Spectral multipliers in a general Gaussian setting

Abstract

We investigate a class of spectral multipliers for an Ornstein-Uhlenbeck operator $\mathcal{L}$ in ${\mathbb{R}}^n$, with drift given by a real matrix $B$ whose eigenvalues have negative real parts. We prove that if $m$ is a function of Laplace transform type defined in the right half-plane, then $m(\mathcal{L})$ is of weak type $(1,1)$ with respect to the invariant measure in ${\mathbb{R}}^n$. The proof involves many estimates of the relevant integral kernels and also a bound for the number of zeros of the time derivative of the Mehler kernel, as well as an enhanced version of the Ornstein-Uhlenbeck maximal operator theorem.

Section 7 revised and split into two new sections (7 and 8)