Spaces of functions of mixed smoothness from the decomposition point of view. (English. Russian original) Zbl 0707.46025

Proc. Steklov Inst. Math. 187, 163-184 (1990); translation from Tr. Mat. Inst. Steklova 187, 143-161 (1989).
This survey deals with the spaces \(S^ r_ pL\) and \(S^{\alpha}_{p,\theta}B\) of Sobolev and Besov type, respectively, having dominating mixed smoothness properties. The authors describe decomposition procedures, characterizations via approximation and Fourier-analytical methods of Paley-Littlewood type. Finally the peiodic case is considered.
Reviewer: H.Triebel


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)