Equivalent (quasi)norms for certain function spaces of generalized mixed smoothness. (English. Russian original) Zbl 1129.46023

Bazarkhanov, D. B.

Studies on function theory and differential equations. Collected papers dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol’skii. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica. Proceedings of the Steklov Institute of Mathematics 248, 21-34 (2005); translation from Tr. Mat. Inst. Steklova 248, 26-39 (2005).

Summary: For certain (quasi)Banach function spaces of generalized mixed positive smoothness defined by (mixed) weighted norms of smooth dyadic decompositions of their Fourier transforms, characterizations are obtained in terms of rather general averages, their Peetre maximal functions, atomic and molecular representations, as well as in terms of the so-called \(\varphi\)-transforms.
For the entire collection see [Zbl 1116.00015].

MSC:

46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B03	Isomorphic theory (including renorming) of Banach spaces