Commutators of Riesz potential in the vanishing generalized weighted Morrey spaces with variable exponent

TL;DR

This paper investigates the boundedness of Riesz potential and its commutators on generalized and vanishing weighted Morrey spaces with variable exponents, extending existing results and unifying different Morrey space theories.

Contribution

It establishes the boundedness of Riesz potential and commutators on these spaces, generalizing and unifying prior results in Morrey space theory.

Findings

Boundedness of Riesz potential on generalized Morrey spaces.

Boundedness of commutators on vanishing Morrey spaces.

Unified framework for variable and classical Morrey spaces.

Abstract

Let $\Omega \subset \mathbb{R}^n$ be an unbounded open set. We consider the generalized weighted Morrey spaces $\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ and the vanishing generalized weighted Morrey spaces $V\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ with variable exponent $p(x)$ and a general function $\varphi(x,r)$ defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces $\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ and $V\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$. This result generalizes several existing results for Riesz potential and its commutators on Morrey type spaces. Especially, it gives a unified result for generalized Morrey spaces and variable Morrey spaces which currently gained a lot of attentions from researchers in theory of function spaces.