Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup

Abstract

We find necessary and sufficient conditions on a Banach space X in order for the vector-valued extensions of several operators associated to the Ornstein-Uhlenbeck semigroup to be of weak type (1,1) or strong type (p,p) in the range 1 < p < ∞. In this setting, we consider the Riesz transforms and the Littlewood-Paley g-functions. We also deal with vector-valued extensions of some maximal operators like the maximal operators of the Ornstein-Uhlenbeck and the corresponding Poisson semigroups and the maximal function with respect to the gaussian measure. In all cases, we show that the condition on X is the same as that required for the corresponding harmonic operator: UMD, Lusin cotype 2 and Hardy-Littlewood property. In doing so, we also find some new equivalences even for the harmonic case.