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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: H. Aimar, L. Forzani and R. Scotto
Journal: Trans. Amer. Math. Soc. 359 (2007), 2137-2154
MSC (2000): Primary 42B20, 42B25; Secondary 42C10, 47D06, 42A50, 60H07
Published electronically: December 15, 2006
MathSciNet review: 2276615
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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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Additional Information

H. Aimar
Affiliation: Departamento de Matemáticas, Universidad Nacional del Litoral–CONICET, Santa Fe 3000, Argentina
Email: haimar@ceride.gov.ar

L. Forzani
Affiliation: Departamento de Matemáticas, Universidad Nacional del Litoral–CONICET, Santa Fe 3000, Argentina – and – School of Statistics, University of Minnesota, Minneapolis, Minnesota 55455
Email: lforzani@math.unl.edu.ar

R. Scotto
Affiliation: Departamento de Matemáticas, Universidad Nacional del Litoral, Santa Fe 3000, Argentina
Email: scotto@math.unl.edu.ar

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04100-6
PII: S 0002-9947(06)04100-6
Keywords: Gaussian measure, Fourier analysis, Fourier analysis in several variables, singular integrals, maximal functions, Hermite expansions
Received by editor(s): May 5, 2004
Received by editor(s) in revised form: March 4, 2005
Published electronically: December 15, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.