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1998 Functional calculi for linear operators in vector-valued $L^p$-spaces via thetransference principle
Matthias Hieber, Jan Prüss
Adv. Differential Equations 3(6): 847-876 (1998).

Abstract

Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class $\cal H \cal T$ and $1<p<\infty$. If $Y=\mathbb{C}$, it is shown by means of the transference principle due to Coifman and Weiss that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda\pm\pi/2|<\theta\}$, where $\theta>0$ and on each sector $\Sigma_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda|<\theta\}$ with $\theta<\pi/2$, respectively. Several extensions of these results to the vector-valued case $L^p(\Omega,\mu,Y)$ are presented. In particular, let $-A$ be the generator of a bounded group on a Banach spaces of class $\cal H\cal T$. Then it is shown that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta$, $\theta > 0$, and that $-A^2$ admits an $H^\infty$-calculus on each sector $\Sigma_\theta$, where $\theta > 0$. Applications of these results deal with elliptic boundary value problems on cylindrical domains and on domains with non smooth boundary.

Citation

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Matthias Hieber. Jan Prüss. "Functional calculi for linear operators in vector-valued $L^p$-spaces via thetransference principle." Adv. Differential Equations 3 (6) 847 - 876, 1998.

Information

Published: 1998
First available in Project Euclid: 18 April 2013

zbMATH: 0956.47008
MathSciNet: MR1659281

Subjects:
Primary: 47A60
Secondary: 35J25, 35J40, 42B15, 47D06, 47N20

Rights: Copyright © 1998 Khayyam Publishing, Inc.

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Vol.3 • No. 6 • 1998
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