  Under the Ornstein-Uhlenbeck semigroup $\{U_t\}$, any non-negative measurable
$f : \mathbb R^n \to \mathbb R_+$ exhibits a uniform tail bound better than
that implied by Markov's inequality and conservation of mass: For every $\alpha
\geq e^3$, and $t > 0$, \[ \gamma_n\left(\left\{x \in \mathbb R^n : U_t f(x) >
\alpha \int f\,d\gamma_n\right\}\right) \leq C(t) \frac{1}{\alpha}
\sqrt{\frac{\log \log \alpha}{\log \alpha}}\] where $\gamma_n$ is the
$n$-dimensional Gaussian measure and $C(t)$ is a constant depending only on
$t$. This confirms positively the Gaussian limiting case of Talagrand's
convolution conjecture (1989).
  This is shown to follow from a more general phenomenon. Suppose that $f :
\mathbb{R}^n \to \mathbb{R}_+$ is {\em semi-log-convex} in the sense that for
some $\beta > 0$, for all $x \in \mathbb{R}^n$, the eigenvalues of $\nabla^2
\log f(x)$ are at least $-\beta$. Then $f$ satisfies a tail bound
asymptotically better than that implied by Markov's inequality.
