One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces

Abstract

We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},\omega )$, where $\omega$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $\alpha \to 1^{-}$ of Marchaud fractional derivatives of order $\alpha$. Similar results for weighted Sobolev spaces $W^{2,p}({\mathbb{R}}^n,\nu )$, where $\nu$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to 1^{-}$ of fractional Laplacians $(-\Delta {)}^s$. These are Bourgain--Brezis--Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e.~and weighted $L^p$ limits as $\alpha ,s\to 0^{+}$.

28 pages. We are grateful to Francisco J. Martín-Reyes for pointing out to us that Lemma 1.3 was not correct, so we erased it and updated the related results accordingly. To appear in Nonlinear Analysis