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Zbl 1175.46024
Schneider, Cornelia
On dilation operators in Besov spaces. (English)
[J] Rev. Mat. Complut. 22, No. 1, 111-128 (2009). ISSN 1139-1138; ISSN 1988-2807/e

Summary: We consider dilation operators $T_k: f\mapsto f(2^k\cdot)$ in the framework of Besov spaces $B^s_{p,q}(\Bbb R^n)$ when $0 <p \leq 1 $. If $ s> n (\frac 1 p - 1)$, then $T_k $ is a bounded linear operator from $B^s_{p,q}(\Bbb R^n)$ into itself and there are optimal bounds for its norm. We study the situation on the line $s=n(\frac 1 p - 1)$, an open problem mentioned in [{\it D.\,E.\thinspace Edmunds} and {\it H.\,Triebel}, ``Function spaces, entropy numbers, differential operators'' (Cambridge Tracts in Mathematics 120; Cambridge Univ.\ Press) (1996; Zbl 0865.46020)]. It turns out that the results shed new light upon the diversity of different approaches to Besov spaces on this line, associated to definitions by differences, Fourier-analytical methods, and subatomic decompositions.

MSC 2000:: *46E35 Sobolev spaces and generalizations

Keywords: Besov spaces; dilation operators; moment conditions

Citations: Zbl 0865.46020

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