On the Search for weighted inequalities for Operators related to the Ornstein-Uhlenbeck Semigroup

Abstract

In this paper,for each given p, 1 < p < ∞, 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 Lp(vdγ ) 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 Lp(vdγ ) into Lp(udγ ). 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.