119863

119897(119909 119910)

10038161003816100381610038161003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

int119883

1

119881120575(119896and119897) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

times1003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

119872(119863119897(119891)) (119909)

(40)


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891) (119909)1003817100381710038171003817119901)

119902

1119902

le 119862sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(41)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

≲

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(42)


sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891) (119909)1003816100381610038161003816)

119902

le 119862sum119896ge1

(sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119896ge1

sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119897ge1

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

(43)





Let 120595119896


119896and

let 1206011


1as shown


119891 = sum

120572isinX1

⟨1206011

120572 119891⟩ 120601

1

120572+ sum

119896ge1

sum

120572isinY119896

⟨120595119896

120572 119891⟩120595

119896

120572 (44)



120572 120595119896



sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(45)


sim

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

( sum

120572isinX119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

100381610038161003816100381610038161205941198761120572)

119902

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572

]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(46)


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

119881120575(1199091

120572) + 119881 (119909 1199091

120572)(

120575

120575 + 119889 (119909 1199091

120572))

120598


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119895

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le119862120575

|119895minus119896|120578

119881120575(119895and119896) (119909

119895

120572) + 119881 (119909 119909119895

120572)(

120575(119895and119896)

120575(119895and119896) + 119889 (119909 119909119895

120572))

120598

(47)


10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(48)


10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816

le sum

120572isinX1

120583 (1198761

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(1199091

120572) + 119881 (119909 1199091

120572)

times (120575

120575 + 119889 (119909 1199091

120572))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

= 119862 sum

120572isinX1

int119883

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572(119911) 119889119911

le 119862119872( sum

120572isinX1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572)(119909)

(49)

so that

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 le 119862[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

1119901

(50)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum

120572isinY119896

120583 (119876119896

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119896

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)(119909)

(51)

and hence

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

120594119876119896120572

]

]

119902119901

(52)


This implies that

10038171003817100381710038171198751

(119891)1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(53)


sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

le sum

120572isinX1

120583 (1198761

120572)

times (int119883

10038161003816100381610038161198751 (119891) (119910)1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910)

119901

le 11986210038171003817100381710038171198751(119891)

1003817100381710038171003817119901

119901

(54)

as well as

sum

120572isinY119894

(10038161003816100381610038161003816⟨120595

119895

120572 119891⟩

10038161003816100381610038161003816120583 (119876

119895

120572)1119901minus12

)119901

le sum

120572isinY119894

120583 (119876119895

120572)(sum

119896ge1

int119883

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int119883

120595119895

120572(119909)

radic120583 (119876119895

120572)

119876119896(119909 119910)119889120583(119909)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

le 119862int119883

(sum119896ge1

120575|119895minus119896|120578

int119883

1

119881120575(119895and119896)(119911) + 119881(119910 119911)

(120575(119895and119896)

120575(119895and119896) + 119889 (119910 119911))

120598

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

119889120583 (119911)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

120575|119895minus119896|120578

119872(119876119896(119891))

1003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

le 119862(sum119896ge1

120575|119895minus119896|120578119901 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119901

(55)


[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119895ge1

120575minus119895119904119902 [

[

sum

120572isinY119895

(120583 (119876119895

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119895

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

le 119862

10038171003817100381710038171198751(119891)1003817100381710038171003817119902

119901

+sum119895ge1

(sum119896ge1

120575(119896minus119895)119904

120575|119895minus119896|120578

120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(56)




119891 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119891⟩ 120593

119896

120572 (57)

where Z0 = X1 1205930

120572= 1206011

120572 Z119896 = Y119896 and 120593119896

120572= 120595119896

120572for 119896 ge



Definition 14 Let 120582 = 120582119896



119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)


119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)


119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)


119901 as well as119865

119904119902

119901



119901 for 1 lt 119901 lt infin



119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902


119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)


isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902


that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)


1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)




1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)


rarr 119891 in 119861119904119902



119901



11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840


given119892 isin 119861minus1199041199021015840

1199011015840 then m


on 119861119904119902


10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)


119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840



1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)



119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m


119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840


Proof of Theorem 17


1199011015840 and 119891 isin 119861

119904119902


119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)


1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)


10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)


119901 cap119866(1205781015840 1205781015840)Clearly m



119901 cap 119866(1205781015840 1205781015840) in 119861119904119902


1199011015840 sub (119861

119904119902

119901 )lowast





1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896and


119896and 119882

119896



119891 = sum119896isinZ

119876119896(119891) = sum

119896isinZ

1198762

119896(119891) (28)


119891 = 119875119897(119891) +sum

119896ge119897

119876119896(119891) = 119875

2

119897(119891) +sum

119896ge119897

1198762

119896(119891) (29)




∘


∘

119866(1205731015840 1205741015840))1015840




119896ge1119876

119896(119891) and 119875

1(119891) in


119896(119909 119910) and 119875


119891 = 1198751(119891) + sum

119896ge1

119876119896(119891) (30)




119901 (119883) is defined by

119861119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

lt infin (31)

where10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(32)


119901 (119883) is defined by

119865119904119902

119901(119883) = 119891 isin (

∘

119866 (120573 120574))1015840

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

lt infin (33)

where

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

=10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(34)


sim 10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

119891 isin 119861119904119902

119901

(35)

as well as

10038171003817100381710038171198911003817100381710038171003817119865119904119902119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

1198751(119891)

119902

+ sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

119891 isin 119865119904119902

119901

(36)




119896


120572120572isinY119896119896isinZ be


1and 119863


1and 119882

119896 respectively


120572120572isinY119896119896isinZ


119866(120573 120574))1015840 For all


10038171003817100381710038171198751(119891)1003817100381710038171003817119901 + sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 + sum

119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(37)


10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim10038171003817100381710038171198641

(119891)1003817100381710038171003817119901 +

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(38)



119901and 119864

1(119891)

119901is a


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

≲ sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(39)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum119897ge1

int119883

1003816100381610038161003816119876119896119863

119897(119909 119910)

10038161003816100381610038161003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

int119883

1

119881120575(119896and119897) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

times1003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

119872(119863119897(119891)) (119909)

(40)


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891) (119909)1003817100381710038171003817119901)

119902

1119902

le 119862sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(41)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

≲

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(42)


sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891) (119909)1003816100381610038161003816)

119902

le 119862sum119896ge1

(sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119896ge1

sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119897ge1

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

(43)





Let 120595119896


119896and

let 1206011


1as shown


119891 = sum

120572isinX1

⟨1206011

120572 119891⟩ 120601

1

120572+ sum

119896ge1

sum

120572isinY119896

⟨120595119896

120572 119891⟩120595

119896

120572 (44)



120572 120595119896



sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(45)


sim

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

( sum

120572isinX119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

100381610038161003816100381610038161205941198761120572)

119902

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572

]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(46)


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

119881120575(1199091

120572) + 119881 (119909 1199091

120572)(

120575

120575 + 119889 (119909 1199091

120572))

120598


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119895

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le119862120575

|119895minus119896|120578

119881120575(119895and119896) (119909

119895

120572) + 119881 (119909 119909119895

120572)(

120575(119895and119896)

120575(119895and119896) + 119889 (119909 119909119895

120572))

120598

(47)


10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(48)


10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816

le sum

120572isinX1

120583 (1198761

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(1199091

120572) + 119881 (119909 1199091

120572)

times (120575

120575 + 119889 (119909 1199091

120572))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

= 119862 sum

120572isinX1

int119883

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572(119911) 119889119911

le 119862119872( sum

120572isinX1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572)(119909)

(49)

so that

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 le 119862[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

1119901

(50)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum

120572isinY119896

120583 (119876119896

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119896

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)(119909)

(51)

and hence

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

120594119876119896120572

]

]

119902119901

(52)


This implies that

10038171003817100381710038171198751

(119891)1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(53)


sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

le sum

120572isinX1

120583 (1198761

120572)

times (int119883

10038161003816100381610038161198751 (119891) (119910)1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910)

119901

le 11986210038171003817100381710038171198751(119891)

1003817100381710038171003817119901

119901

(54)

as well as

sum

120572isinY119894

(10038161003816100381610038161003816⟨120595

119895

120572 119891⟩

10038161003816100381610038161003816120583 (119876

119895

120572)1119901minus12

)119901

le sum

120572isinY119894

120583 (119876119895

120572)(sum

119896ge1

int119883

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int119883

120595119895

120572(119909)

radic120583 (119876119895

120572)

119876119896(119909 119910)119889120583(119909)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

le 119862int119883

(sum119896ge1

120575|119895minus119896|120578

int119883

1

119881120575(119895and119896)(119911) + 119881(119910 119911)

(120575(119895and119896)

120575(119895and119896) + 119889 (119910 119911))

120598

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

119889120583 (119911)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

120575|119895minus119896|120578

119872(119876119896(119891))

1003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

le 119862(sum119896ge1

120575|119895minus119896|120578119901 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119901

(55)


[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119895ge1

120575minus119895119904119902 [

[

sum

120572isinY119895

(120583 (119876119895

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119895

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

le 119862

10038171003817100381710038171198751(119891)1003817100381710038171003817119902

119901

+sum119895ge1

(sum119896ge1

120575(119896minus119895)119904

120575|119895minus119896|120578

120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(56)




119891 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119891⟩ 120593

119896

120572 (57)

where Z0 = X1 1205930

120572= 1206011

120572 Z119896 = Y119896 and 120593119896

120572= 120595119896

120572for 119896 ge



Definition 14 Let 120582 = 120582119896



119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)


119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)


119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)


119901 as well as119865

119904119902

119901



119901 for 1 lt 119901 lt infin



119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902


119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)


isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902


that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)


1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)




1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)


rarr 119891 in 119861119904119902



119901



11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840


given119892 isin 119861minus1199041199021015840

1199011015840 then m


on 119861119904119902


10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)


119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840



1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)



119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m


119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840


Proof of Theorem 17


1199011015840 and 119891 isin 119861

119904119902


119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)


1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)


10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)


119901 cap119866(1205781015840 1205781015840)Clearly m



119901 cap 119866(1205781015840 1205781015840) in 119861119904119902


1199011015840 sub (119861

119904119902

119901 )lowast





1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901and 119864

1(119891)

119901is a


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

≲ sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(39)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum119897ge1

int119883

1003816100381610038161003816119876119896119863

119897(119909 119910)

10038161003816100381610038161003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

int119883

1

119881120575(119896and119897) + 119881 (119909 119910)

(120575(119896and119897)

120575(119896and119897) + 119889 (119909 119910))

120598

times1003816100381610038161003816119863119897

(119891) (119910)1003816100381610038161003816 119889120583 (119910)

le 119862sum119897ge1

120575|119896minus119897|120578

119872(119863119897(119891)) (119909)

(40)


sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891) (119909)1003817100381710038171003817119901)

119902

1119902

le 119862sum119897ge1

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(41)



10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

≲

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897ge1

(120575minus119897119904 1003816100381610038161003816119863119897

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(42)


sum119896ge1

(120575minus119896119904 1003816100381610038161003816119876119896

(119891) (119909)1003816100381610038161003816)

119902

le 119862sum119896ge1

(sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119896ge1

sum119897ge1

120575(119897minus119896)119904

120575|119896minus119897|120578

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

le 119862sum119897ge1

(120575minus119897119904119872(119863

119897(119891)) (119909))

119902

(43)





Let 120595119896


119896and

let 1206011


1as shown


119891 = sum

120572isinX1

⟨1206011

120572 119891⟩ 120601

1

120572+ sum

119896ge1

sum

120572isinY119896

⟨120595119896

120572 119891⟩120595

119896

120572 (44)



120572 120595119896



sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(45)


sim

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

( sum

120572isinX119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

100381610038161003816100381610038161205941198761120572)

119902

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572

]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(46)


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 1198621

119881120575(1199091

120572) + 119881 (119909 1199091

120572)(

120575

120575 + 119889 (119909 1199091

120572))

120598


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119895

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le119862120575

|119895minus119896|120578

119881120575(119895and119896) (119909

119895

120572) + 119881 (119909 119909119895

120572)(

120575(119895and119896)

120575(119895and119896) + 119889 (119909 119909119895

120572))

120598

(47)


10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(48)


10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816

le sum

120572isinX1

120583 (1198761

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(1199091

120572) + 119881 (119909 1199091

120572)

times (120575

120575 + 119889 (119909 1199091

120572))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

= 119862 sum

120572isinX1

int119883

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572(119911) 119889119911

le 119862119872( sum

120572isinX1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572)(119909)

(49)

so that

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 le 119862[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

1119901

(50)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum

120572isinY119896

120583 (119876119896

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119896

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)(119909)

(51)

and hence

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

120594119876119896120572

]

]

119902119901

(52)


This implies that

10038171003817100381710038171198751

(119891)1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(53)


sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

le sum

120572isinX1

120583 (1198761

120572)

times (int119883

10038161003816100381610038161198751 (119891) (119910)1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910)

119901

le 11986210038171003817100381710038171198751(119891)

1003817100381710038171003817119901

119901

(54)

as well as

sum

120572isinY119894

(10038161003816100381610038161003816⟨120595

119895

120572 119891⟩

10038161003816100381610038161003816120583 (119876

119895

120572)1119901minus12

)119901

le sum

120572isinY119894

120583 (119876119895

120572)(sum

119896ge1

int119883

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int119883

120595119895

120572(119909)

radic120583 (119876119895

120572)

119876119896(119909 119910)119889120583(119909)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

le 119862int119883

(sum119896ge1

120575|119895minus119896|120578

int119883

1

119881120575(119895and119896)(119911) + 119881(119910 119911)

(120575(119895and119896)

120575(119895and119896) + 119889 (119910 119911))

120598

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

119889120583 (119911)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

120575|119895minus119896|120578

119872(119876119896(119891))

1003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

le 119862(sum119896ge1

120575|119895minus119896|120578119901 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119901

(55)


[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119895ge1

120575minus119895119904119902 [

[

sum

120572isinY119895

(120583 (119876119895

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119895

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

le 119862

10038171003817100381710038171198751(119891)1003817100381710038171003817119902

119901

+sum119895ge1

(sum119896ge1

120575(119896minus119895)119904

120575|119895minus119896|120578

120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(56)




119891 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119891⟩ 120593

119896

120572 (57)

where Z0 = X1 1205930

120572= 1206011

120572 Z119896 = Y119896 and 120593119896

120572= 120595119896

120572for 119896 ge



Definition 14 Let 120582 = 120582119896



119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)


119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)


119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)


119901 as well as119865

119904119902

119901



119901 for 1 lt 119901 lt infin



119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902


119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)


isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902


that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)


1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)




1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)


rarr 119891 in 119861119904119902



119901



11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840


given119892 isin 119861minus1199041199021015840

1199011015840 then m


on 119861119904119902


10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)


119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840



1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)



119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m


119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840


Proof of Theorem 17


1199011015840 and 119891 isin 119861

119904119902


119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)


1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)


10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)


119901 cap119866(1205781015840 1205781015840)Clearly m



119901 cap 119866(1205781015840 1205781015840) in 119861119904119902


1199011015840 sub (119861

119904119902

119901 )lowast





1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119895

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le119862120575

|119895minus119896|120578

119881120575(119895and119896) (119909

119895

120572) + 119881 (119909 119909119895

120572)(

120575(119895and119896)

120575(119895and119896) + 119889 (119909 119909119895

120572))

120598

(47)


10038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(48)


10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816

le sum

120572isinX1

120583 (1198761

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(1199091

120572) + 119881 (119909 1199091

120572)

times (120575

120575 + 119889 (119909 1199091

120572))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

le 119862 sum

120572isinX1

int1198761120572

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119911

= 119862 sum

120572isinX1

int119883

1

119881120575(119911) + 119881 (119909 119911)

(120575

120575 + 119889 (119909 119911))

120598

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572(119911) 119889119911

le 119862119872( sum

120572isinX1

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨1206011

120572

radic120583 (1198761

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205941198761120572)(119909)

(49)

so that

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 le 119862[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

1119901

(50)


1003816100381610038161003816119876119896(119891) (119909)

1003816100381610038161003816

le sum

120572isinY119896

120583 (119876119896

120572)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

times

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119876119896(

120595119896

120572

radic120583 (119876119896

120572)

) (119909)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)(119909)

(51)

and hence

sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119872( sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572)

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum

120572isinY119896

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

⟨120595119896

120572

radic120583 (119876119896

120572)

119891⟩

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120594119876119896120572

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

119902

119901

le 119862sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

120594119876119896120572

]

]

119902119901

(52)


This implies that

10038171003817100381710038171198751

(119891)1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(53)


sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

le sum

120572isinX1

120583 (1198761

120572)

times (int119883

10038161003816100381610038161198751 (119891) (119910)1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910)

119901

le 11986210038171003817100381710038171198751(119891)

1003817100381710038171003817119901

119901

(54)

as well as

sum

120572isinY119894

(10038161003816100381610038161003816⟨120595

119895

120572 119891⟩

10038161003816100381610038161003816120583 (119876

119895

120572)1119901minus12

)119901

le sum

120572isinY119894

120583 (119876119895

120572)(sum

119896ge1

int119883

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int119883

120595119895

120572(119909)

radic120583 (119876119895

120572)

119876119896(119909 119910)119889120583(119909)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

le 119862int119883

(sum119896ge1

120575|119895minus119896|120578

int119883

1

119881120575(119895and119896)(119911) + 119881(119910 119911)

(120575(119895and119896)

120575(119895and119896) + 119889 (119910 119911))

120598

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

119889120583 (119911)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

120575|119895minus119896|120578

119872(119876119896(119891))

1003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

le 119862(sum119896ge1

120575|119895minus119896|120578119901 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119901

(55)


[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119895ge1

120575minus119895119904119902 [

[

sum

120572isinY119895

(120583 (119876119895

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119895

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

le 119862

10038171003817100381710038171198751(119891)1003817100381710038171003817119902

119901

+sum119895ge1

(sum119896ge1

120575(119896minus119895)119904

120575|119895minus119896|120578

120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(56)




119891 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119891⟩ 120593

119896

120572 (57)

where Z0 = X1 1205930

120572= 1206011

120572 Z119896 = Y119896 and 120593119896

120572= 120595119896

120572for 119896 ge



Definition 14 Let 120582 = 120582119896



119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)


119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)


119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)


119901 as well as119865

119904119902

119901



119901 for 1 lt 119901 lt infin



119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902


119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)


isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902


that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)


1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)




1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)


rarr 119891 in 119861119904119902



119901



11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840


given119892 isin 119861minus1199041199021015840

1199011015840 then m


on 119861119904119902


10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)


119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840



1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)



119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m


119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840


Proof of Theorem 17


1199011015840 and 119891 isin 119861

119904119902


119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)


1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)


10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)


119901 cap119866(1205781015840 1205781015840)Clearly m



119901 cap 119866(1205781015840 1205781015840) in 119861119904119902


1199011015840 sub (119861

119904119902

119901 )lowast





1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










This implies that

10038171003817100381710038171198751

(119891)1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le

[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+ sum119896ge1

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(53)


sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

le sum

120572isinX1

120583 (1198761

120572)

times (int119883

10038161003816100381610038161198751 (119891) (119910)1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198751(

1206011

120572

radic120583 (1198761

120572)

) (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119889119910)

119901

le 11986210038171003817100381710038171198751(119891)

1003817100381710038171003817119901

119901

(54)

as well as

sum

120572isinY119894

(10038161003816100381610038161003816⟨120595

119895

120572 119891⟩

10038161003816100381610038161003816120583 (119876

119895

120572)1119901minus12

)119901

le sum

120572isinY119894

120583 (119876119895

120572)(sum

119896ge1

int119883

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

int119883

120595119895

120572(119909)

radic120583 (119876119895

120572)

119876119896(119909 119910)119889120583(119909)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

le 119862int119883

(sum119896ge1

120575|119895minus119896|120578

int119883

1

119881120575(119895and119896)(119911) + 119881(119910 119911)

(120575(119895and119896)

120575(119895and119896) + 119889 (119910 119911))

120598

1003816100381610038161003816119876119896(119891) (119910)

1003816100381610038161003816 119889120583 (119910))

119901

119889120583 (119911)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge1

120575|119895minus119896|120578

119872(119876119896(119891))

1003817100381710038171003817100381710038171003817100381710038171003817

119901

119901

le 119862(sum119896ge1

120575|119895minus119896|120578119901 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119901

(55)


[ sum

120572isinX1

(120583 (1198761

120572)1119901minus12 10038161003816100381610038161003816

⟨1206011

120572 119891⟩

10038161003816100381610038161003816)119901

]

119902119901

+sum119895ge1

120575minus119895119904119902 [

[

sum

120572isinY119895

(120583 (119876119895

120572)1119901minus12 10038161003816100381610038161003816

⟨120595119895

120572 119891⟩

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

le 119862

10038171003817100381710038171198751(119891)1003817100381710038171003817119902

119901

+sum119895ge1

(sum119896ge1

120575(119896minus119895)119904

120575|119895minus119896|120578

120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198751 (119891)

1003817100381710038171003817119902

119901+ sum

119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(56)




119891 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119891⟩ 120593

119896

120572 (57)

where Z0 = X1 1205930

120572= 1206011

120572 Z119896 = Y119896 and 120593119896

120572= 120595119896

120572for 119896 ge



Definition 14 Let 120582 = 120582119896



119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)


119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)


119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)


119901 as well as119865

119904119902

119901



119901 for 1 lt 119901 lt infin



119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902


119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)


isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902


that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)


1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)




1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)


rarr 119891 in 119861119904119902



119901



11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840


given119892 isin 119861minus1199041199021015840

1199011015840 then m


on 119861119904119902


10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)


119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840



1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)



119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m


119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840


Proof of Theorem 17


1199011015840 and 119891 isin 119861

119904119902


119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)


1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)


10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)


119901 cap119866(1205781015840 1205781015840)Clearly m



119901 cap 119866(1205781015840 1205781015840) in 119861119904119902


1199011015840 sub (119861

119904119902

119901 )lowast





1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Definition 14 Let 120582 = 120582119896



119901 = 120582 120582119887119904119902

119901lt infin with

120582119887119904119902

119901

=

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(58)


119901 is defined by 119891119904119902

119901 = 120582

120582119891119904119902

119901lt infin with

120582119891119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge0

120575minus119896119904119902 [

[

sum

120572isinZ119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(59)


119901 hArr ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901

and 119891 isin 119865119904119902

119901 hArr ⟨120593119896

120572 119891⟩ isin 119891

119904119902

119901 due to (57) (45) and (46)


119901 as well as119865

119904119902

119901



119901 for 1 lt 119901 lt infin



119901

and 119865119904119902

119901 Let 119891 isin 119861119904119902


119891119872= 119875

2

1(119891) +

119872

sum119896=1

1198762

119896(119891) (60)


isin

119866(1205781015840 1205781015840) and 119891119872119861119904119902

119901le 119891

119861119904119902


that

lim119872rarrinfin

1003817100381710038171003817100381710038171003817100381710038171003817

sum119896gt119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

= 0 (61)


1003816100381610038161003816119863119897119876

119896(119891)

1003816100381610038161003816 le 119862120575|119896minus119897|120578

1015840

119872(119876119896(119891)) (119909) (62)




1003817100381710038171003817100381710038171003817100381710038171003817

sum119896ge119872

1198762

119896(119891)

1003817100381710038171003817100381710038171003817100381710038171003817119861119904119902119901

le 119862sum119896gt119872

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(63)


rarr 119891 in 119861119904119902



119901



11199011015840 = 1119902 + 11199021015840 = 1 Then (119861119904119902

119901 )lowast= 119861

minus1199041199021015840


given119892 isin 119861minus1199041199021015840

1199011015840 then m


on 119861119904119902


10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

(64)


119901 with norm atmost 119862119892

119861minus1199041199021015840

1199011015840



1015840

1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (65)



119892with 119892

119861minus1199041199021015840

1199011015840

le 119862m


119901 and 119861minus1199041199021015840

1199011015840

replaced by 119865119904119902

119901 and 119865minus1199041199021015840


Proof of Theorem 17


1199011015840 and 119891 isin 119861

119904119902


119892 = sum119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ 120593

119896

120572 (66)


1199011015840 and hence ⟨120593119896

120572 119892⟩ isin 119887

minus1199041199021015840

1199011015840 Also

119891 = sum119896ge0

sum

120572isinZ119896

120593119896

120572⟨120593

119896

120572 119891⟩ (67)

is an element of 119861119904119902

119901 cap 119866(1205781015840 1205781015840) and hence ⟨120593119896

120572 119891⟩ isin 119887

119904119902

119901 Write now

m119892(119891) = sum

119896ge0

sum

120572isinZ119896

⟨120593119896

120572 119892⟩ ⟨120593

119896

120572 119891⟩ (68)


10038161003816100381610038161003816le10038171003817100381710038171003817⟨120593

119896

120572 119892⟩

10038171003817100381710038171003817119887minus1199041199021015840

1199011015840

10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901le10038171003817100381710038171198921003817100381710038171003817119861minus119904119902

1015840

1199011015840

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(69)


119901 cap119866(1205781015840 1205781015840)Clearly m



119901 cap 119866(1205781015840 1205781015840) in 119861119904119902


1199011015840 sub (119861

119904119902

119901 )lowast





1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ 119891 isin 119861

119904119902

119901cap 119866 (120578

1015840 120578

1015840) (70)

Note that1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (71)


120572 119891⟩ 119891 isin 119861

119904119902

119901 by

m (⟨120593119896

120572 119891⟩) = m (119891) (72)


120572 119891⟩)

10038161003816100381610038161003816=1003816100381610038161003816m (119891)

1003816100381610038161003816 le m10038171003817100381710038171003817⟨120593

119896

120572 119891⟩

10038171003817100381710038171003817119887119904119902119901 (73)


119904119902

119901 )lowast= 119887

minus1199041199021015840


119892119896 isin 119887

minus1199041199021015840

1199011015840 such that m

119887minus1199041199021015840

1199011015840

le 119862m and

m (119886119896) = sum

119896ge0

119886119896119892119896 119886

119896 isin 119887

119904119902

119901 (74)


119901 cap 119866(1205781015840 1205781015840) we have

m (119891) = m (⟨120593119896

120572 119891⟩) = sum

119896ge0

⟨120593119896

120572 119891⟩ 119892

119896= ⟨sum

119896ge0

120593119896

120572119892119896 119891⟩

(75)

Let 119892 = sum119896ge0

120593119896


120572 we have

10038171003817100381710038171198921003817100381710038171003817119887minus119904119902

1015840

1199011015840

=

sum119896ge0

120575minus1198961199041199021015840

times [

[

sum120572

(120583 (119876119896

120572)1119901minus12

times

1003816100381610038161003816100381610038161003816100381610038161003816

⟨120593119896

120572sum119897ge0

119892119897120593119897

120572⟩

1003816100381610038161003816100381610038161003816100381610038161003816

)

1199011015840

]

]

11990210158401199011015840

11199021015840

le 119862

sum119896ge0

120575minus1198961199041199021015840

times [sum120572

(120583 (119876119896

120572)1119901minus12

times1003816100381610038161003816119892119896

1003816100381610038161003816 )1199011015840

]

11990210158401199011015840

11199021015840

le 119862 m

(76)

and |m(119891)| = |m119892(119891)| = |⟨119892 119891⟩| le m119891

119861119904119902

119901 Note that

119861119904119902

119901 cap 119866(1205781015840 1205781015840) is dense in 119861119904119902



119904119902

119901




and |119904| lt 120578 Then

1198611199041199020

119901sub 119861

1199041199021

119901 119865

1199041199020

119901sub 119865

1199041199021

119901 (77)

(ii) Let 1 lt 1199020le 119902


119861119904+1205791199020

119901sub 119861

1199041199021

119901 119865

119904+1205791199020

119901sub 119865

1199041199021

119901 (78)


119861119904min(119901119902)119901

sub 119865119904119902

119901sub 119861

119904max(119901119902)119901

(79)









= 119891119866(120573120574)



1003816100381610038161003816m (119891)1003816100381610038161003816 le 119862

100381710038171003817100381711989110038171003817100381710038171198660(120573120574) (80)



119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (81)

with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

= sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

(82)


119901(119883) is defined by

119904119902

119901(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (83)


with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










with

10038171003817100381710038171198911003817100381710038171003817119904119902119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904 1003816100381610038161003816119876119896

(119891)1003816100381610038161003816)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(84)


119896for 119896 isin Z



sum119896isinZ

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

sim sum119897isinZ

(120575minus119897119904 1003817100381710038171003817119863119897

(119891)1003817100381710038171003817119901)

119902

1119902

(85)


10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

(120575minus119896119904|119876

119896(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

sim

10038171003817100381710038171003817100381710038171003817100381710038171003817

sum119897isinZ

(120575minus119897119904|119863

119897(119891)|)

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817119901

(86)




119861119904119902

119901=

119904119902

119901cap 119871

119901 (87)


119865119904119902

119901=

119904119902

119901cap 119871

119901 (88)



119901 sub 119904119902

119901cap 119871

119901 Let 119891 isin 119861119904119902


119901 and

10038171003817100381710038171198911003817100381710038171003817119871119901 le

10038171003817100381710038171198751(119891)1003817100381710038171003817119901 +

infin

sum119896=1

1003817100381710038171003817119876119896(119891)

1003817100381710038171003817119901

le10038171003817100381710038171198751(119891)

1003817100381710038171003817119901 + 119862

infin

sum119896=1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(89)


1198961198751(26) we also have 119891 isin

119904119902

119901and

10038171003817100381710038171198911003817100381710038171003817119904119902119901

le sum119896lt1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

+ sum119896ge1

(120575minus119896119904 1003817100381710038171003817119876119896

(119891)1003817100381710038171003817119901)

119902

1119902

le 11986210038171003817100381710038171198911003817100381710038171003817119901 +

10038171003817100381710038171198911003817100381710038171003817119861119904119902119901

(90)

Thus 119891 isin 119904119902

119901cap 119871119901 and 119891

119871119901 + 119891

119904119902

119901

≲ 119891119861119904119902

119901


119901cap 119871119901 sub 119861

119904119902


119904119902


1 we have

10038161003816100381610038161198751 (119891) (119909)1003816100381610038161003816 le 119862119872(119891) (119909) (91)

and hence 1198751(119891)

119901le 119862119891


119861119904119902

119901le

119862(119891119901+ 119891

119904119902

119901






119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (92)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120595119896

120572 11989110038161003816100381610038161003816)119901

]

]

119902119901

1119902

(93)


119904119902


119904119902

119901(2)(119883) = 119891 isin (119866

0(120573 120574))

1015840

10038171003817100381710038171198911003817100381710038171003817119904119902119901

lt infin (94)

where10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

⟨120595119896

120572 119891⟩

10038161003816100381610038161003816120594119876119896120572)119902

11199021003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

(95)




sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)


10038171003817100381710038171198911003817100381710038171003817119904119902119901

sim10038171003817100381710038171198911003817100381710038171003817119904119902119901(2)

1 lt 119901 lt infin(96)


and 119904119902


120572 119896 isin Z 120572 isin


is defined by 119904119902119901= 120582 120582

119904119902

119901

lt infin with

120582119904119902

119901

=

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)1119901minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816)119901

]

]

119902119901

1119902

(97)


119901is defined by 119904119902

119901= 120582

120582119904119902

119901

lt infin with

120582119904119902

119901

=

10038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sum119896isinZ

120575minus119896119904119902 [

[

sum

120572isinY119896

(120583 (119876119896

120572)minus12 10038161003816100381610038161003816

120582119896

120572

10038161003816100381610038161003816120594119876119896120572)]

]

119902

111990210038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119901

(98)




119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901and 119891 isin

119904119902

119901hArr ⟨120593119896

120572 119891⟩ isin

119904119902

119901






119901for 1 lt 119901 119902 le infin





119901)lowast =

minus1199041199021015840


given119892 isin minus1199041199021015840

1199011015840 then m


on 119904119902


10038161003816100381610038161003816m119892(119891)

10038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119904119902119901

10038171003817100381710038171198921003817100381710038171003817minus1199041199021015840

1199011015840

(99)


119901with norm at

most 119862119892minus1199041199021015840

1199011015840


119901 then there


1199011015840 such that

m119892(119891) = ⟨119891 119892⟩ (100)


119901cap1198660(1205781015840 1205781015840)with |119904| lt 1205781015840 lt 120578


minus1199041199021015840

1199011015840

le 119862m


119901and

minus1199041199021015840

1199011015840


119901and minus119904119902

1015840





and |119904| lt 120578 Then

1199041199020

119901sub

1199041199021

119901

1199041199020

119901sub

1199041199021

119901 (101)


119904min(119901119902)119901

sub 119904119902

119901sub

119904max(119901119902)119901

(102)



Acknowledgments


References





119901 120579rdquo




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article some function spaces via orthonormal...

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