Abstract
In this paper, we study the boundedness of the fractional integral with variable kernel. Under some assumptions, we prove that such kind of operators is bounded from the variable exponent Herz-Morrey spaces to the variable exponent Herz-Morrey spaces.
Highlights
When μ ≡ 1, the above integral takes the Cauchy principal value
In 1991, Kováčik and Rákosník [6] introduced variable exponent Lebesgue and Sobolev spaces as a new method for dealing with nonlinear Dirichet boundary value problem
The class of Herz-Morrey spaces with variable exponent is initially defined by the author [7], and the boundedness of vector-valued sub-linear operator and fractional integral on Herz-Morrey spaces with variable exponent was introduced by authors [7] and [8]
Summary
( ) ( ) Let 0 < μ < n , Ω ∈ L∞ n × Lr S n−1 (r ≥ 1) is homogenous of degree zero on n , S n−1 denotes the unit sphere in n. The fractional integral operator with variable kernel TΩ,μ is defined by TΩ,μ f (x). (2016) The Boundedness of Fractional Integral with Variable Kernel on Variable Exponent Herz-Morrey Spaces. The corresponding fractional maximal operator with variable kernel is defined by. The class of Herz-Morrey spaces with variable exponent is initially defined by the author [7], and the boundedness of vector-valued sub-linear operator and fractional integral on Herz-Morrey spaces with variable exponent was introduced by authors [7] and [8]. Wang Zijian and Zhu Yueping [11] proved the boundedness of multilinear fractional integral operators on Herz-Morrey spaces with variable exponent. The main purpose of this paper is to establish the boundedness of the fractional integral with variable kernel ( ) ( ) from. C always means a positive constant independent of the main parameters and may change from one occurrence to another
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