Nonlocal equations with degenerate weights

TL;DR

This paper develops fractional weighted Sobolev spaces with degenerate weights, establishing fundamental inequalities and regularity results for solutions to weighted nonlocal equations, extending classical local results to a nonlinear, nonlocal context.

Contribution

It introduces new fractional weighted Sobolev spaces with degenerate weights and proves key inequalities and regularity results, extending classical local theory to the nonlocal, weighted setting.

Findings

Established embeddings and Poincaré inequalities for fractional weighted Sobolev spaces.

Proved interior Hölder continuity for solutions to weighted nonlocal equations.

Extended classical results like Harnack inequalities to a nonlinear, nonlocal framework.

Abstract

We introduce fractional weighted Sobolev spaces with degenerate weights. For these spaces we provide embeddings and Poincar\'e inequalities. When the order of fractional differentiability goes to $0$ or $1$, we recover the weighted Lebesgue and Sobolev spaces with Muckenhoupt weights, respectively. Moreover, we prove interior H\"older continuity and Harnack inequalities for solutions to the corresponding weighted nonlocal integro-differential equations. This naturally extends a classical result by Fabes, Kenig, and Serapioni to the nonlinear, nonlocal setting.