  Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$ and let
$(Q_t)$ be the Ornstein--Ulhenbeck semigroup. Eldan and Lee recently
established that for every non--negative function $f$ of integral $1$ and any
time $t$ the following tail inequality holds true: \[ \gamma_n ( \{ Q_t f > r
\} ) \leq C_t \, \frac{ (\log \log r)^4 }{r \sqrt{\log r}} , \quad \forall r>1
\] where $C_t$ is a constant depending on $t$ but not on the dimension. The
purpose of the present paper is to simplify parts of their argument and to
remove the $(\log \log r)^4$ factor.
