## Séminaire Orléans

**RECENT RESULTS ON THE ANALYSIS OF THE ORNSTEIN–UHLENBECK OPERATOR**

Stefano Meda (Università di Milano Bicocca)

Tuesday 29 September 2009 14:00 - Orléans - Salle de Séminaire

**Résumé :**

Denote by g the Gauss measure on the Euclidean space R^d , and by L the Ornstein–Uhlenbeck operator, formally deﬁned by L = − 1/2 ∆ + x · ∇, where ∆ and ∇ denote the standard Laplacian and gradient operators on R^d. It is well known that L is essentially self adjoint in L^2(g) and that the imaginary powers of L are bounded on L^p(g) for all p in (1,∞) and unbounded on L^1(g). In this talk we shall deﬁne two subspaces of L^1(g), denoted by H^1(g) and h^1(g), which may be thought of as analogues for Gaussian analysis of the classical Hardy space and the local Hardy space of D. Goldberg respectively. We shall then discuss the boundedness of singular integral operators, speciﬁcally of the “imaginary powers” of L, from each of these spaces to L^1(g). This is an account of recent works with G. Mauceri and with A. Carbonaro and Mauceri.

**Liens :**