A MAXIMAL THEOREM FOR HOLOMORPHIC SEMIGROUPS ON VECTOR-VALUED SPACES GORDON BLOWER, IAN DOUST, AND ROBERT J. TAGGART 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 f belongs to the domain of A α then the maximal function sup z ‖T z f ‖ 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 [BD]. 1. Introduction Suppose that {T t } t≥0 is a C 0 -semigroup of bounded linear operators on a Banach space E. In the case that E is a space of functions f from a set Ω to a normed space X , an important tool in the analysis of such a semigroup is the maximal function Mf where Mf (ω) = ess-sup t≥0 ‖T t f (ω)‖ X . The classical theorems of Stein [St] and Cowling [Co] apply to sym- metric diffusion semigroups on E, where E = L p (Ω) and 1 <p< ∞, and show that in this case ‖Mf ‖ p ≤ c ‖f ‖ p for all f in L p (Ω). Taggart [Ta] extended this to the vector-valued context where E = L p (Ω; X ) and X satisfies a geometric condition weak enough to include, for example, many of the classical reflexive function spaces. Under much weaker hypotheses, Blower and Doust [BD] showed that in the scalar-valued case, if the semigroup {T t } t>0 can be extended to a bounded holomorphic semigroup on sector of the complex plane, then Mf lies in L p (Ω) at least for f in a large submanifold of L p (Ω). Date : 15 October 2009. 2000 Mathematics Subject Classification. 47D06 (47A60, 34G10, 46E40). Key words and phrases. Holomorphic semigroup, maximal theorem, sectorial operator, vector-valued Lebesgue–Bochner spaces. The third author was supported by the Australian Research Council and the Centre for Mathematics and its Applications, Canberra. 1