On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup

In this paper, for each given p, 1 < p < infinity, we characterize the weights v for which the centered maximal function with respect to the gaussian measure and the Ornstein-Uhlenbeck maximal operator are well defined for every function in L-p (vd gamma) and their means converge almost everywhere. In doing so, we find that this condition is also necessary and sufficient for the existence of a weight u such that the operators are bounded from L-p(vd gamma) into L-p(ud gamma). We approach the problem by proving some vector valued inequalities. As a byproduct we obtain the strong type (1, 1) for the "global" part of the centered maximal function.