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On Talagrand's convolution conjecture in Gaussian space

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We consider the Ornstein-Uhlenbeck convolution operator in Gaussian space, $f\to P_t[f]$. An easy fact is that every function, when convoluted with a small Gaussian noise, becomes $C_\infty$ smooth. This raises the question: is there any quantitative way of characterizing how quickly smoothing occurs under convolution? One natural way to quantify this is the so-called hypercontractivity property of the operator $P_t$: for every $t.0$ and $p.1$ there exists $q>p$ such that $P_t$ is a contraction from $L_p$ to $L_q$. This property, which is equivalent to a Log-Sobolev inequality has turned out to be extremely useful in several fields such as analysis of PDEs and quantum information theory. However, this is only meaningful when one has some a priori bound the $L_p$ norm of the initial function, for some $p>1$, and it is not clear if one can say anything about singular measures, for example. IN 1989 , Talagrand conjectured that for any non-negative function $f$ normalized to have integral $1$ over Gaussian space, the function $P_t[f]$ becomes smooth in the sense that the Gaussian measure of the set ${P_t[f](x) > \alpha}$ has Gaussian measure $o(1/\alpha)$, hence $P_t[f]$ satisfies an improved Markov inequality (this is dual to a certain isoperimetric-type bound). We prove this conjecture (this is joint work with James Lee).

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