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arXiv · 1810.13305

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

P. R. Stinga·M. Vaughan
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Abstract

We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},ω)$, where $ω$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $α\to1^-$ of Marchaud fractional derivatives of order $α$. Similar results for weighted Sobolev spaces $W^{2,p}(\mathbb{R}^n,ν)$, where $ν$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to1^-$ of fractional Laplacians $(-Δ)^s$. 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 $α,s\to0^+$.

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P. R. Stinga, M. Vaughan. 2019-06-28. One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces. https://doi.org/10.1016/j.na.2019.04.004

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