Maximal operators for the holomorphic Ornstein-Uhlenbeck semigroup

For each p in [1, infinity) let E-p denote the-closure of the region of holomorphy of the Ornstein-Uhlenbeck semigroup {H-t : t > 0} on LP with respect to the Gaussian measure. Sharp weak type and strong type estimates are proved for the maximal operator f --> H-p(*) f = sup{\H(z)f\ : z is an element of E-p} and for a class of related operators. As a consequence, a new and simpler proof of the weak type 1 estimate is given for the maximal operator associated to the Mehler kernel.