A maximal theorem for holomorphic semigroups.

Blower, Gordon and Doust, Ian (2005) A maximal theorem for holomorphic semigroups. The Quarterly Journal of Mathematics, 56 (1). pp. 21-30. ISSN 0033-5606

Abstract

Let X be a closed linear subspace of the Lebesgue space L^p(Omega ; mu); let -A be an invertible linear operator that is the generator of abounded holomorphic semigroup T_t on X. The for each 0<a<1 the maximal operator sup |T_tf(x)| belongs to L^p for each f in the domain of A^a. If moreover iA generates a bounded C_0 group and A has spectrum contained in the positive real semi axis, then A has a bounded H infinity functional calculus.