L-p-L-q OFF-DIAGONAL ESTIMATES FOR THE ORNSTEIN-UHLENBECK SEMIGROUP: SOME POSITIVE AND NEGATIVE RESULTS

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