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

Hugo A. Aimar, Liliana Forzani, Roberto Scotto


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.

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