A characterization of Sobolev spaces via local derivatives

Tom 119 / 2010

David Swanson Colloquium Mathematicum 119 (2010), 157-167 MSC: Primary 46E35, 26B35. DOI: 10.4064/cm119-1-11


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.


  • David SwansonDepartment of Mathematics
    University of Louisville
    Louisville, KY 40292, U.S.A.

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