MAXIMAL OPERATORS FOR THE HOLOMORPHIC ORNSTEIN–UHLENBECK SEMIGROUP

Abstract

For each p in [1, ∞) let Ep denote the closure of the region of holomorphy of the Ornstein-Uhlenbeck semigroup {Ht : t> 0} on L p with respect to the Gaussian measure. We prove sharp weak type and strong type estimates for the maximal operator f �→ H ∗ f = sup{|Hzf | : z ∈ Ep} and for a class of related operators. As a consequence of our methods, we give a new and simpler proof of the weak type 1 estimate for the maximal operator associated to the Mehler kernel.