ON Riesz Transforms and Maximal Functions in the Context of Gaussian Harmonic Analysis

Abstract

The purpose of this paper is twofold. We introduce a general maximal function on the Gaussian setting which dominates the Ornstein-Uhlenbeck maximal operator and prove its weak type (1; 1) by using a covering lemma which is halfway between Besicovitch and Wiener. On the other hand, by taking as starting point the generalized Cauchy-Riemann equations, we introduce a new class of Gaussian Riesz Transforms. We prove, using the maximal function de¯ned in the ¯rst part of the paper, that unlike the ones already studied these new Riesz Transforms are weak type (1; 1) independently of their orders.