MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 315-328 (1999)

A second look on definition and equivalent norms of Sobolev spaces

J. Naumann, C. G. Simader

J. Naumann, Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany, e-mail:; C. G. Simader, Lehrstuhl III für Mathematik, Universität Bayreuth, D-95440 Bayreuth, Germany, e-mail:

Abstract: Sobolev's original definition of his spaces $L^{m,p}(\Omega)$ is revisited. It only assumed that $\Omega\subseteq\Bbb R^n$ is a domain. With elementary methods, essentially based on Poincaré's inequality for balls (or cubes), the existence of intermediate derivates of functions $u\in L^{m,p}(\Omega)$ with respect to appropriate norms, and equivalence of these norms is proved.

Keywords: Sobolev spaces, Poincaré's inequality

Classification (MSC2000): 46E35

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