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 $(\BR^d,\rho,\ga)$, where $\rho$ denotes the Euclidean distance and $\ga$ the Gauss measure. Our theory plays for the Ornstein-Uhlenbeck operator the same r\^ole that the classical Calder\`on-Zygmund theory plays for the Laplacian on $(\BR^d,\rho,\lambda),$ where $\la$ is the Lebesgue measure. Our method requires the introduction of two new function spaces: the space $BMO(\ga)$ of functions with ``bounded mean oscillation'' and its predual, the atomic Hardy space $H^1(\ga)$. We show that if $p$ is in $(2,\infty)$, then $\lp\ga$ is an intermediate space between $\ld\ga$ and $BMO(\ga)$, and that an inequality of John--Nirenberg type holds for functions in $BMO(\ga)$. Then we show that if $\cM$ is a bounded operator on $\ld\ga$ and the Schwartz kernels of $\cM$ and of its adjoint satisfy a \lq\lq local integral condition of H\"ormander type'', then $\cM$ extends to a bounded operator from $H^1(\ga)$ to $\lu\ga$, from $\ly\ga$ to $BMO(\ga)$ and on $\lp\ga$ for all $p$ in $(1,\infty)$. As an application, we show that certain singular integral operators related to the Ornstein--Uhlenbeck operator, which are unbounded on $\lu\ga$ and on $\ly\ga$, turn out to be bounded from $H^1(\ga)$ to $\lu\ga$ and from $\ly\ga$ to $BMO(\ga)$.