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

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.

Final version. To appear in the Bulletin of the Australian Mathematical Society