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A second look on definition and equivalent norms of Sobolev spaces. (English) Zbl 0941.46019
Summary: Sobolev’s original definition of his spaces \(L^{m,p}(\Omega)\) is revisited. It is only assumed that \(\Omega \subseteq \mathbb R^n\) 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.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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