*Ian
Doust's *i*nformation pages*

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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.
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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*_{z}ƒ||_{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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