A second look on definition and equivalent norms of Sobolev spaces. (English) Zbl 0941.46019

Naumann, J.; Simader, C. G.

Math. Bohem. 124, No. 2-3, 315-328 (1999).

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

Keywords:

Sobolev spaces; Poincaré’s inequality; existence of intermediate derivates