## A $$\varphi$$-transform result for spaces with dominating mixed smoothness properties.(English)Zbl 0905.46017

The main result of this note is a sequence space characterization of certain Banach spaces with dominating mixed smoothness properties, in the spirit of the work of M. Frazier and B. Jawerth [J. Funct. Anal. 93, No. 1, 34-170 (1990; Zbl 0716.46031)]. More precisely, the author describes the corresponding Triebel-Lizorkin spaces [cf. the book by H.-J. Schmeisser and H. Triebel: “Topics in Fourier analysis and function spaces” (1987; Zbl 0661.46024) for details on these spaces] in terms of appropriate Fourier decompositions and derives therefrom the fact that these spaces are retracts of certain solid sequence spaces, for the full range of parameters, i.e., indices $$p,q$$ down to $$0$$. Although the general approach follows the lines of previous work the details are technically quite involved.

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 46E15 Banach spaces of continuous, differentiable or analytic functions 42B25 Maximal functions, Littlewood-Paley theory 42C15 General harmonic expansions, frames 46F99 Distributions, generalized functions, distribution spaces

### Citations:

Zbl 0661.46024; Zbl 0716.46031
Full Text:

### References:

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