$L^p$-Taylor approximations characterize the Sobolev space $W^{1,p}$

TL;DR

This paper introduces an $L^p$-Taylor approximation concept that characterizes Sobolev spaces $W^{1,p}$, linking local and nonlocal function space characterizations and providing a new criterion for Sobolev regularity.

Contribution

It establishes that $L^p$-Taylor approximations precisely characterize Sobolev spaces $W^{1,p}$, connecting various existing characterizations and offering a simple test for Sobolev membership.

Findings

$L^p$-Taylor approximation exists for functions in $W^{1,p}$

Characterization of $W^{1,p}$ via $L^p$-Taylor approximation

A simple criterion for Sobolev membership of BV functions

Abstract

In this note, we introduce a variant of Calder\'on and Zygmund's notion of $L^p$-differentiability - an \emph{$L^p$-Taylor approximation}. Our first result is that functions in the Sobolev space $W^{1,p}(\mathbb{R}^N)$ possess a first order $L^p$-Taylor approximation. This is in analogy with Calder\'on and Zygmund's result concerning the $L^p$-differentiability of Sobolev functions. In fact, the main result we announce here is that the first order $L^p$-Taylor approximation characterizes the Sobolev space $W^{1,p}(\mathbb{R}^N)$, and therefore implies $L^p$-differentiability. Our approach establishes connections between some characterizations of Sobolev spaces due to Swanson using Calder\'on-Zygmund classes with others due to Bourgain, Brezis, and Mironescu using nonlocal functionals with still others of the author and Mengesha using nonlocal gradients. That any two characterizations of Sobolev spaces are related is not surprising, however, one consequence of our analysis is a simple condition for determining whether a function of bounded variation is in a Sobolev space.