Revista Matemática Iberoamericana


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Volume 28, Issue 1, 2012, pp. 77–91
DOI: 10.4171/RMI/667

Published online: 2012-01-20

Endpoint estimates for first-order Riesz transforms associated to the Ornstein–Uhlenbeck operator

Giancarlo Mauceri[1], Stefano Meda[2] and Peter Sjögren[3]

(1) Università di Genova, Italy
(2) Università degli Studi di Milano-Bicocca, Italy
(3) Chalmers University of Technology, Göteborg, Sweden

In the setting of Euclidean space with the Gaussian measure $\gamma$, we consider all first-order Riesz transforms associated to the infinitesimal generator of the Ornstein–Uhlenbeck semigroup. These operators are known to be bounded on $L^p(\gamma)$, for $ 1< p< \infty$. We determine which of them are bounded from $H^1(\gamma)$ to $L^1(\gamma)$ and from $L^\infty(v)$ to BMO($\gamma$). Here $H^1(\gamma)$ and BMO($\gamma$) are the spaces introduced in this setting by the first two authors. Surprisingly, we find that the results depend on the dimension of the ambient space.

Keywords: Operator, Riesz transforms, Hardy spaces, BMO, Gaussian measure

Mauceri Giancarlo, Meda Stefano, Sjögren Peter: Endpoint estimates for first-order Riesz transforms associated to the Ornstein–Uhlenbeck operator. Rev. Mat. Iberoam. 28 (2012), 77-91. doi: 10.4171/RMI/667