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# $H^p$-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces

Published:2007-12-01
Printed: Dec 2007
• Shangquan Bu
• Christian Le Merdy
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## Abstract

We consider maximal regularity in the $H^p$ sense for the Cauchy problem $u'(t) + Au(t) = f(t)\ (t\in \R)$, where $A$ is a closed operator on a Banach space $X$ and $f$ is an $X$-valued function defined on $\R$. 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 $H^p(\R;X)$, in the case when $X$ is an AUMD Banach space.
 Keywords: $L^p$-maximal regularity, $H^p$-maximal regularity, Rademacher boundedness
 MSC Classifications: 42B30 - $H^p$-spaces 47D06 - One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

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