Hp -Maximal Regularity and Operator Valued Multipliers on Hardy Spaces

Bu, Shangquan ; Merdy, Christian Le

Canadian journal of mathematics, Tome 59 (2007) no. 6, pp. 1207-1222

Résumé

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.

Détail

DOI : 10.4153/CJM-2007-051-5

Mots-clés : 42B30, 47D06, Lp -maximal regularity, Hp -maximal regularity, Rademacher boundedness

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