Journal of Function Spaces
Volume 2019 (2019), Article ID 7152346, 11 pages
https://doi.org/10.1155/2019/7152346

Research Article

Weak Type Estimates of Variable Kernel Fractional Integral and Their Commutators on Variable Exponent Morrey Spaces

Xukui Shao1,2 and Shuangping Tao¹

¹College of Mathematics and Statistics, Northwest Normal University, Gansu, Lanzhou 730070, China
²College of Mathematics and Statistics, LongDong University, Gansu, Qingyang 745000, China

Correspondence should be addressed to Shuangping Tao; taosp@nwnu.edu.cn

Received 10 April 2019; Revised 29 May 2019; Accepted 4 June 2019; Published 27 June 2019

Academic Editor: Alberto Fiorenza

Copyright © 2019 Xukui Shao and Shuangping Tao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 satisfies

where 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 as

where 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:

and

It 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 , , and

with

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 by

The weak Morrey space with variable exponent is defined by

For any , one has

That 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 if

In 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 that

Fix a with satisfies Let . Then . Thus, we have

By Lemma 11 and Young’s inequality, it has

Noting that , , then

Referring to the argument used in the proofs of [16, heorems 1.8 and 1.9], we can obtain the following inequalities:

Then, we have

Thus

Lemma 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 as

If 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 then

where

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

Since

then we have

On 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 that

Then

According to (63), for some independent of , we have that

Remark 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 obtain

From (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).

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