## Proceedings of the Centre for Mathematics and its Applications

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

#### 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