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-calculus for submarkovian generators

Authors:	Peer Christian Kunstmann Zeljko Strkalj

Institution:	Institute of Mathematics I, University of Karlsruhe, Englerstrasse 2, D-76128 Karlsruhe, Germany ; Department of Mathematics, 202 Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65211

Abstract:	Let 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 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.

Keywords:	Submarkovian semigroups functional calculus

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