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