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arXiv · 2202.01547

Spectral multipliers in a general Gaussian setting

Valentina Casarino·Paolo Ciatti·Peter Sjögren
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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.

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Valentina Casarino, Paolo Ciatti, Peter Sjögren. 2022-02-03. Spectral multipliers in a general Gaussian setting. https://arxiv.org/abs/2202.01547

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