Fractional weighted Sobolev spaces associated to the Riesz fractional gradient

TL;DR

This paper introduces weighted fractional Sobolev spaces using the Riesz fractional gradient, explores their properties, and applies them to degenerate fractional elliptic PDEs, extending classical unweighted theories to weighted settings.

Contribution

The paper defines and analyzes a new class of weighted fractional Sobolev spaces associated with the Riesz fractional gradient, extending unweighted space theory to weighted contexts.

Findings

Spaces coincide with weighted Bessel potential spaces

Established structural properties and embeddings

Studied degenerate fractional elliptic PDEs

Abstract

In this work, we introduce a new family of functions spaces, the weighted fractional Sobolev spaces $X^{s,p}_{0,w}(\Omega)$, where $w$ is a weight in the Muckenhoupt class $A_p$. This space is a natural extension of the fractional Sobolev spaces $H^{s,p}_0$, obtained by means of the Riesz fractional gradient $D^s$, to the setting of the weighted Lebesgue spaces $L^p_w$. As it happened in the unweighted space, the spaces $X^{s,p}_{0,w}(\Omega)$ coincide with the weighted version of the Bessel potential space. We obtaien several structural properties for these spaces, as well as continuous and compact embeddings. We conclude with the study of a family of degenerate fractional elliptic partial differential equations.