Weak Type Estimates of Variable Kernel Fractional Integral and Their Commutators on Variable Exponent Morrey Spaces
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  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  subsequently introduced the fractional integral operator with variable kernels, which is defined by
Muckenhoupt and Wheeden  also gave the boundedness with the power weight of .
Theorem A (see ). 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  obtained the boundedness of Marcinkiewicz integral operator with variable kernels; Wang  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 , 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 , the authors considered the boundedness of higher order commutators of Marcinkiewicz integral on the Lebesgue space with variable exponent. Ho  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  proved the boundedness of Marcinkiewicz integral and it is commutators on Morrey space with variable exponent. In , 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 .
Remark 1. For any and , by Jensen’s inequality, we have . See emark 2.13 in .
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 .
Definition 3 (see ). Let and be a Lebesgue measurable function; we say if there exists a constant such that for any and , fulfills
Definition 4 (see ). 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 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 ). Let . Define by ; then there exists a constant such that for any ball , we have
Lemma 9 (see ). Let and . Defined by , for all measurable functions and ; we have
Lemma 10 (see ). 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
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 16 (see ). Let ; if , then there exists a constant such that for all balls ,
Lemma 17 (see ). Suppose , to be a positive integer, with , and thenwhere
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