BMO and H^1 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 (R^d,rho, gamma), where rho denotes the Euclidean distance and gamma the Gauss measure. Our theory plays for the Ornstein-Uhlenbeck operator the same role that the classical Calderon-Zygmund theory plays for the Laplacian on (R^d, rho,lambda), where lambda is the Lebesgue measure. Our method requires the introduction of two new function spaces: the space BMO(gamma) of functions with "bounded mean oscillation" and its predual, the atomic Hardy space H^1(gamma). We show that if p is in (2, infinity), then L^p (gamma) is an intermediate space between L2(gamma) and BMO(gamma), and that an inequality of John-Nirenberg type holds for functions in BMO(gamma). Then we show that if M is a bounded operator on L2(gamma) and the Schwartz kernels of M and of its adjoint satisfy a "local integral condition of Hormander type," then M extends to a bounded operator from H^1(gamma) to L^1 (gamma), from L^infinity (gamma) to BMO(gamma) and on L^p(gamma) for all p in (1, infinity). As an application, we show that certain singular integral operators related to the Ornstein-Uhlenbeck operator, which are unbounded on L^1(gamma) and on L^infinity (gamma), turn out to be bounded from H^1 (gamma) to L^1 (gamma) and from L^infinity (gamma) to BMO(gamma).