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Hp -Maximal Regularity and Operator Valued Multipliers on Hardy Spaces

Published online by Cambridge University Press:  20 November 2018

Shangquan Bu  and
Christian Le Merdy
Shangquan Bu
Affiliation:
Department of Mathematical Science, University of Tsinghua, Beijing 100084, P.R. China email: sbu@math.tsinghua.edu.cn
Christian Le Merdy
Affiliation:
Département de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France email: clemerdy@univ-fcomte.fr
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Abstract

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We consider maximal regularity in the ${{H}^{p}}$ sense for the Cauchy problem ${{u}^{\prime }}(t)+Au(t)=f(t)(t\,\in \mathbb{R})$ , where $A$ is a closed operator on a Banach space $X$ and $f$ is an $X$ -valued function defined on $\mathbb{R}$ . We prove that if $X$ is an AUMD Banach space, then $A$ satisfies ${{H}^{p}}$ -maximal regularity if and only if $A$ is Rademacher sectorial of type $<\frac{\pi }{2}.$ Moreover we find an operator $A$ with ${{H}^{p}}$ -maximal regularity that does not have the classical ${{L}^{p}}$ -maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces ${{H}^{p}}(\mathbb{R};\,X)$ , in the case when $X$ is an AUMD Banach space.

Keywords

42B3047D06Lp -maximal regularityHp -maximal regularityRademacher boundedness

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 6 , 01 December 2007 , pp. 1207 - 1222
Copyright
Copyright © Canadian Mathematical Society 2007

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