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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
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References:

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