Maximal operators related to the Ornstein-Uhlenbeek semigroup with complex time parameter

摘要

Let gamma be the Gaussian measure in R-d and H-t, t > 0, the corresponding Ornstein-Uhlenbeck semigroup, whose infinitesimal generator is -1/2 Delta + x . grad. For each p with 1 < p < infinity, let E-p subset of C be the closure of the region of holomorphy of the map t bar right arrow H-l taking values in the space of bounded operators on L-p(gamma). We examine the maximal operator f bar right arrow H-p* f = sup{H(z)f: z is an element of E-p}. The known results about H-p* concern mainly the case p < 2. We prove that for p > 2 this operator is of weak type but not of strong type (p, p) for gamma. However, if a neighbourhood of the origin is deleted from E-p in the definition of H-p*, the resulting operator is shown to be of strong type. (C) 2005 Elsevier Inc. All rights reserved.