𝐻^{∞}-calculus for submarkovian generators

Kunstmann, Peer; Štrkalj, Željko

doi:10.1090/S0002-9939-03-06956-9

$H^\infty$-calculus for submarkovian generators
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by Peer Christian Kunstmann and Željko Štrkalj PDF

Proc. Amer. Math. Soc. 131 (2003), 2081-2088 Request permission

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 $\Sigma (\phi )=\{z\in \mathbb {C}\setminus \{0\}:|\mbox {arg} z|<\phi \}$ for each $\phi >\psi _p^*$ with \[ \psi _p^*=\frac {\pi }{2}|\frac {1}{p}-\frac {1}{2}| +(1-|\frac {1}{p}-\frac {1}{2}|)\arcsin (\frac {|p-2|}{2p-|p-2|}). \] 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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