Blower, Gordon and Doust, Ian and Taggart, Robert (2010) A maximal theorem for holomorphic semigroups on vector-valued spaces. In: The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis : The Australian National University, Canberra, July 2009. Proceedings of the Centre for Mathematics and its Applications, 44 (44). Australian National University, Canberra, pp. 105-114. ISBN 0 7315 5208 3
Suppose that 1<p\leq \infty (\Omega ,\mu) is a \sigma finite measure space and E is a closed subspace of Labesgue Bochner space L^p(\Omega; E) consisting of function oon \Omega that take their values in some complex Banach space X. Suppose that -A is invertible and generates a bounded hlomorphic semigroup T_z on E. If 0<\alpha <1, and f belongs to the domain of A^\alpha, then the maximal function \sup_z|T_zf|, where the supremum is taken over any sector contained in the sector of holomorphy, belongs to L^p. This extends an earlier result of Blower and Doust.
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