Regularization in $L_1$ for the Ornstein--Uhlenbeck semigroup

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.