On a generalization of $L^p$-differentiability

TL;DR

This paper bridges classical $L^p$-differentiability with modern non-local Sobolev space characterizations, providing new insights, characterizations, and conditions for functions in Sobolev and BV spaces.

Contribution

It generalizes Calderón and Zygmund's $L^p$-differentiability results to non-local Sobolev space characterizations, strengthening existing theorems and introducing new criteria.

Findings

New characterizations of Sobolev spaces

A novel condition for BV functions to be in $W^{1,1}$

Completion of a key Sobolev space characterization

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 claimed in the paper "Characterization of Sobolev and BV spaces".