Spectral multipliers in a general Gaussian setting

TL;DR

This paper studies spectral multipliers for a generalized Ornstein-Uhlenbeck operator in n, establishing weak type (1,1) bounds for Laplace transform type functions and providing kernel estimates and zero bounds for the Mehler kernel.

Contribution

It extends spectral multiplier results to a broad Gaussian setting with drift, including new kernel estimates and zero bounds for the Mehler kernel.

Findings

Spectral multipliers of Laplace transform type are weak type (1,1).

Derived new estimates for integral kernels of the Ornstein-Uhlenbeck semigroup.

Bound the number of zeros of the time derivative of the Mehler kernel.

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.