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 $\alpha\to1^-$ 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 $s\to1^-$ 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 $\alpha,s\to0^+$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2018
- DOI:
- 10.48550/arXiv.1810.13305
- arXiv:
- arXiv:1810.13305
- Bibcode:
- 2018arXiv181013305S
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- Mathematics - Analysis of PDEs;
- Mathematics - Functional Analysis
- E-Print:
- 28 pages. We are grateful to Francisco J. Mart\'in-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