A characterization of Sobolev spaces via local derivatives
Volume 119 / 2010
Colloquium Mathematicum 119 (2010), 157-167
MSC: Primary 46E35, 26B35.
DOI: 10.4064/cm119-1-11
Abstract
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.
