A characterization of Sobolev spaces via local derivatives

David Swanson¹

¹Department of Mathematics University of Louisville Louisville, KY 40292, U.S.A.

Colloquium Mathematicum, Tome 119 (2010) no. 1, pp. 157-167

Résumé

Let $1 \le p \infty$, $k \ge 1$, and let $\Omega \subset \mathbb R^n$ be an arbitrary open set. We prove a converse of the Calderón–Zygmund theorem that a function $f \in W^{k,p}(\Omega)$ possesses an $L^p$ derivative of order $k$ at almost every point $x \in \Omega$ and obtain a characterization of the space $W^{k,p}(\Omega)$. Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

Détail

DOI : 10.4064/cm119-1-11

Keywords: infty omega subset mathbb arbitrary set prove converse calder zygmund theorem function omega possesses derivative order almost every point omega obtain characterization space omega method based distributional arguments pointwise inequality due bojarski haj asz