$L^p$-$L^q$ off-diagonal estimates for the Ornstein--Uhlenbeck   semigroup: some positive and negative results
  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. 
Alex Amenta, Jonas Teuwen
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