research article on the riesz potential and its ... · research article on the riesz potential and...

Research ArticleOn the Riesz Potential and Its Commutators onGeneralized Orlicz-Morrey Spaces

Vagif S. Guliyev1,2 and Fatih Deringoz1

1 Department of Mathematics, Ahi Evran University, 40200 Kirsehir, Turkey2 Institute of Mathematics and Mechanics, 1141 Baku, Azerbaijan

Correspondence should be addressed to Vagif S. Guliyev; [email protected]

Received 23 October 2013; Revised 24 December 2013; Accepted 25 December 2013; Published 21 January 2014

Academic Editor: Yoshihiro Sawano

Copyright © 2014 V. S. Guliyev and F. Deringoz.This is an open access article distributed under the Creative CommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.

We consider generalized Orlicz-Morrey spaces 𝑀Φ,𝜑

(R𝑛) including their weak versions 𝑊𝑀Φ,𝜑

(R𝑛). In these spaces we prove theboundedness of the Riesz potential from 𝑀

Φ,𝜑1(R𝑛) to 𝑀

Ψ,𝜑2(R𝑛) and from 𝑀

Φ,𝜑1(R𝑛) to 𝑊𝑀

Ψ,𝜑2(R𝑛). As applications of those

results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all thecases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on (𝜑

1, 𝜑2), which do not

assume any assumption on monotonicity of 𝜑1(𝑥, 𝑟), 𝜑

2(𝑥, 𝑟) in 𝑟.

1. Introduction

The theory of boundedness of classical operators of the realanalysis, such as the maximal operator, fractional maximaloperator, Riesz potential, and the singular integral operators,and so forth, has been extensively investigated in variousfunction spaces. Results on weak and strong type inequalitiesfor operators of this kind in Lebesgue spaces are classicaland can be found for example in [1–3]. This boundednessextended to several function spaces which are generalizationsof 𝐿

𝑝-spaces, for example, Orlicz spaces, Morrey spaces,

Lorentz spaces, Herz spaces, and so forth.Orlicz spaces, introduced in [4, 5], are generalizations of

Lebesgue spaces 𝐿𝑝. They are useful tools in harmonic analy-

sis and its applications. For example, the Hardy-Littlewoodmaximal operator is bounded on 𝐿

𝑝for 1 < 𝑝 < ∞,

but not on 𝐿1. Using Orlicz spaces, we can investigate the

boundedness of the maximal operator near 𝑝 = 1 moreprecisely (see [6–8]).

It is well known that the Riesz potential 𝐼𝛼of order 𝛼 (0 <

𝛼 < 𝑛) plays an important role in harmonic analysis, PDE,and potential theory (see [2]). Recall that 𝐼

𝛼is defined by

𝐼𝛼𝑓 (𝑥) = ∫

R𝑛

𝑓 (𝑦)

𝑥 − 𝑦

𝑛−𝛼𝑑𝑦, 𝑥 ∈ R

𝑛. (1)

The classical result by Hardy-Littlewood-Sobolev statesthat, if 1 < 𝑝 < 𝑞 < ∞, then the operator 𝐼

𝛼is bounded from

𝐿𝑝(R𝑛) to 𝐿

𝑞(R𝑛) if and only if 𝛼 = 𝑛((1/𝑝) − (1/𝑞)) and, for

𝑝 = 1 < 𝑞 < ∞, the operator 𝐼𝛼is bounded from 𝐿

1(R𝑛) to

𝑊𝐿𝑞(R𝑛) if and only if 𝛼 = 𝑛(1 − (1/𝑞)). For boundedness of

𝐼𝛼on Morrey spaces 𝑀

𝑝,𝜆(R𝑛), see Peetre (Spanne) [9] and

Adams [10].The boundedness of 𝐼

𝛼from Orlicz space 𝐿

Φ(R𝑛) to

𝐿Ψ(R𝑛) was studied by O’Neil [11] and Torchinsky [12]

under some restrictions involving the growths and certainmonotonicity properties of Φ and Ψ. Moreover Cianchi [6]gave a necessary and sufficient condition for the boundednessof 𝐼

𝛼from𝐿

Φ(R𝑛) to𝐿

Ψ(R𝑛) and from𝐿

Φ(R𝑛) toweakOrlicz

space 𝑊𝐿Ψ(R𝑛), which contain results above.

In [13] the authors study the boundedness of the maximaloperator𝑀 and the Calderón-Zygmund operator𝑇 from onegeneralized Orlicz-Morrey space 𝑀

Φ,𝜑1

(R𝑛) to 𝑀Φ,𝜑2

(R𝑛)

and from𝑀Φ,𝜑1

(R𝑛) to the weak space 𝑊𝑀Φ,𝜑2

(R𝑛).Our definition of Orlicz-Morrey spaces (see Section 3) is

different from that of Sawano et al. [14] and Nakai [15, 16].The main purpose of this paper is to find sufficient

conditions on general Young functions Φ,Ψ and functions𝜑1, 𝜑

2which ensure the boundedness of the Riesz potential

𝐼𝛼from one generalized Orlicz-Morrey spaces 𝑀

Φ,𝜑1

(R𝑛) to

Hindawi Publishing CorporationJournal of Function SpacesVolume 2014, Article ID 617414, 11 pageshttp://dx.doi.org/10.1155/2014/617414

2 Journal of Function Spaces

another 𝑀Ψ,𝜑2

(R𝑛) and from 𝑀Φ,𝜑1

(R𝑛) to weak generalizedOrlicz-Morrey spaces 𝑊𝑀

Ψ,𝜑2

(R𝑛) and the boundedness ofthe commutator of the Riesz potential [𝑏, 𝐼

𝛼] from𝑀

Φ,𝜑1

(R𝑛)

to𝑀Ψ,𝜑2

(R𝑛).In the next section we recall the definitions of Orlicz and

Morrey spaces and give the definition of Orlicz-Morrey andgeneralized Orlicz-Morrey spaces in Section 3. In Section 4and Section 5 the results on the boundedness of the Rieszpotential and its commutator on generalized Orlicz-Morreyspaces are obtained.

By 𝐴 ≲ 𝐵 we mean that 𝐴 ≤ 𝐶𝐵 with some positiveconstant 𝐶 independent of appropriate quantities. If 𝐴 ≲ 𝐵and 𝐵 ≲ 𝐴, we write 𝐴 ≈ 𝐵 and say that 𝐴 and 𝐵 areequivalent.

2. Some Preliminaries on Orlicz andMorrey Spaces

In the study of local properties of solutions of partial dif-ferential equations, together with weighted Lebesgue spaces,Morrey spaces 𝑀

𝑝,𝜆(R𝑛) play an important role; see [17].

Introduced by Morrey Jr. [18] in 1938, they are defined by thenorm

𝑓𝑀𝑝,𝜆

:= sup𝑥,𝑟>0

𝑟−𝜆/𝑝𝑓

𝐿𝑝(𝐵(𝑥,𝑟))

, (2)

where 0 ≤ 𝜆 ≤ 𝑛, 1 ≤ 𝑝 < ∞. Here and everywhere in thesequel 𝐵(𝑥, 𝑟) stands for the ball in R𝑛 of radius 𝑟 centeredat 𝑥. Let |𝐵(𝑥, 𝑟)| be the Lebesgue measure of the ball 𝐵(𝑥, 𝑟)and |𝐵(𝑥, 𝑟)| = V

𝑛𝑟𝑛, where V

𝑛is the volume of the unit ball in

R𝑛.Note that 𝑀

𝑝,0= 𝐿

𝑝(R𝑛) and 𝑀

𝑝,𝑛= 𝐿

∞(R𝑛). If 𝜆 < 0

or 𝜆 > 𝑛, then 𝑀𝑝,𝜆

= Θ, where Θ is the set of all functionsequivalent to 0 on R𝑛.

We also denote by 𝑊𝑀𝑝,𝜆

≡ 𝑊𝑀𝑝,𝜆

(R𝑛) the weakMorrey space of all functions 𝑓 ∈ 𝑊𝐿loc

𝑝(R𝑛) for which

𝑓𝑊𝑀

𝑝,𝜆

= sup𝑥∈R𝑛,𝑟>0

𝑟−𝜆/𝑝𝑓

𝑊𝐿𝑝(𝐵(𝑥,𝑟))

< ∞, (3)

where𝑊𝐿𝑝(𝐵(𝑥, 𝑟)) denotes the weak 𝐿

𝑝-space.

We refer in particular to [19] for the classical Morreyspaces.

We recall the definition of Young functions.

Definition 1. A function Φ : [0, +∞) → [0,∞] is calleda Young function if Φ is convex and left-continuous,lim

𝑟→+0Φ(𝑟) = Φ(0) = 0, and lim

𝑟→+∞Φ(𝑟) = ∞.

From the convexity and Φ(0) = 0 it follows that anyYoung function is increasing. If there exists 𝑠 ∈ (0, +∞) suchthatΦ(𝑠) = +∞, thenΦ(𝑟) = +∞ for 𝑟 ≥ 𝑠.

LetY be the set of all Young functionsΦ such that

0 < Φ (𝑟) < +∞ for 0 < 𝑟 < +∞. (4)

If Φ ∈ Y, then Φ is absolutely continuous on every closedinterval in [0, +∞) and bijective from [0, +∞) to itself.

Definition 2 (Orlicz space). For a Young functionΦ, the set

𝐿Φ(R

𝑛) = {𝑓 ∈ 𝐿

loc1

(R𝑛) : ∫

R𝑛Φ(𝑘

𝑓 (𝑥)) 𝑑𝑥

< +∞ for some 𝑘 > 0}(5)

is called Orlicz space. If Φ(𝑟) = 𝑟𝑝, 1 ≤ 𝑝 < ∞, then𝐿Φ(R𝑛) = 𝐿

𝑝(R𝑛). If Φ(𝑟) = 0 (0 ≤ 𝑟 ≤ 1) and Φ(𝑟) =

∞ (𝑟 > 1), then 𝐿Φ(R𝑛) = 𝐿

∞(R𝑛). The space 𝐿loc

Φ(R𝑛)

endowed with the natural topology is defined as the set of allfunctions 𝑓 such that 𝑓𝜒

𝐵∈ 𝐿

Φ(R𝑛) for all balls 𝐵 ⊂ R𝑛. We

refer to the books [20–22] for the theory of Orlicz spaces.𝐿Φ(R𝑛) is a Banach space with respect to the norm

𝑓𝐿Φ

= inf {𝜆 > 0 : ∫R𝑛

Φ(

𝑓 (𝑥)

𝜆) 𝑑𝑥 ≤ 1} . (6)

We note that

∫R𝑛

Φ(

𝑓 (𝑥)

𝑓𝐿Φ

)𝑑𝑥 ≤ 1. (7)

For a measurable set Ω ⊂ R𝑛, a measurable function 𝑓, and𝑡 > 0, let

𝑚(Ω, 𝑓, 𝑡) ={𝑥 ∈ Ω :

𝑓 (𝑥) > 𝑡}

. (8)

In the case Ω = R𝑛, we shortly denote it by 𝑚(𝑓, 𝑡).

Definition 3. The weak Orlicz space

𝑊𝐿Φ(R

𝑛) := {𝑓 ∈ 𝐿

1

loc (R𝑛) :

𝑓𝑊𝐿Φ

< +∞} (9)

is defined by the norm

𝑓𝑊𝐿Φ

= inf {𝜆 > 0 : sup𝑡>0

Φ (𝑡)𝑚(𝑓

𝜆, 𝑡) ≤ 1} . (10)

For a Young functionΦ and 0 ≤ 𝑠 ≤ +∞, let

Φ−1

(𝑠) = inf {𝑟 ≥ 0 : Φ (𝑟) > 𝑠} (inf 0 = +∞) . (11)

If Φ ∈ Y, then Φ−1 is the usual inverse function of Φ. Wenote that

Φ(Φ−1

(𝑟)) ≤ 𝑟 ≤ Φ−1

(Φ (𝑟)) for 0 ≤ 𝑟 < +∞. (12)

A Young function Φ is said to satisfy the Δ2-condition,

denoted by Φ ∈ Δ2, if

Φ (2𝑟) ≤ 𝑘Φ (𝑟) for 𝑟 > 0 (13)

for some 𝑘 > 1. If Φ ∈ Δ2, then Φ ∈ Y. A Young function Φ

is said to satisfy the ∇2-condition, denoted also by Φ ∈ ∇

2, if

Φ (𝑟) ≤1

2𝑘Φ (𝑘𝑟) , 𝑟 ≥ 0, (14)

for some 𝑘 > 1. The function Φ(𝑟) = 𝑟 satisfies the Δ2-

condition but does not satisfy the∇2-condition. If 1 < 𝑝 < ∞,

Journal of Function Spaces 3

then Φ(𝑟) = 𝑟𝑝 satisfies both the conditions. The functionΦ(𝑟) = 𝑒

𝑟−𝑟−1 satisfies the∇

2-condition but does not satisfy

the Δ2-condition.

For a Young function Φ, the complementary functionΦ̃(𝑟) is defined by

Φ̃ (𝑟) = {

sup {𝑟𝑠 − Φ (𝑠) : 𝑠 ∈ [0,∞)} , 𝑟 ∈ [0,∞) ,+∞, 𝑟 = +∞.

(15)

The complementary function Φ̃ is also a Young function and̃̃Φ = Φ. If Φ(𝑟) = 𝑟, then Φ̃(𝑟) = 0 for 0 ≤ 𝑟 ≤ 1 and Φ̃(𝑟) =+∞ for 𝑟 > 1. If 1 < 𝑝 < ∞, 1/𝑝 + 1/𝑝 = 1, and Φ(𝑟) =𝑟𝑝/𝑝, then Φ̃(𝑟) = 𝑟𝑝

/𝑝. If Φ(𝑟) = 𝑒𝑟 − 𝑟 − 1, then Φ̃(𝑟) =

(1 + 𝑟) log(1 + 𝑟) − 𝑟. Note that Φ ∈ ∇2if and only if Φ̃ ∈ Δ

2.

It is known that

𝑟 ≤ Φ−1

(𝑟) Φ̃−1

(𝑟) ≤ 2𝑟 for 𝑟 ≥ 0. (16)

Note that Young functions satisfy the properties

Φ (𝛼𝑡) ≤ 𝛼Φ (𝑡) , if 0 ≤ 𝛼 ≤ 1,

Φ (𝛼𝑡) ≥ 𝛼Φ (𝑡) , if 𝛼 > 1,

Φ−1

(𝛼𝑡) ≥ 𝛼Φ−1

(𝑡) , if 0 ≤ 𝛼 ≤ 1,

Φ−1

(𝛼𝑡) ≤ 𝛼Φ−1

(𝑡) , if 𝛼 > 1.

(17)

The following analogue of the Hölder inequality isknown; see [23].

Theorem 4 (see [23]). For a Young functionΦ and its comple-mentary function Φ̃, the following inequality is valid:

𝑓𝑔𝐿1(R𝑛)

≤ 2𝑓

𝐿Φ

𝑔𝐿Φ̃

. (18)

The following lemma is valid.

Lemma 5 (see [1, 24]). Let Φ be a Young function and 𝐵 a setin R𝑛 with finite Lebesgue measure. Then

𝜒𝐵𝑊𝐿Φ(R𝑛)

=𝜒𝐵

𝐿Φ(R𝑛)

=1

Φ−1 (|𝐵|−1)

. (19)

In the next sections where we prove our main estimates,we use the following lemma, which follows fromTheorem 4,Lemma 5, and (16).

Lemma 6. For a Young function Φ and 𝐵 = 𝐵(𝑥, 𝑟), thefollowing inequality is valid:

𝑓𝐿1(𝐵)

≤ 2 |𝐵|Φ−1

(|𝐵|−1)𝑓

𝐿Φ(𝐵)

. (20)

3. Orlicz-Morrey and GeneralizedOrlicz-Morrey Spaces

Definition 7 (Orlicz-Morrey space). For a Young function Φand 0 ≤ 𝜆 ≤ 𝑛, one denotes by 𝑀

Φ,𝜆(R𝑛) the Orlicz-Morrey

space, the space of all functions 𝑓 ∈ 𝐿locΦ(R𝑛) with finite

quasinorm𝑓

𝑀Φ,𝜆


Φ−1

(𝑟−𝜆

)𝑓

𝐿Φ(𝐵(𝑥,𝑟))

. (21)

Note that 𝑀Φ,0

= 𝐿Φ(R𝑛) and if Φ(𝑟) = 𝑟𝑝, 1 ≤ 𝑝 < ∞,

then𝑀Φ,𝜆

(R𝑛) = 𝑀𝑝,𝜆

(R𝑛).We also denote by 𝑊𝑀

Φ,𝜆(R𝑛) the weak Orlicz-Morrey

space of all functions 𝑓 ∈ 𝑊𝐿locΦ(R𝑛) for which

𝑓𝑊𝑀

Φ,𝜆


Φ−1

(𝑟−𝜆

)𝑓

𝑊𝐿Φ(𝐵(𝑥,𝑟))

< ∞, (22)

where 𝑊𝐿Φ(𝐵(𝑥, 𝑟)) denotes the weak 𝐿

Φ-space of measur-

able functions 𝑓 for which𝑓


≡𝑓𝜒𝐵(𝑥,𝑟)

𝑊𝐿Φ(R𝑛)

. (23)

Definition 8 (generalized Orlicz-Morrey space). Let 𝜑(𝑥, 𝑟)be a positive measurable function on R𝑛 × (0,∞) and Φ anyYoung function. One denotes by 𝑀

Φ,𝜑(R𝑛) the generalized

Orlicz-Morrey space, the space of all functions 𝑓 ∈ 𝐿locΦ(R𝑛)

with finite quasinorm𝑓

𝑀Φ,𝜑


𝜑(𝑥, 𝑟)−1Φ−1

(|𝐵(𝑥, 𝑟)|−1)𝑓


.

(24)

Also by𝑊𝑀Φ,𝜑

(R𝑛) one denotes theweak generalizedOrlicz-Morrey space of all functions 𝑓 ∈ 𝑊𝐿loc

Φ(R𝑛) for which

𝑓𝑊𝑀

𝑝,𝜑


𝜑(𝑥, 𝑟)−1Φ−1

(|𝐵 (𝑥, 𝑟)|−1)𝑓


< ∞.

(25)

According to this definition, we recover the spaces 𝑀Φ,𝜆

and𝑊𝑀Φ,𝜆

under the choice 𝜑(𝑥, 𝑟) = (Φ−1(𝑟−𝑛)/Φ−1(𝑟−𝜆)):

𝑀Φ,𝜆

= 𝑀Φ,𝜑

𝜑(𝑥,𝑟)=(Φ−1(𝑟−𝑛)/Φ−1(𝑟−𝜆)),

𝑊𝑀Φ,𝜆

= 𝑊𝑀Φ,𝜑

(Φ−1(𝑟−𝑛)/Φ−1(𝑟−𝜆)).

(26)

According to this definition, we recover the generalizedMorrey spaces 𝑀

𝑝,𝜑and weak generalized Morrey spaces

𝑊𝑀𝑝,𝜑

under the choice Φ(𝑟) = 𝑟𝑝, 1 ≤ 𝑝 < ∞:

𝑀𝑝,𝜑

= 𝑀Φ,𝜑

Φ(𝑟)=𝑟𝑝,

𝑊𝑀𝑝,𝜑

= 𝑊𝑀Φ,𝜑

Φ(𝑟)=𝑟𝑝.

(27)

Remark 9. There are different kinds of Orlicz-Morrey spacesin the literature.Wewant tomake some comment about thesespaces.

Let 𝜑 : (0,∞) → (0,∞) be a function andΦ : (0,∞) →(0,∞) a Young function.


(1) For a cube 𝑄, define (𝜑, Φ)-average over 𝑄 by𝑓

(𝜑,Φ);𝑄

:= inf {𝜆 > 0 :𝜑 (|𝑄|)

|𝑄|∫𝑄

Φ(

𝑓 (𝑥)

𝜆) 𝑑𝑥 ≤ 1}

(28)

and define its Φ-average over 𝑄 by

𝑓Φ;𝑄

:= inf {𝜆 > 0 : 1|𝑄|

∫𝑄

Φ(

𝑓 (𝑥)

𝜆) 𝑑𝑥 ≤ 1} .

(29)

(2) Define𝑓

L𝜑,Φ

:= sup𝑄∈Q

𝑓(𝜑,Φ);𝑄

. (30)

The function space L𝜑,Φ

is defined to be the Orlicz-Morrey space of the first kind as the set of allmeasurable functions 𝑓 for which the norm ‖𝑓‖L

𝜑,Φ

is finite.(3) Define

𝑓M𝜑,Φ

:= sup𝑄∈Q

𝜑 (|𝑄|)𝑓

Φ;𝑄. (31)

The function space M𝜑,Φ

is defined to be the Orlicz-Morrey space of the second kind as the set of allmeasurable functions 𝑓 for which the norm ‖𝑓‖M

𝜑,Φ

is finite.

According to our best knowledge, it seems that L𝜑,Φ

ismore investigated thanM

𝜑,Φ. The spaceL

𝜑,Φis investigated

in [15, 16, 25–34] and the spaceM𝜑,Φ

is investigated in [14, 35–37].

4. Boundedness of the Riesz Potential inGeneralized Orlicz-Morrey Spaces

In this section sufficient conditions on the pairs (𝜑1, 𝜑2)

and (Φ,Ψ) for the boundedness of 𝐼𝛼from one generalized

Orlicz-Morrey spaces 𝑀Φ,𝜑1

(R𝑛) to another 𝑀Ψ,𝜑2

(R𝑛) andfrom 𝑀

Φ,𝜑1

(R𝑛) to the weak space 𝑊𝑀Ψ,𝜑2

(R𝑛) have beenobtained.

Necessary and sufficient conditions on (Φ,Ψ) for theboundedness of 𝐼

𝛼from 𝐿

Φ(R𝑛) to 𝐿

Ψ(R𝑛) and 𝐿

Φ(R𝑛)

to 𝑊𝐿Ψ(R𝑛) have been obtained in [6, Theorem 2]. In the

statement of the theorem,Ψ𝑝is the Young function associated

with the Young function Ψ and 𝑝 ∈ (1,∞] whose Youngconjugate is given by

Ψ̃𝑝 (𝑠) = ∫

𝑠

0

𝑟𝑝−1(B

−1

𝑝(𝑟𝑝

))

𝑝

𝑑𝑟, (32)

where

B𝑝 (𝑠) = ∫

𝑠

0

Ψ (𝑡)

𝑡1+𝑝𝑑𝑡, (33)

and 𝑝, the Holder conjugate of 𝑝, equals either 𝑝/(𝑝 − 1) or1, according to whether 𝑝 < ∞ or 𝑝 = ∞ andΦ

𝑝denotes the

Young function defined by

Φ𝑝 (𝑠) = ∫

𝑠

0

𝑟𝑝−1(A

−1

𝑝(𝑟𝑝

))

𝑝

𝑑𝑟, (34)

where

A𝑝 (𝑠) = ∫

𝑠

0

Φ̃ (𝑡)

𝑡1+𝑝𝑑𝑡. (35)

Recall that, if Φ and Ψ are functions from [0,∞) into[0,∞], then Ψ is said to dominate Φ globally if a positiveconstant 𝑐 exists such thatΦ(𝑠) ≤ Ψ(𝑐𝑠) for all 𝑠 ≥ 0.

Theorem 10 (see [6]). Let 0 < 𝛼 < 𝑛. Let Φ and Ψ Youngfunctions and letΦ

𝑛/𝛼andΨ

𝑛/𝛼be the Young functions defined

as in (34) and (32), respectively. Then(i) the Riesz potential 𝐼

𝛼is bounded from 𝐿

Φ(R𝑛) to

𝑊𝐿Ψ(R𝑛) if and only if

∫

1

0

Φ̃ (𝑡)

𝑡1+𝑛/(𝑛−𝛼)𝑑𝑡 < ∞ 𝑎𝑛𝑑 Φ

𝑛/𝛼𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑠 Ψ 𝑔𝑙𝑜𝑏𝑎𝑙𝑙𝑦.

(36)

(ii) The Riesz potential 𝐼𝛼is bounded from 𝐿

Φ(R𝑛) to

𝐿Ψ(R𝑛) if and only if

∫

1

0

Φ̃ (𝑡)

𝑡1+𝑛/(𝑛−𝛼)𝑑𝑡 < ∞, ∫

1

0

Ψ (𝑡)

𝑡1+𝑛/(𝑛−𝛼)𝑑𝑡 < ∞,

Φ𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑠 Ψ𝑛/𝛼

𝑔𝑙𝑜𝑏𝑎𝑙𝑙𝑦, 𝑎𝑛𝑑Φ𝑛/𝛼

𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑠 Ψ 𝑔𝑙𝑜𝑏𝑎𝑙𝑙𝑦.

(37)

We will use the following statement on the boundednessof the weighted Hardy operator

𝐻𝑤𝑔 (𝑡) := ∫

∞

𝑡

𝑔 (𝑠) 𝑤 (𝑠) 𝑑𝑠, 0 < 𝑡 < ∞, (38)

where 𝑤 is a weight.The following theorem was proved in [38] (see, also [13]).

Theorem 11. Let V1, V2, and 𝑤 be weights on (0,∞) and V

1(𝑡)

bounded outside a neighborhood of the origin. The inequality

ess sup𝑡>0

V2 (𝑡)𝐻𝑤𝑔 (𝑡) ≤ 𝐶 ess sup

𝑡>0

V1 (𝑡) 𝑔 (𝑡) (39)

holds for some 𝐶 > 0 for all nonnegative and nondecreasing 𝑔on (0,∞) if and only if

𝐵 := sup𝑡>0

V2 (𝑡) ∫

∞

𝑡

𝑤 (𝑠) 𝑑𝑠

ess sup𝑠


Proof. If ∫10Φ̃(𝑡)/𝑡

1+𝑝

𝑑𝑡 < ∞, then

1 ≤ 2𝑟−1/𝑝

Φ̃−1

(𝑟)Φ−1

𝑝(𝑟) , for 𝑟 > 0. (42)

For the proof of this claim see [39, page 50].If Φ

𝑝dominates Ψ globally, then a positive constant 𝐶

exists such that

Φ−1

𝑝(𝑟) ≤ 𝐶Ψ

−1(𝑟) , for 𝑟 > 0. (43)

Indeed,

Ψ−1

(𝑟) = inf {𝑡 ≥ 0 : Ψ (𝑡) > 𝑟}

≥ inf {𝑡 ≥ 0 : Φ𝑝 (𝐶𝑡) > 𝑟}

=1

𝐶inf {𝐶𝑡 ≥ 0 : Φ

𝑝 (𝐶𝑡) > 𝑟}

=1

𝐶Φ−1

𝑝(𝑟) .

(44)

Thus, (41) follows from (42), (43), and (16).

The following lemma is valid.

Lemma 13. Let 0 < 𝛼 < 𝑛, Φ and Ψ Young functions, 𝑓 ∈𝐿locΦ(R𝑛), and𝐵 = 𝐵(𝑥

0, 𝑟). If (Φ,Ψ) satisfy the conditions (37),

then

𝐼𝛼𝑓𝐿Ψ(𝐵)

≲1

Ψ−1 (𝑟−𝑛

)∫

∞

2𝑟

𝑓𝐿Φ(𝐵(𝑥0,𝑡))

Ψ−1

(𝑡−𝑛

)𝑑𝑡

𝑡

(45)

and if (Φ,Ψ) satisfy the conditions (36), then

𝐼𝛼𝑓𝑊𝐿Ψ(𝐵)

≲1

Ψ−1 (𝑟−𝑛

)∫

∞

2𝑟


Ψ−1

(𝑡−𝑛

)𝑑𝑡

𝑡.

(46)

Proof. Suppose that the conditions (37) are satisfied. Forarbitrary 𝑥

0∈ R𝑛, set 𝐵 = 𝐵(𝑥

0, 𝑟) for the ball centered at

𝑥0and of radius 𝑟, 2𝐵 = 𝐵(𝑥

0, 2𝑟). We represent 𝑓 as

𝑓 = 𝑓1+ 𝑓

2, 𝑓

1(𝑦) = 𝑓 (𝑦) 𝜒

2𝐵(𝑦) ,

𝑓2(𝑦) = 𝑓 (𝑦) 𝜒∁

(2𝐵)(𝑦) , 𝑟 > 0,

(47)

and have𝐼𝛼𝑓

𝐿Ψ(𝐵)

≤𝐼𝛼𝑓1

𝐿Ψ(𝐵)

+𝐼𝛼𝑓2

𝐿Ψ(𝐵)

. (48)

Since 𝑓1

∈ 𝐿Φ(R𝑛), 𝐼

𝛼𝑓1

∈ 𝐿Ψ(R𝑛), and from the

boundedness of 𝐼𝛼from 𝐿

Φ(R𝑛) to 𝐿

Ψ(R𝑛) (seeTheorem 10)

it follows that𝐼𝛼𝑓1

𝐿Ψ(𝐵)

≤𝐼𝛼𝑓1

𝐿Ψ(R𝑛)

≤ 𝐶𝑓1

𝐿Φ(R𝑛)

= 𝐶𝑓

𝐿Φ(2𝐵)

,

(49)

where constant 𝐶 > 0 is independent of 𝑓.

It is clear that 𝑥 ∈ 𝐵, 𝑦 ∈ ∁(2𝐵) implies (1/2)|𝑥0− 𝑦| ≤

|𝑥 − 𝑦| ≤ (3/2)|𝑥0− 𝑦|. We get

𝐼𝛼𝑓2 (𝑥) ≤ 2

𝑛−𝛼∫∁(2𝐵)

𝑓 (𝑦)

𝑥0 − 𝑦

𝑛−𝛼𝑑𝑦. (50)

By Fubini’s theorem we have

∫∁(2𝐵)

𝑓 (𝑦)

𝑥0 − 𝑦

𝑛−𝛼𝑑𝑦 ≈ ∫

∁(2𝐵)

𝑓 (𝑦) ∫

∞

|𝑥0−𝑦|

𝑑𝑡

𝑡𝑛+1−𝛼𝑑𝑦

≈ ∫

∞

2𝑟

∫2𝑟≤|𝑥

0−𝑦|


Suppose that the conditions (37) are satisfied. Fromthe boundedness of 𝐼

𝛼from 𝐿

Φ(R𝑛) to 𝑊𝐿

Ψ(R𝑛) (see

Theorem 10) and (56) it follows that𝐼𝛼𝑓1

𝑊𝐿Ψ(𝐵)

≤𝐼𝛼𝑓1

𝑊𝐿Ψ(R𝑛)

≲𝑓1

𝐿Φ(R𝑛)

=𝑓

𝐿Φ(2𝐵)

≲1

Ψ−1 (𝑟−𝑛

)∫

∞

2𝑟


Ψ−1

(𝑡−𝑛

)𝑑𝑡

𝑡.

(58)

Then by (53) and (58) we get the inequality (46).

Theorem 14. Let 0 < 𝛼 < 𝑛 and the functions (𝜑1, 𝜑2) and

(Φ,Ψ) satisfy the condition

∫

∞

𝑟

ess inf𝑡0

𝜑1(𝑥, 𝑟)

−1Φ−1

(𝑟−𝑛

)𝑓


=𝑓

𝑀Φ,𝜑1

,

(60)

if (37) is satisfied and𝐼𝛼𝑓

𝑊𝑀Ψ,𝜑2

≲ sup𝑥∈R𝑛,𝑟>0

𝜑2(𝑥, 𝑟)

−1∫

∞

𝑟

Ψ−1

(𝑡−𝑛

)𝑓

𝐿Φ(𝐵(𝑥,𝑡))

𝑑𝑡

𝑡

≲ sup𝑥∈R𝑛,𝑟>0

𝜑1(𝑥, 𝑟)

−1Φ−1

(𝑟−𝑛

)𝑓


=𝑓

𝑀Φ,𝜑1

,

(61)

if (36) is satisfied.

Remark 15. Recall that, for 0 < 𝛼 < 𝑛,

𝑀𝛼𝑓 (𝑥) ≤ V(𝛼/𝑛)−1

𝑛𝐼𝛼(𝑓

) (𝑥) ;(62)

hence Theorem 14 implies the boundedness of the fractionalmaximal operator 𝑀

𝛼from 𝑀

Φ,𝜑1

(R𝑛) to 𝑀Ψ,𝜑2

(R𝑛) andfrom𝑀

Φ,𝜑1

(R𝑛) to𝑊𝑀Ψ,𝜑2

(R𝑛).If we take Φ(𝑡) = 𝑡𝑝, Ψ(𝑡) = 𝑡𝑞, 1 ≤ 𝑝, 𝑞 < ∞, at

Theorem 14 we get following corollary which was proved in[40] and containing results obtained in [41–45].

Corollary 16. Let 0 < 𝛼 < 𝑛, 1 ≤ 𝑝 < 𝑛/𝛼, 1/𝑞 = (1/𝑝) −(𝛼/𝑛) and (𝜑

1, 𝜑2) satisfy the condition

∫

∞

𝑟

ess inf𝑡 1 the Riesz potential 𝐼

𝛼

is bounded from 𝑀𝑝,𝜆

(R𝑛) to 𝑀𝑞,𝜇

(R𝑛) and for 𝑝 = 1, 𝐼𝛼is

bounded from𝑀1,𝜆

(R𝑛) to𝑊𝑀𝑞,𝜇

(R𝑛).

5. Commutators of Riesz Potential inthe Spaces 𝑀

Φ,𝜑

For a function 𝑏 ∈ 𝐿loc1(R𝑛), let 𝑀

𝑏be the corresponding

multiplication operator defined by 𝑀𝑏𝑓 = 𝑏𝑓 for measur-

able function 𝑓. Let 𝑇 be the classical Calderón-Zygmundsingular integral operator; then the commutator between 𝑇and 𝑀

𝑏is denoted by [𝑏, 𝑇] := 𝑀

𝑏𝑇 − 𝑇𝑀

𝑏. A famous

theorem of Coifman et al. [46] gave a characterization of𝐿𝑝-boundedness of [𝑏, 𝑇] when 𝑇 are the Riesz transforms

𝑅𝑗(𝑗 = 1, . . . , 𝑛). Using this characterization, the authors of

[46] got a decomposition theorem of the real Hardy spaces.The boundedness result was generalized to other contexts andimportant applications to somenonlinear PDEswere given byCoifman et al. [47].

We recall the definition of the space of BMO(R𝑛).

Definition 19. Suppose that 𝑓 ∈ 𝐿loc1(R𝑛); let

𝑓∗


1

|𝐵 (𝑥, 𝑟)|∫𝐵(𝑥,𝑟)

𝑓 (𝑦) − 𝑓𝐵(𝑥,𝑟) 𝑑𝑦 < ∞,

(65)

where

𝑓𝐵(𝑥,𝑟)

=1

|𝐵 (𝑥, 𝑟)|∫𝐵(𝑥,𝑟)

𝑓 (𝑦) 𝑑𝑦. (66)


Define

BMO (R𝑛) = {𝑓 ∈ 𝐿loc1

(R𝑛) :

𝑓∗

< ∞} . (67)

Modulo constants, the space BMO(R𝑛) is a Banach spacewith respect to the norm ‖ ⋅ ‖

∗.

Remark 20. (1) The John-Nirenberg inequality: there areconstants 𝐶

1, 𝐶

2> 0, such that for all 𝑓 ∈ BMO(R𝑛) and

𝛽 > 0

{𝑥 ∈ 𝐵 :𝑓 (𝑥) − 𝑓𝐵

> 𝛽}

≤ 𝐶1 |𝐵| 𝑒

−𝐶2𝛽/‖𝑓‖

∗ , ∀𝐵 ⊂ R𝑛.

(68)

(2) The John-Nirenberg inequality implies that𝑓

∗

≈ sup𝑥∈R𝑛,𝑟>0

(1

|𝐵 (𝑥, 𝑟)|∫𝐵(𝑥,𝑟)

𝑓(𝑦) − 𝑓𝐵(𝑥,𝑟)

𝑝𝑑𝑦)

1/𝑝

(69)

for 1 < 𝑝 < ∞.(3) Let 𝑓 ∈ BMO(R𝑛). Then there is a constant 𝐶 > 0

such that

𝑓𝐵(𝑥,𝑟) − 𝑓𝐵(𝑥,𝑡) ≤ 𝐶

𝑓∗

ln 𝑡𝑟

for 0 < 2𝑟 < 𝑡, (70)

where 𝐶 is independent of 𝑓, 𝑥, 𝑟, and 𝑡.

Definition 21. A Young functionΦ is said to be of upper typep (resp., lower type 𝑝) for some 𝑝 ∈ [0,∞), if there exists apositive constant 𝐶 such that, for all 𝑡 ∈ [1,∞) (resp., 𝑡 ∈[0, 1]) and 𝑠 ∈ [0,∞),

Φ (𝑠𝑡) ≤ 𝐶𝑡𝑝Φ (𝑠) . (71)

Remark 22. We know that if Φ is lower type 𝑝0and upper

type 𝑝1with 1 < 𝑝

0≤ 𝑝

1< ∞, thenΦ ∈ Δ

2∩∇

2. Conversely

if Φ ∈ Δ2∩ ∇

2, then Φ is lower type 𝑝

0and upper type 𝑝

1

with 1 < 𝑝0≤ 𝑝

1< ∞ (see [20]).

Lemma 23 (see [48]). Let Φ be a Young function which islower type 𝑝


1with 1 ≤ 𝑝

0≤ 𝑝

1< ∞. Let

𝐶 be a positive constant.Then there exists a positive constant 𝐶such that for any ball 𝐵 of R𝑛 and 𝜇 ∈ (0,∞)

∫𝐵

Φ(

𝑓 (𝑥)

𝜇) 𝑑𝑥 ≤ 𝐶 (72)

implies that ‖𝑓‖𝐿Φ(𝐵)

≤ 𝐶𝜇.

In the following lemmawe provide a generalization of theproperty (69) from 𝐿

𝑝-norms to Orlicz norms.

Lemma 24. Let 𝑓 ∈ BMO(Rn) and Φ a Young function. LetΦ is lower type 𝑝


1with 1 ≤ 𝑝

0≤ 𝑝

1< ∞;

then𝑓

∗≈ sup𝑥∈R𝑛,𝑟>0

Φ−1

(𝑟−𝑛

)𝑓 (⋅) − 𝑓𝐵(𝑥,𝑟)


. (73)

Proof. By Hölder’s inequality, we have𝑓

∗≲ sup𝑥∈R𝑛,𝑟>0

Φ−1

(𝑟−𝑛

)𝑓(⋅) − 𝑓𝐵(𝑥,𝑟)


. (74)

Now we show that

sup𝑥∈R𝑛,𝑟>0

Φ−1

(𝑟−𝑛

)𝑓(⋅) − 𝑓𝐵(𝑥,𝑟)


≲𝑓

∗. (75)

Without loss of generality, we may assume that ‖𝑓‖∗

= 1;otherwise, we replace𝑓 by𝑓/‖𝑓‖

∗. By the fact thatΦ is lower

type 𝑝0and upper type 𝑝

1and (12) it follows that

∫𝐵(𝑥,𝑟)

Φ(

𝑓 (𝑦) − 𝑓𝐵(𝑥,𝑟) Φ

−1(|𝐵(𝑥, 𝑟)|

−1)

𝑓∗

)𝑑𝑦

= ∫𝐵(𝑥,𝑟)

Φ(𝑓 (𝑦) − 𝑓𝐵(𝑥,𝑟)

Φ−1

(|𝐵 (𝑥, 𝑟)|−1)) 𝑑𝑦

≲1

|𝐵 (𝑥, 𝑟)|

× ∫𝐵(𝑥,𝑟)

[𝑓 (𝑦) − 𝑓𝐵(𝑥,𝑟)

𝑝0

+𝑓 (𝑦) − 𝑓𝐵(𝑥,𝑟)

𝑝1

] 𝑑𝑦

≲ 1.

(76)

By Lemma 23 we get the desired result.

Remark 25. Note that statements of type of Lemma 24 areknown in a more general case of rearrangement invariantspaces and also variable exponent Lebesgue spaces 𝐿𝑝(⋅),see [49, 50], but we gave a short proof of Lemma 24 forcompleteness of presentation.

Definition 26. LetΦ be a Young function. Let

𝑎Φ

:= inf𝑡∈(0,∞)

𝑡Φ(𝑡)

Φ (𝑡), 𝑏

Φ:= sup

𝑡∈(0,∞)

𝑡Φ(𝑡)

Φ (𝑡). (77)

Remark 27. It is known that Φ ∈ Δ2∩ ∇

2if and only if 1 <

𝑎Φ

≤ 𝑏Φ

< ∞ (see [21]).

Remark 28. Remarks 27 and 22 showus that aYoung functionΦ is lower type 𝑝


1with 1 < 𝑝

0≤ 𝑝

1< ∞

if and only if 1 < 𝑎Φ

≤ 𝑏Φ

< ∞.

The characterization of (𝐿𝑝, 𝐿

𝑞) boundedness of the

commutator [𝑏, 𝐼𝛼] between𝑀

𝑏and 𝐼

𝛼was given by Chanillo

[51].

Theorem 29 (see [51]). Let 0 < 𝛼 < 𝑛, 1 < 𝑝 < 𝑛/𝛼, and1/𝑞 = (1/𝑝) − (𝛼/𝑛). Then [𝑏, 𝐼

𝛼] is a bounded operator from

𝐿𝑝(R𝑛) to 𝐿

𝑞(R𝑛) if and only if 𝑏 ∈ BMO(R𝑛).

The (𝐿Φ, 𝐿

Ψ) boundedness of the commutator [𝑏, 𝐼

𝛼]was

given by Fu et al. [52].

Theorem 30 (see [52]). Let 0 < 𝛼 < 𝑛 and 𝑏 ∈ BMO(R𝑛). LetΦ be a Young function andΨ defined, via its inverse, by setting,


for all 𝑡 ∈ (0,∞), Ψ−1(𝑡) := Φ−1(𝑡)𝑡−𝛼/𝑛. If 1 < 𝑎Φ

≤ 𝑏Φ

< ∞,and 1 < 𝑎

Ψ≤ 𝑏

Ψ< ∞ then [𝑏, 𝐼

𝛼] is bounded from 𝐿

Φ(R𝑛) to

𝐿Ψ(R𝑛).

We will use the following statement on the boundednessof the weighted Hardy operator

𝐻∗

𝑤𝑔 (𝑟) := ∫

∞

𝑟

(1 + ln 𝑡𝑟) 𝑔 (𝑡) 𝑤 (𝑡) 𝑑𝑡, 𝑟 ∈ (0,∞) ,

(78)

where 𝑤 is a weight.The following theorem was proved in [53].

Theorem 31. Let V1, V2, and 𝑤 be weights on (0,∞) and V

1(𝑡)

bounded outside a neighborhood of the origin. The inequality

ess sup𝑟>0

V2 (𝑟)𝐻

∗

𝑤𝑔 (𝑟) ≤ 𝐶 ess sup

𝑟>0

V1 (𝑟) 𝑔 (𝑟) (79)

holds for some 𝐶 > 0 for all nonnegative and nondecreasing 𝑔on (0,∞) if and only if

𝐵 := sup𝑟>0

V2 (𝑟) ∫

∞

𝑟

(1 + ln 𝑡𝑟)

𝑤 (𝑡) 𝑑𝑡

ess sup𝑡


By Lemma 24, we get

𝐽2≲ ‖𝑏‖∗

1

Ψ−1 (𝑟−𝑛

)∫∁(2𝐵)

𝑓 (𝑦)

𝑥0 − 𝑦

𝑛−𝛼𝑑𝑦. (89)

Thus, by (52)

𝐽2≲ ‖𝑏‖∗

1

Ψ−1 (𝑟−𝑛

)∫

∞

2𝑟


Ψ−1

(𝑡−𝑛

)𝑑𝑡

𝑡. (90)

Summing 𝐽1and 𝐽

2we get

[𝑏, 𝐼𝛼] 𝑓2𝐿Ψ(𝐵)

≲ ‖𝑏‖∗

1

Ψ−1 (𝑟−𝑛

)

× ∫

∞

2𝑟

(1 + ln 𝑡𝑟)𝑓

𝐿Φ(𝐵(𝑥0,𝑡))

Ψ−1

(𝑡−𝑛

)𝑑𝑡

𝑡.

(91)

Finally,[𝑏, 𝐼𝛼] 𝑓

𝐿Ψ(𝐵)

≲ ‖𝑏‖∗𝑓

𝐿Φ(2𝐵)

+ ‖𝑏‖∗

1

Ψ−1 (𝑟−𝑛

)

× ∫

∞

2𝑟

(1 + ln 𝑡𝑟)𝑓

𝐿Φ(𝐵(𝑥0,𝑡))

Ψ−1

(𝑡−𝑛

)𝑑𝑡

𝑡,

(92)

and the statement of Lemma 33 follows by (56).

Theorem 34. Let 0 < 𝛼 < 𝑛 and 𝑏 ∈ BMO(R𝑛). Let Φ be aYoung function andΨ defined, via its inverse, by setting, for all𝑡 ∈ (0,∞), Ψ−1(𝑡) := Φ−1(𝑡)𝑡−𝛼/𝑛, and 1 < 𝑎

Φ≤ 𝑏

Φ< ∞ and

1 < 𝑎Ψ

≤ 𝑏Ψ

< ∞. (𝜑1, 𝜑2) and (Φ,Ψ) satisfy the condition

∫

∞

𝑟

(1 + ln 𝑡𝑟) ess inf𝑡


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