Advances in the Boundedness of Fractional Integrals on Grand Variable Weighted Herz-Morrey Spaces

Abstract

In this paper, we first introduce and rigorously define the concept of grand variable weighted Herz-Morrey spaces. The principal aim of this study is to establish the boundedness of the fractional integral operator of variable order on these newly defined function spaces, under suitable assumptions on the associated variable exponents. To achieve this, we employ generalized Hölder-type and Minkowski inequalities, alongside a combination of analytical techniques. The core strategy of the proof involves decomposing the relevant summation into multiple terms and estimating each term under distinct conditions. By systematically combining these estimates, we demonstrate the boundedness of the fractional integral operator of variable order on the grand variable weighted Herz-Morrey spaces. Furthermore, the regularity of solutions to certain elliptic partial differential equations (PDEs) with smooth boundaries is known to be linked to the boundedness of corresponding commutators with smooth kernels. As an application of our main result, we highlight that the boundedness of the fractional integral operator of variable order in the aforementioned function spaces can be effectively applied to the study of regularity properties of solutions to elliptic PDEs.