Calculus of Variations and Geometric Measure Theory

D. Spector

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

created by spector on 03 Jun 2014
modified on 12 Oct 2015


Accepted Paper

Inserted: 3 jun 2014
Last Updated: 12 oct 2015

Journal: C. R. Acad. Sci. Paris Sér. I Math
Pages: 7
Year: 2014

To appear in Comptes Rendus - the article is substantially different in exposition than the preprint.


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.