Fractional weighted Sobolev spaces associated to the Riesz fractional gradient

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.