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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
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References:

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