Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $\mathbb {L}^{1}$

It is well known that some important Markov semi-groups have a “regularization effect” – as for example th hypercontractivity property of the noise operator on Boolean hypercube or Ornstein-Uhlenbeck semi-group real line, which applies to functions in Lp p > 1. Talagrand had conjectured 1989 has further subtle regularization are just integrable, but this conjecture remains open. Nonetheless, Gaussian analogue was proven recent years by Eldan-Lee and Lehec, combining an inequality log-Hessian with new deviation log-semi-convex under measure. In work, we explore question how much more general phenomenon is. Specifically, our first goal validity both these ingredients diffusion $\mathbb {R}^{n}$ , $M/M/\infty $ queue non-negative integers Laguerre positive line. Our second prove one-dimensional effect settings, even those cases where not valid.