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VOL. 44 | 2010 A Maximal Theorem for Holomorphic Semigroups on Vector-Valued Spaces
Gordon Blower, Ian Doust, Robert J. Taggart

Editor(s) Andrew Hassell, Alan McIntosh, Robert Taggart


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.


Published: 1 January 2010
First available in Project Euclid: 18 November 2014

zbMATH: 1235.47040
MathSciNet: MR2655388

Rights: Copyright © 2010, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


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