## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- The AMSI-ANU workshop on spectral theory and harmonic analysis. Andrew Hassell, Alan McIntosh and Robert Taggart, eds. Proceedings of the Centre for Mathematics and its Applications, v. 44. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2010), 105 - 114

### A Maximal Theorem for Holomorphic Semigroups on Vector-Valued Spaces

Gordon Blower, Ian Doust, and Robert J. Taggart

#### Abstract

Suppose that $1 \lt p \geq \infy, (\Omega, \mu)$ is a $\sigma$-finite measure space and $E$ is a closed subspace of a Lebesgue-Bochner space $L^p(\Omega; X)$, consisting of functions on $\Omega$ that take their values in some complex Banach space $X$. Suppose also that $- A$ is injective and generates a grounded holomorphic semigroup ${T_z}$ on $E$. If $0 \lt \alpha \lt 1$ and $f$ belongs to the domain of $A^\alpha$ then the maximal function $\sup_z \|T_zf\|_x$, where the supremum is taken over any given sector contained in the sector of holomorphy, belongs to $L^p$. A similar result holds for generators that are not injective. This extends earlier work of Blower and Doust.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416320873

**Mathematical Reviews number (MathSciNet)**

MR2655388

**Zentralblatt MATH identifier**

1235.47040

#### Citation

Blower, Gordon; Doust, Ian; Taggart, Robert J. A Maximal Theorem for Holomorphic Semigroups on Vector-Valued Spaces. The AMSI–ANU Workshop on Spectral Theory and Harmonic Analysis, 105--114, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 2010. https://projecteuclid.org/euclid.pcma/1416320873