Xukui SHAO Shuangping TAO

Abstract In this paper， the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces.And the corresponding commutators generated by BMO function are also considered.

Keywords Fractional integral，Commutator，Variable kernel，Vanishing generalized weighted Morrey space with variable exponent， BMO space

1 Introduction and Main Results

Let Ω(x，z′)∈L∞(Rn)×Ld(Sn-1) (1 ＜d ≤∞) satisfying

where Sn-1= {x ∈Rn: |x| = 1} equipped with Lebesgue measure dz′.For 0 ＜α ＜n and d ≥1， the fractional integral operator with variable kernel is defined by

In 1955，Calder´on and Zygmund[1]investigated the L2boundedness of the singular integral operator with variable kernels.They found that these operators TΩare closely related to the problem about the second order linear elliptic equations with variable coefficients.They proved the following result.

Theorem A(see [1])Suppose thatsatisfies(1.1)–(1.2).Then there exists a constantC ＞0independent offsuch that

In 1971， Muckenhoupt and Wheeden [2] gave the (Lp，Lq) boundedness of TΩ，α.

Theorem B(see [2])Let0 ＜α ＜n，Suppose thatΩ(x，z) ∈L∞(Rn)×Ld(Sn-1)withs ＞p′.Then there exists a constantC ＞0independent offsuch that

Suppose that b ∈Lloc(Rn)，the corresponding m-order commutator generated by b and TΩ，αis defined by

As it is known， in the last two decades there has been an increasing interest to the study of singular integral operators with variable kernels.For instance， Ding et al.[3] obtained the Lpboundedness of Marcinkiewicz integral operators μΩwith variable kernels; Chen and Ding[4]proved the Lpboundedness of Littlewood-Paley operator with variable kernel;Tao and Shao[5]proved the boundedness of Marcinkiewicz integral operator with variable kernel on the homogeneous Morrey-Herz spaces and the weak homogeneous Morrey-Herz spaces.Wang [6]proved the boundedness properties of singular integral operators TΩ， fractional integral TΩ，αand parametric Marcinkiewicz integral μρΩwith variable kernels on the Hardy spaces Hp(Rn)and weak Hardy spaces WHp(Rn).Recently，Shao and Tao[7]obtained the boundedness of the fractional integral operators with variable kernels and its commutators on the variable exponent weak Morrey spaces as the infmum of exponent function p(·) equals 1.For further details on recent developments on this field， we refer the readers to [8–11].

After Kov´aˇcik and R´akosn´ık[12]introduced the spaces Lp(x)and Wk，p(x)in high dimensional Euclidean spaces，many mathematicians have been involved in this field.The theory of function spaces with variable exponent has made great progress during the past 20 years.Due to their applications to PDE with nonstandard growth conditions and so on， we may refer to [13–16].

On the other hand，variable exponent Morrey spaces were introduced and studied in[17–18]in the Euclidean setting.Morrey type spaces have attracted considerable attention in recent years because the interesting norm includes explicitly both local and global information of the function.The authors of[19]established the boundedness of fractional integrals and oscillatory fractional integrals and their commutators on some generalized weighted Morrey spaces.Ho[20]gave some sufficient conditions for the boundedness of fractional integral operators and singular integral operators in Morrey space with variable exponent Mp(·)，u， and he also obtained the weak type estimates of fractional integral operators on Morrey space with variable exponent(see[21]).In[22]，Tao and Li proved the boundedness of Marcinkiewicz integral and its commutators on Morrey space with variable exponent.Guliyev[23]et al.obtained the boundedness of Riesz potential in the vanishing generalized weighted Morrey spaces with variable exponent.Long and Han [24] considered the boundedness of maximal operators， potential operators and singular integral operators on the vanishing generalized Morrey space with variable exponent.

Inspired by the statements above， in this paper， we continue to develop the results from[23–24].The boundedness of the fractional integral operators with variable kernels and their commutators on the variable exponent generalized weighted Morrey spaces and the vanishing generalized weighted Morrey spaces with variable exponent were considered，where the smoothness condition on Ω has been removed.

Before stating the main results of this article， we first recall some necessary definitions and notations.

For any x ∈Rnand r ＞0， let B(x，r) ={z ∈Rn:|z-x| ≤r}.|E| denotes the Lebesgue measure of E ⊂Rnand χEdenotes its characteristic function.Define P(Rn) to be the set of p(·):Rn→(1，∞) such that

Let p(·) ∈P(Rn).The Lebesgue space with variable exponent Lp(·)(Rn) consists of all Lebesgue measurable functions f satisfying

It is easy to know that Lp(·)(Rn)becomes a Banach function space when equipped with the Luxemburg-Nakano norm above.

Given a measurable function b， the maximal commutator is defined by

For 0 ≤α ＜n， fractional maximal operator with variable kernel is defined as

It is easy to see when Ω(x，y)=1， MΩ，αis just the fractional maximal operator

The sharp maximal function is defined by

Let B(Rn) denote the set of p(·)∈P(Rn) which satisfies the following conditions

and

It is proved that the Hardy-Littlewood maximal operater M is bounded on Lp(·)(Rn) as p(·)∈B(Rn) in [25].

Remark 1.1For any p(·) ∈B(Rn) and λ ＞1， by Jensen’s inequality， we have λp(·) ∈B(Rn).See [26， Remark 2.13].

We say an order pair of variable exponents function (p(·)，q(·)) ∈Bα(Rn)， if p(·) ∈P(Rn)，0 ＜α ＜and

Definition 1.1The spaceBMO(Rn)consists of all functionssuch that

where

Definition 1.2Define theBMOp(·)，w(Rn)space as the set of all functionssuch that

Remark 1.2Let p(·)∈B(Rn)and w be a Lebesgue measurable function.If w ∈Ap(·)(Rn)，then the norms ‖·‖BMOp(·)，wand ‖·‖BMOare mutually equivalent (see [27]).

Definition 1.3(see [28])Letwbe a positive， locally integrable function.We say that a weight functionwbelongs to the classAp(·)(Rn)if

A weight functionwbelongs to the classAp(·)，q(·)(Rn)if

Remark 1.3Let w ∈Ap(·)，q(·)(Rn).Then w-1∈Ap′(·)，q′(·)(Rn) (see [23]).

Definition 1.4(see [23])Letλ(·) : Rn→(0，n)be a measurable function，p(·) ∈P(Rn).The Morrey space with variable exponentsLp(·)，λ(·)(Rn)and weighted Morrey space with variable exponentsare defined by

Throughout this paper， u(x，r)， u1(x，r) and u2(x，r) are non-negative measurable functions on Rn×(0，∞).

Definition 1.5(see [23])Letp(·)∈P(Rn)andu(x，r):Rn×(0，∞)→(0，∞).The variable exponent generalized Morrey spaceMp(·)，u(Rn)and variable exponent generalized weighted Morrey spaceare defined by

whereθp(x，r)=

Remark 1.4According to Definition 1.4，if u(x，r)=，then the variable exponent generalized Morrey space Mp(·)，u(Rn) is exactly the Morrey space with variable exponent Lp(·)，λ(·)(Rn).

Definition 1.6(see [23])Letu1(x，r):Rn×(0，∞)→(0，∞).The vanishing generalized weighted Morrey space with variable exponentis defined as the space of functionssuch that

In this paper we assume that

and

The main results of this paper are stated as follows.

Theorem 1.1Suppose thatΩ(x，z)satisfies(1.1)–(1.2).Let0 ＜α ＜n，p(·) ∈B(Rn)，Ifw ∈Ap(·)，q(·)(Rn)，u1(x，t)andu2(x，t)satisfy the condition

Then there exists a constantC ＞0such that for any

Theorem 1.2Suppose thatΩ(x，z)satisfies(1.1)–(1.2).Letb ∈BMO(Rn)，0 ＜α ＜n，p(·) ∈B(Rn)，u1(x，t)andu2(x，t)satisfy the condition

Then there exists a constantC ＞0such that for any

Theorem 1.3Suppose thatΩ(x，z)satisfies(1.1)–(1.2).Let0 ＜α ＜n，p(·) ∈B(Rn)，Ifw ∈Ap(·)，q(·)(Rn)，u1(x，t)andu2(x，t)satisfy the condition

for anyτ0＞0， and

Then there exists a constantC ＞0such that for any

Theorem 1.4Suppose thatΩ(x，z)satisfies(1.1)–(1.2).Letb ∈BMO(Rn)，0 ＜α ＜n，p(·) ∈B(Rn)，Ifw ∈Ap(·)，q(·)(Rn)，u1(x，t)andu2(x，t)satisfy the condition

for anyκ ＞0， and

Then there exists a constantC ＞0such that for any

Throughout this paper， the letter C stands for a positive constant that is independent of the essential variables and not necessarily the same one in each occurrence.

2 Preliminaries Lemmas

In this section we shall give some lemmas which will be used in the proofs of our main theorems.

Lemma 2.1(see [12]) (Generalized H¨older Inequality)Letq(·) : Rn→[1，∞).Iff ∈Lq(·)(Rn)andg ∈Lq′(·)(Rn)，thenf，gare integrable onRnand

where

Lemma 2.2Suppose thatp(·) ∈B(Rn)，0 ＜α ＜n，w ∈Ap(·)，q(·)(Rn).Then there exists a constantC ＞0such that for any

By applying the similar method with the proof of [29]， we can obtain the above result， the details are omitted here.

Lemma 2.3(see [7])Suppose that0 ＜θ ＜min{α，n-α}，x ∈Rn.Then

Lemma 2.4Letp(·)∈B(Rn)，0 ＜α ＜n，IfΩ(x，z)satisfies(1.1)–(1.2)， then there exists a constantC ＞0such that for anyf ∈

ProofWe first prove (2.1).LetUsing H¨older’s inequality， we obtain

According to above inequality， we have

Noting that

so

Then we have

Therefore

By Lemma 2.2， we have

Now we pay attention to the proof of (2.2).

It is enough to prove that the inequality

holds for every function f such that ‖f‖Lp(·)(Rn)≤C.

Fix a θ with 0 ＜θ ＜min{α，n-α} satisfyingDefineThus，we have

By Lemmas 2.1 and 2.3， it has

Without loss of generality， we may assume that the infimum is taken over values of η greater than 1.Since η ＞1 and x ∈Rn，we have

Therefore， by (2.1) and (2.3)， we can obtain

Similarly， we have

So it follows from (2.1) and (2.3) that

Thus

Lemma 2.5(see [30])Letv1，v2andϖbe weights on(0，∞)andv1(t)be bounded outside a neighborhood of the origin.The inequality

holds for someC ＞0and all non-negative and non-decreasinggif and only if

where

Lemma 2.6(see [31])Letp(·) ∈B(Rn)andw ∈Ap(·)(Rn).Then there exists a constantC ＞0independent offsuch that

Lemma 2.7Suppose thatΩ(x，z)satisfies(1.1)–(1.2).If1 ＜s1，s2＜∞，b ∈BMO(Rn)，then there exists a constantC ＞0independent offsuch that， for anyf ∈Lp(Rn)，

With the similar argument in the proof of [32， Lemmas 2.4.1 and 3.5.1]， it is easy to draw the above conclusion.

Lemma 2.8(see [33])Letp(·) ∈B(Rn).Thenif and only ifw ∈Ap(·)(Rn).

Lemma 2.9Letb ∈BMO(Rn)， 0 ＜α ＜n， p(·) ∈B(Rn)withIfΩ(x，z)satisfies(1.1)–(1.2)，q(·)as defined in(1.8)，w ∈Ap(·)，q(·)，then there exists a constantC ＞0such that for any

ProofLetLemma 2.6 implies that

By Lemma 2.7， we have

It follows from Lemmas 2.2 and 2.4 that

Thus

Lemma 2.10(see [23])Letb ∈BMO(Rn)，p(·)∈B(Rn)andw ∈Ap(·)(Rn).ThenMbis bounded on

Lemma 2.11(see [34])Letv1，v2andϖbe weights on(0，∞)andv1be bounded outside a neighborhood of the origin.The inequality

holds for someC ＞0for all non-negative and non-decreasinggif and only if

where0 ＜t ＜∞.

3 Proofs of Theorems 1.1–1.4

Proof of Theorem 1.1LetFor any t ＞0， write

where f1=fχB(x，2t)， f2=fχB(x，2t)c， and

For I1， Lemma 2.4 immediately implies that

where the constant C ＞0 is independent of f.

On the other hand， by (1.9)–(1.10) we have

Taking into account that

we have

Now turn to estimate I2.Note that|x-z|≤t， |z-y|≥2t，and|z-y|≤2|x-y|≤3|z-y|.By H¨older’s inequality， we have

Defined p1(·) with(1.10) implies that

It follows from (3.4) and Lemma 2.1 that

Write p(·)=d′p1(·) and q(·)=d′q1(·)， we haveTherefore

From (3.3) and (3.5)， we can obtain

Let

and

Lemma 2.5 and (1.11) yield that

This completes the proof of Theorem 1.1.

Proof of Theorem 1.2Let b ∈BMO(Rn)，As in the proof of Theorem 1.1， for any t ＞0， write

Let us prove the following inequality:

First we have that

By Lemma 2.9， we can obtain

where C is a constant independent of f.

From (3.5)， we get

Then

Noting that |x-z|≤t， |x-z|≥2t， we have |z-y|≤2|x-y|≤3|z-y|， therefore

To estimate J1， we have

For J11， we can obtain by using H¨older’s inequality and Lemma 2.1 that

Noting that p(·)=d′p1(·)， q(·)=d′q1(·) andwe have

It follows from the above inequality that

For J12， the H¨older’s inequality assures that

By the estimates of J11and J12， we can get

Now turn to estimate J2.By H¨older’s inequality，

where C ＞0 is the constant independent of x and t.

Combining estimates of J1and J2yields that

By Lemma 2.10， we have

Hence

This finishes the proof of (3.7).

Note that w ∈Ap(·)，q(·).Let

and

Then by Lemma 2.11 and (1.12)， we can obtain

The proof of Theorem 1.2 is completed.

Proof of Theorem 1.3We have proved the following inequality in Theorem 1.1，

so we only have to prove that

when

Next we prove， for any small r ＞0， that

We split the right-hand side of (3.6) as

where 0 ＜τ0＜1.

the constants C and C′come from (1.14) and the inequality above， respectively.

Then for 0 ＜t ＜τ0， by (1.14)， we have

To the estimation of K2， from (1.13)，

Now we can choose t small enough.It follows from (1.14) that

Thus

The proof of Theorem 1.3 is completed.

Proof of Theorem 1.4We have proved the following inequality in Theorem 1.2，

so we only have to prove that if

then

Next， we show that

for a sufficiently small r.We may decompose the right-hand side of (3.7) as

We estimate G1.Since， for all 0 ＜t ＜κ， we choose any fixed κ ＞0 such that

For 0 ＜t ＜κ， it follows from (1.16) that

To the estimation of G2， we can choose t small enough.It follows from (1.15)–(1.16)that

Hence we can get

which completes the proof of Theorem 1.4.

AcknowledgementThe authors are very grateful to the referees for their valuable comments.