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

where Sn−1 = {x ∈ R : |x| = 1} is equipped with the Lebesgue measure dz󸀠. In 1955, Calderón and Zygmund [1] investigated the Lp 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

where −1 = { ∈ R : | | = 1} 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 Ω, .
Theorem A (see [2]). Let 0 < < , 1 < < / , and 1/ = 1/ − / . Suppose that Ω( , ) ∈ ∞ (R ) × ( −1 ) with > . Then there exists a constant > 0 independent of such that Ω, 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 Ω with variable kernels; Wang [4] proved the boundedness properties of 2 Journal of Function Spaces singular integral operators Ω , fractional integral Ω, , and parametric Marcinkiewicz integral Ω with variable kernels on the Hardy spaces (R ) and weak Hardy spaces (R ). For the related results of the singular integral operator with variable kernels, the reader is refereed to [5][6][7][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 M (⋅), ; 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 ( (⋅) (R ), (⋅) (R ))−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 Ω has been removed.
Before stating the main results of this article, we first recall some necessary definitions and notations.
For 0 < ≤ 1, the Lipschitz space Lip (R ) is defined as BMO(R ) space is defined as where the supreme is taken over all cubes ⊂ R , and = (1/| |) ∫ ( ) .
Define P(R ) to be the set of (⋅) : R → [1, ∞) such that Let (⋅) ∈ P(R ). The Lebesgue space with variable exponent (⋅) (R ) consists of all Lebesgue measurable function satisfying (⋅) (R ) becomes a Banach function space when equipped with the Luxemburg-Nakano norm above.
The weak Lebesgue space with variable exponent (⋅) (R ) consists of all Lebesgue measurable function satisfying It is easy to see that ‖ ⋅ ‖ (⋅) (R ) is a quasi-norm; that is, for any 1 , 2 ∈ (⋅) (R ), we have Let B(R ) denote the set of (⋅) ∈ P(R ) which satisfies the following conditions: and It is proved that the Hardy-Littlewood maximal operator M is bounded on (⋅) (R ) as (⋅) ∈ B(R ) satisfies − > 1 in [16].

Preliminaries Lemmas
In this section we shall give some lemmas which will be used in the proofs of our main theorems.
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, Lemma 4], we can obtain the following result.