Open Access
2005 An extrapolation theorem for the $H^\infty$calculus on $L^p(\Omega;X)$
Robert Haller-Dintelmann
Differential Integral Equations 18(3): 263-280 (2005). DOI: 10.57262/die/1356060218


Let $(A_p)_{1 < p < \infty}$ be a consistent family of sectorial operators on $L^p(\Omega; X)$, where $\Omega$ is a homogeneous space with doubling property and $X$ is a Banach space having the Radon-Nykodým property. If $A_{p_0}$ has a bounded {$H^\infty$ calculus}{} for some $1 < p_0 < \infty$ and the resolvent or the semigroup generated by $A_{p_0}$ fulfills a Poisson estimate, then it is proved that $A_p$ has a bounded {$H^\infty$ calculus}{} for all $1 < p \le p_0$ and even for $1 < p < \infty$ if $X$ is reflexive. In order to do so, the Calderón-Zygmund decomposition is generalized to the vector-valued setting.


Download Citation

Robert Haller-Dintelmann. "An extrapolation theorem for the $H^\infty$calculus on $L^p(\Omega;X)$." Differential Integral Equations 18 (3) 263 - 280, 2005.


Published: 2005
First available in Project Euclid: 21 December 2012

zbMATH: 1212.47009
MathSciNet: MR2122719
Digital Object Identifier: 10.57262/die/1356060218

Primary: 47A60
Secondary: 43A85

Rights: Copyright © 2005 Khayyam Publishing, Inc.

Vol.18 • No. 3 • 2005
Back to Top