Abstract
We investigate $L^{p}(\unicode[STIX]{x1D6FE})$–$L^{q}(\unicode[STIX]{x1D6FE})$ 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.
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