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Hp-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Copyright © Canadian Mathematical Society 2007
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