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$H^\infty$-calculus for submarkovian generators

Author(s): Peer Christian Kunstmann; Zeljko Strkalj
Journal: Proc. Amer. Math. Soc. 131 (2003), 2081-2088.
MSC (2000): Primary 47A60, 47D03, 47D07
Posted: February 5, 2003
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Abstract: Let $-A$ be the generator of a symmetric submarkovian semigroup in $L_2(\Omega)$. In this note we show that on $L_p(\Omega), 1<p<\infty,$ the operator $A$ admits a bounded $H^\infty$ functional calculus on the sector $\Sigma(\phi)=\{z\in\mathbb{C}\setminus \{0\}:\vert\mbox{arg}\,z\vert<\phi\}$ for each $\phi>\psi_p^*$ with

\begin{displaymath}\psi_p^*=\frac{\pi}{2}\vert\frac{1}{p}-\frac{1}{2}\vert +(1-\... ...{1}{2}\vert)\arcsin(\frac{\vert p-2\vert}{2p-\vert p-2\vert}). \end{displaymath}

This improves a result due to M. Cowling. We apply our result to obtain maximal regularity for parabolic equations and evolutionary integral equations.

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Additional Information:

Peer Christian Kunstmann
Affiliation: Institute of Mathematics I, University of Karlsruhe, Englerstrasse 2, D-76128 Karlsruhe, Germany
Email: peer.kunstmann@math.uni-karlsruhe.de

Zeljko Strkalj
Affiliation: Department of Mathematics, 202 Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65211
Address at time of publication: Institute of Mathematics I, University of Karlsruhe, Englerstrasse 2, D-76128 Karlsruhe, Germany
Email: zeljko.strkalj@math.uni-karlsruhe.de

DOI: 10.1090/S0002-9939-03-06956-9
PII: S 0002-9939(03)06956-9
Keywords: Submarkovian semigroups, functional calculus
Received by editor(s): March 19, 2001
Received by editor(s) in revised form: December 12, 2001
Posted: February 5, 2003
Additional Notes: This work has been partially supported by the ``Landesforschungsschwerpunkt Evolutionsgleichungen'' of the Land Baden-Württemberg
The second author acknowledges support from DAAD. Die Arbeit wurde mit Unterstützung eines Stipendiums im Rahmen des Gemeinsamen Hochschulsonderprogramms III von Bund und Ländern über den DAAD ermöglicht
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2003, American Mathematical Society

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