On Riesz transforms and maximal functions in the context of Gaussian Harmonic Analysis

Aimar, H.; Forzani, L.; Scotto, R.

doi:10.1090/S0002-9947-06-04100-6

On Riesz transforms and maximal functions in the context of Gaussian Harmonic Analysis
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by H. Aimar, L. Forzani and R. Scotto PDF

Trans. Amer. Math. Soc. 359 (2007), 2137-2154 Request permission

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 a starting point the generalized Cauchy-Riemann equations, we introduce a new class of Gaussian Riesz Transforms. We prove, using the maximal function defined in the first 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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