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

Abstract.

In this paper, for each given $$p, 1 < p < \infty,$ 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 poblem 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.