Maximal operators for the holomorphic Ornstein-Uhlenbeck semigroup

For each p in [1,∞) let Ep denote the region of holomorphy of the Ornstein-Uhlenbeck semigroup {Ht : t > 0} on Lp with respect to the Gaus- sian measure. We prove sharp weak type and strong type estimates for the maximal operator f → Hp∗f = sup{|Hzf| : z ∈ Ep} and for a class of re- lated 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