Sobolev meets Riesz: a characterization of weighted Sobolev spaces via weighted Riesz bounded variation space

TL;DR

This paper introduces weighted Riesz bounded variation spaces and characterizes weighted Sobolev spaces with Muckenhoupt weights, also extending the characterization to variable exponent Sobolev spaces using extrapolation theory.

Contribution

It provides a novel characterization of weighted Sobolev spaces through Riesz bounded variation spaces, including variable exponent cases, advancing the understanding of these function spaces.

Findings

Characterization of weighted Sobolev spaces via Riesz bounded variation spaces

Extension of the characterization to variable exponent Sobolev spaces

Application of Rubio de Francia's extrapolation theory

Abstract

We introduce weighted Riesz bounded variation spaces defined on an open subset of the $n$-dimensional Euclidean space and use them to characterize weighted Sobolev spaces when the weight belongs to the Muckenhoupt class. As an application, using Rubio de Francia's extrapolation theory, a similar characterization of the variable exponent Sobolev spaces via variable exponent Riesz bounded variation spaces is obtained.