Spectral multipliers in a general Gaussian setting

Journal article, 2026

We investigate a class of spectral multipliers for an Ornstein–Uhlenbeck
operator L in R^n, with drift given by a real matrix B whose eigenvalues have nega-
tive real parts. We prove that if m is a function of Laplace transform type defined
in the right half-plane, then m(L) is of weak type (1, 1) with respect to the in-
variant measure in 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.