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$%C. Schneider, On dilation operators in Besov spaces, Rev. Mat. Complut. 22 (2009), no. 1, 111–128.%$

On Dilation Operators in Besov Spaces

Cornelia SCHNEIDER

Leipzig University
Department of Mathematics
PF 100920
D-04009 Leipzig — Germany
schneider@math.uni-leipzig.de

Received: April 22, 2008
Accepted: May 6, 2008

ABSTRACT

We consider dilation operators $k Tk: f → f(2 ⋅)$ in the framework of Besov spaces $s n Bp,q(ℝ )$ when $0 <p ≤ 1$ . If $(1 ) s> n p - 1$ , $Tk$ is a bounded linear operator from $s n Bp,q(ℝ )$ into itself and there are optimal bounds for its norm. We study the situation on the line $s= n(1 - 1) p$ , an open problem mentioned in "Function spaces, entropy numbers, differential operators" (Edmunds and Triebel, 1996). 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.

Key words: Besov spaces, dilation operators, moment conditions.
2000 Mathematics Subject Classification: 46E35.