On singular integrals with respect to the Gaussian measure.(English)Zbl 0737.42018

Let $$R_\alpha=D^\alpha(-L)^{-| \alpha|/2}$$ be the Riesz transform of order $$\alpha$$ associated with the Ornstein-Uhlenbeck operator $$L$$. The author gives a new analytic proof of the fact that $$R_\alpha$$ is bounded in $$L^p(\gamma_n(x)\,dx)$$, where $$\gamma_n(x)\,dx$$ is the Gaussian measure in $$\mathbb{R}^n$$. The case $$n=1$$, $$\alpha=1$$ was treated by B. Muckenhoupt [Trans. Am. Math. Soc. 139, 243–260 (1969; Zbl 0175.12701)] and the general case was treated by P. A. Meyer [Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 179–193 (1984; Zbl 0543.60078)] and also by R. Gundy [C. R. Acad. Sci., Paris, Sér. I 303, 967–970 (1986; Zbl 0606.60063)] using probabilistic methods. G. Pisier [Sémin. Probabilités, Strasbourg/France, Lect. Notes Math. 1321, 485–501 (1988; Zbl 0645.60061)] has also given a proof of Meyer’s theorem using transform methods.
The methods developed in the paper under review are also applied to deal with other singular integrals.

MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 60H99 Stochastic analysis
Full Text:

References:

 [1] C.P. Calderón , Some remarks on the multiple Weierstrass Transform and Abel summability of multiple Fourier Series , Studia Mathematica XXXII ( 1969 ), 119 - 148 . Article | Zbl 0182.15901 · Zbl 0182.15901 [2] I.S. Gradshteeyn - S.M. Ryzhik , Table of Integrals, Series and Products , Academic Press ( 1986 ). [3] R. Gundy , Sur les Transformations de Riesz pour le semigroup d’Ornstein-Uhlenbeck , C.R. Acad. Sci. Paris 303 (Série I ) ( 1986 ), 967 - 970 . Zbl 0606.60063 · Zbl 0606.60063 [4] E. Harboure De Aguilera , Non-standard Truncations of Singular Integrals , Indiana Univ. Math. J. Vol. 28 , 5 ( 1979 ), 779 - 790 . Zbl 0395.42011 · Zbl 0395.42011 [5] P.A. Meyer , Transformations de Riesz pour les lois Gaussiens , Sem. Prob. XVIII Springer Lecture Notes Math . 1059 ( 1984 ), 179 - 193 . Numdam | Zbl 0543.60078 · Zbl 0543.60078 [6] B. Muckenhoupt , Poisson Integrals for Hermite and Laguerre expansions , Trans. Amer. Math. Soc. 139 ( 1969 ), 231 - 242 . Zbl 0175.12602 · Zbl 0175.12602 [7] B. Muckenhoupt , Hermite conjugated expansions , Trans. Amer. Math. Soc. 139 ( 1969 ), 243 - 260 . Zbl 0175.12701 · Zbl 0175.12701 [8] I.P. Natanson , Theory of Functions of a real variable , Vol. II Ungar , New York , ( 1960 ). · Zbl 0091.05404 [9] G. Pisier , Riesz Transforms: a simpler analytic proof of P.A. Meyer inequality , preprint. · Zbl 0645.60061 [10] E.M. Stein , Singular Integrals and differentiability properties of functions , Princeton University Press , Princeton, New Jersey , ( 1970 ). Zbl 0207.13501 · Zbl 0207.13501 [11] D. Stroock , Notes on Malliavin Calculus , manuscript. · Zbl 0633.60078 [12] S. Watanabe - N. Ikeda , An Introduction to Malliavin Calculus , Taniguchi Symp. S A Katata ( 1983 ), 1 - 52 . Zbl 0546.60055 · Zbl 0546.60055 [13] S. Watanabe , Lecture on Stochastic Differential Equations and Malliavin Calculus , Tata Institute Springer Verlag ( 1984 ). Zbl 0546.60054 · Zbl 0546.60054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.