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OPERATOR-VALUED FOURIER MULTIPLIERS ON PERIODIC BESOV SPACES AND APPLICATIONS

Published online by Cambridge University Press:  27 May 2004

Wolfgang Arendt  and
Shangquan Bu
Wolfgang Arendt
Affiliation:
Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany ([email protected])
Shangquan Bu
Affiliation:
Department of Mathematical Science, University of Tsinghua, Beijing 100084, People's Republic of China ([email protected])
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Abstract

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Let $1\leq p,q\leq\infty$, $s\in\mathbb{R}$ and let $X$ be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on $L^p(\mathbb{T})$ holds for the Besov space $B_{p,q}^s(\mathbb{T};X)$ if and only if $1\ltp\lt\infty$ and $X$ is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr. 186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.

AMS 2000 Mathematics subject classification: Primary 47D06; 42A45

Keywords

Fourier multipliersBesov spacesperiodic solutionsCauchy problemmaximal regularity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 1 , February 2004 , pp. 15 - 33
Copyright
Copyright © Edinburgh Mathematical Society 2004