Maximal operators related to the Ornstein–Uhlenbeck semigroup with complex time parameter

Abstract: Let γ be the Gaussian measure in and , , the corresponding Ornstein–Uhlenbeck semigroup, whose infinitesimal generator is . For each p with , let be the closure of the region of holomorphy of the map taking values in the space of bounded operators on . We examine the maximal operator . The known results about concern mainly the case . We prove that for this operator is of weak type but not of strong type for γ. However, if a neighbourhood of the origin is deleted from in the definition of , the resulting operator is shown to be of strong type. [Copyright &y& Elsevier]