$L^p$-$L^q$ off-diagonal estimates for the Ornstein--Uhlenbeck semigroup: some positive and negative results

TL;DR

This paper studies the conditions under which $L^p$-$L^q$ off-diagonal estimates hold or fail for the Ornstein-Uhlenbeck semigroup, providing both positive results for large times and counterexamples for small times.

Contribution

It offers new insights into the time-dependent behavior of off-diagonal estimates for the Ornstein-Uhlenbeck semigroup, including explicit counterexamples.

Findings

Estimates hold for large $t$ in an unrestricted sense.

Estimates fail for small $t$ on certain geometric regions.

Counterexamples are constructed using Mehler kernel estimates.

Abstract

We investigate $L^p(\gamma)$-$L^q(\gamma)$ off-diagonal estimates for the Ornstein-Uhlenbeck semigroup $(e^{tL})_{t > 0}$. For sufficiently large $t$ (quantified in terms of $p$ and $q$) these estimates hold in an unrestricted sense, while for sufficiently small $t$ they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.