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 $\alpha ,s\to 0^{+}$.

arXiv comment28 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

References in corpus (2)

• A Parabolic Problem with a Fractional-Time Derivative
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Cited by in corpus (3)

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• A new theory of fractional differential calculus and fractional Sobolev spaces: One-dimensional case
• On a New Class of Fractional Calculus of Variations and Related Fractional Differential Equations