In the setting of Euclidean space with the Gaussian measure , 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 , for . We determine which of them are bounded from to and from to BMO(). Here and BMO() 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.
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Giancarlo Mauceri, Stefano Meda, Peter Sjögren, Endpoint estimates for first-order Riesz transforms associated to the Ornstein–Uhlenbeck operator. Rev. Mat. Iberoam. 28 (2012), no. 1, pp. 77–91DOI 10.4171/RMI/667