Nonlocal equations with degenerate weights

Abstract

We introduce fractional weighted Sobolev spaces with degenerate weights. For these spaces we provide embeddings and Poincaré 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 Hölder 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.