Calculus of Variations and Geometric Measure Theory
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D. Spector

On a generalization of $L^p$-differentiability

created by spector on 14 Oct 2015

[BibTeX]

Preprint

Inserted: 14 oct 2015
Last Updated: 14 oct 2015

Pages: 22
Year: 2015

Abstract:

In this paper we connect Calder\'on and Zygmund's notion of $L^p$\- -differentiability with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu. We show how the results of the former can be generalized to the setting of the latter, while the latter results can be strengthened in the spirit of the former. As a consequence of these results we give several new characterizations of Sobolev spaces, a novel condition for whether a function of bounded variation is in the Sobolev space $W^{1,1}$, and complete the proof of a characterization of the Sobolev spaces recently claimed in "Characterization of Sobolev and BV spaces".


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