## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 3 (2005), 263-280.

### An extrapolation theorem for the $H^\infty$calculus on $L^p(\Omega;X)$

#### Abstract

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.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 3 (2005), 263-280.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060218

**Mathematical Reviews number (MathSciNet)**

MR2122719

**Zentralblatt MATH identifier**

1212.47009

**Subjects**

Primary: 47A60: Functional calculus

Secondary: 43A85: Analysis on homogeneous spaces

#### Citation

Haller-Dintelmann, Robert. An extrapolation theorem for the $H^\infty$calculus on $L^p(\Omega;X)$. Differential Integral Equations 18 (2005), no. 3, 263--280. https://projecteuclid.org/euclid.die/1356060218