Abstract

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent weak Morrey spaces based on the results of Lebesgue space with variable exponent as the infimum of exponent function equals 1. The corresponding commutators generated by BMO and Lipschitz functions are considered, respectively.

1. Introduction and Main Results

Let . It satisfieswhere is equipped with the Lebesgue measure .

In 1955, Calderón and Zygmund [1] investigated the boundedness of the singular integral operator with variable kernels. They found that these operators connect closely with the problem about the second-order linear elliptic equations with variable coefficients. Muckenhoupt and Wheeden [2] subsequently introduced the fractional integral operator with variable kernels, which is defined by

Muckenhoupt and Wheeden [2] also gave the boundedness with the power weight of .

Theorem A (see [2]). Let , , and . Suppose that with . Then there exists a constant independent of such that

It is well known that the fractional integral operators play an important role in harmonic analysis, which greatly promotes the process of the intersection and integration of harmonic analysis and other disciplines.

Given a local integrable function , the corresponding order commutator is defined by

In recent years, the boundedness of singular integral operators with variable kernels has been widely concerned. For example, Ding Lin and Shao [3] obtained the boundedness of Marcinkiewicz integral operator with variable kernels; Wang [4] proved the boundedness properties of singular integral operators , fractional integral , and parametric Marcinkiewicz integral with variable kernels on the Hardy spaces and weak Hardy spaces . For the related results of the singular integral operator with variable kernels, the reader is refereed to [5–8].

After the paper [9], the variable exponent space theory has been rapidly developed in the past 20 years due to its extensive application in the fields of fluid dynamics and differential equations with nongrowth conditions. For example, in [10], the authors considered the boundedness of higher order commutators of Marcinkiewicz integral on the Lebesgue space with variable exponent. Ho [11] has given some sufficient conditions for the boundedness of fractional integral operators and singular integral operators in Morrey space with variable exponent ; he also obtained the weak type estimates of fractional integral operators on Morrey space with variable exponent and singular integral operators on Morrey-Banach space (see [12, 13]). In 2016, Tao and Li [14] proved the boundedness of Marcinkiewicz integral and it is commutators on Morrey space with variable exponent. In [15], the boundedness of the parameterized Littlewood-Paley operators and their commutators was given by Wang and Tao.

Motivated by the above research, in this paper, we will consider the boundedness of the fractional integral operators and their commutators with variable kernels on variable exponent weak Morrey spaces, where the smoothness condition on has been removed.

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

For , the Lipschitz space is defined as

space is defined aswhere the supreme is taken over all cubes , and .

For any and , let . denotes the Lebesgue measure of and its characteristic function.

Define to be the set of such that

Let . The Lebesgue space with variable exponent consists of all Lebesgue measurable function satisfying becomes a Banach function space when equipped with the Luxemburg-Nakano norm above.

The weak Lebesgue space with variable exponent consists of all Lebesgue measurable function satisfying

It is easy to see that is a quasi-norm; that is, for any , we have

Let , the Hardy-Littlewood maximum operator is defined by

Let denote the set of which satisfies the following conditions:andIt is proved that the Hardy-Littlewood maximal operator is bounded on as satisfies in [16].

Remark 1. For any and , by Jensen’s inequality, we have . See emark 2.13 in [17].
We say an order pair of variable exponents function , if , , andwith

Remark 2. (1) The condition is equivalent to saying that there exists with such that .
(2) implies

Definition 3 (see [18]). Let and be a Lebesgue measurable function; we say if there exists a constant such that for any and , fulfills

Definition 4 (see [18]). Let and . The Morrey space with variable exponent is defined byThe weak Morrey space with variable exponent is defined byFor any , one hasThat is, is a quasi-norm.

The main results of this paper are stated as follows.

Theorem 5. Let , satisfying (13), (14), and (15) with . If and satisfies (1) and (2), then there exists a constant such that for any ,In particular, we have the strong type result as : .

Theorem 6. Let with . Then under the hypotheses of Theorem 5, there exists a constant such that for any ,In particular, is bounded from to as .

Theorem 7. Suppose that , , satisfying (13), (14), and (15) with . If satisfies (1) and (2) and meets with the following condition:Then there exists a constant such that for any ,In particular, is bounded from to as .

Throughout this paper, the letter 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 8 (see [19]). Let . Define by ; then there exists a constant such that for any ball , we have

Lemma 9 (see [20]). Let and . Defined by , for all measurable functions and ; we have

Lemma 10 (see [16]). Let . Then if and only ifIn particular, if either constant equals 1 we can take the other equal to 1 as well.

By applying the similar method used in the proof of [21, emma 4], we can obtain the following result.

Lemma 11. Suppose that , . Then where

Lemma 12. Let , . Suppose that satisfies (1) and (2); is defined by (15). Then there exists a constant such that for any ,

Proof. Let . Without loss of generality we may assume that . Noting that , we only need to prove that, for any ,Since , by Lemma 10 it will suffice to prove thatFix a with satisfies Let . Then . Thus, we haveBy Lemma 11 and Young’s inequality, it has Noting that , , thenReferring to the argument used in the proofs of [16, heorems 1.8 and 1.9], we can obtain the following inequalities:Then, we haveThusLemma 12 is proved.

Lemma 13. Suppose that , with . Then, for any ,where

With the similar argument in the proof of [22, emma 2], it is easy to draw the above conclusion; the details are omitted here.

Lemma 14. Let , , with . is defined asIf satisfies (1) and (2), then there exists a constant such that for any ,

Lemma 15. Let with . If satisfies (1) and (2), as defined in (15), then there exists a constant such that for any ,

By applying Lemma 13, we can prove Lemmas 14 and 15 with the similar way used in the proof of Lemma 12. Thus, we omit the details here.

Lemma 16 (see [23]). Let ; if , then there exists a constant such that for all balls ,

Lemma 17 (see [24]). Suppose , to be a positive integer, with , and thenwhere

3. Proofs of Theorems 5–7

Proof of Theorem 5. Let . For any , and , write where and .
Lemma 12 immediately implies that Note that there exists a constant such that In the view of , we have , so the Hardy-Littlewood maximal operator is bounded on ; it follows that Sincethen we haveOn the other hand, for any and , by Hölder’s inequality, we have According to Lemmas 9, 16, and in [20, heorem 4.5.9], define ; it yields that Thus Noting that for any , , we have and Therefore For any , we can get Since , from (1) and (2), it follows that Then Therefore,Applying the quasi-norm on both sides of the above inequality, then Note thatThenAccording to (63), for some independent of , we have thatRemark 1 and (50) yield that By taking the supremum over , we obtain

Proof of Theorem 6. Let and . Using the same decomposition of as in the proof of Theorem 5, by Lemma 14, we have Thus, we get by (49) For any , , we can obtain by using Hölder’s inequality, Note that , ; we have Similar to the estimate of , by the fact of , we have that It follows from Lemma 8 that The above estimates imply that Therefore Applying the quasi-norm on both sides of the above inequality, we get It implies by (63) that From (68), we can arrive at By taking the supremum over and , we conclude that This completes the proof of Theorem 6.

Proof of Theorem 7. Let , . As in the proof in Theorem 5, write By Lemma 15, we can obtainFrom (49), we get That is,Furthermore, for any , we have For , Lemmas 9, 16 and Hölder’s inequality assure that Note that Then according to Lemma 17, we have Hence Now turn to estimate . By Lemmas 9, 17 and Hölder’s inequality, we have Thus applying Lemma 16 leads to The estimates of and imply that We apply the quasi-norm on both sides of the above inequality Denote Then we have It follows from (82) that Thus, by taking the supremum over and , we obtain The proof of Theorem 7 is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant No. 11561062) and Scientific Research Project of Colleges and Universities in Gansu province (2017A-100).