A maximal theorem for holomorphic semigroups on vector-valued spaces

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A maximal theorem for holomorphic semigroups on vector-valued spaces

The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis (Canberra 2009), pp. 105-114, Proc. Centre Math. Appl. Austral. Nat. Univ., 44, Austral. Nat. Univ., Canberra, 2010.

Gordon Blower, Ian Doust and Robert J. Taggart

Contact details:
Blower: Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK.
Doust: School of Mathematics, UNSW Sydney, NSW 2052, Australia
Taggart: Mathematical Sciences Institute, The Australian National University, ACT 0200, Australia

Abstract: Suppose that 1 < p < ∞, (Ω,μ) is a σ-finite measure space and E is a closed subspace of a Lebesgue-Bochner space L^p(Ω;X), consisting of functions on Ω that take their values in some complex Banach space X. Suppose also that -A is invertible and generates a bounded holomorphic semigroup {T_z} on E. If 0 < α < 1 and ƒ belongs to the domain of A^α then the maximal function sup_z || T_zƒ||_X, where the supremum is taken over any given sector contained in the sector of holomorphy, belongs to L^p. This extends an earlier result of Blower and Doust.

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