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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:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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