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 if M is a bounded operator on L2(γ) and the Schwartz kernels of M and of its adjoint satisfy a “local integral condition of Hörmander type,” then M extends 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).