A maximal theorem for holomorphic semigroups

Abstract

Let X be a closed linear subspace of the Lebesgue space L p (Ω, µ) for some 1 < p < ∞, and let −A be an invertible operator that is the generator of a bounded holomorphic semigroup T t on X. Then for each 0 < α < 1 the maximal function sup t>0 |T t f (x)| belongs to L p (Ω, µ) for each f in the domain of A α . If moreover iA generates a bounded C 0 -group and A has spectrum contained in (0, ∞), then A has a bounded H ∞ functional calculus.

H∞Functional Calculus and Maximal Inequalities for Semigroups of Contractions on Vector-ValuedLp-Spaces

“…This proof is a simple application of the transference principle. The argument consists in transferring ergodic inequalities like (2) to the special case where {T t } t>0 is the translation group of R. This powerful technique was invented by Calderón [5] and largely developed by Coifman and Weiss [7]. Since then it is commonly called transference principle and has been widely applied to many different situations.…”

“…Without loss of generality, we can assume that T t is positive for all t. Let A t = A t (T ). By virtue of the maximal inequality (2), it suffices to show the first limit for f in the domain of A, for instance, for f = T s (g) with s > 0 and g ∈ L p (X; E). We can further assume that f ≥ 0.…”

“…The above theorem is proved in [18] for symmetric diffusion semigroups. We also refer to [2,10,12,33] for some works related to the present article.…”