BMO and H1 for the Ornstein–Uhlenbeck operator ✩

In this paper we develop a theory of singular integral operators acting on function spaces over the measured metric space (Rd,ρ,γ ), where ρ denotes the Euclidean distance and γ the Gauss measure. Our theory plays for the Ornstein–Uhlenbeck operator the same rôle that the classical Calderòn–Zygmund theory plays for the Laplacian on (Rd,ρ,λ), where λ is the Lebesgue measure. Our method requires the introduction of two new function spaces: the space BMO(γ ) of functions with “bounded mean oscillation” and its predual, the atomic Hardy space H1(γ ). We show that if p is in (2,∞), then Lp(γ ) is an intermediate space between L2(γ ) and BMO(γ ), and that an inequality of John–Nirenberg type holds for functions in BMO(γ ). Then we show that ifMis a bounded operator on L2(γ ) and the Schwartz kernels ofMand of its adjoint satisfy a “local integral condition of Hörmander type,” thenMextends to a bounded operator from H1(γ ) to L1(γ ), from L∞(γ ) to BMO(γ ) and on Lp(γ ) for all p in (1,∞). As an application, we show that certain singular integral operators related to the Ornstein–Uhlenbeck operator, which are unbounded on L1(γ ) and on L∞(γ ), turn out to be bounded from H1(γ ) to L1(γ ) and from L∞(γ ) to BMO(γ ). © 2007 Elsevier Inc. All rights reserved.