Abstract view
$H^p$Maximal Regularity and Operator Valued Multipliers on Hardy Spaces


Published:20071201
Printed: Dec 2007
Shangquan Bu
Christian Le Merdy
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.