Related papers: Hypercontractive measures, Talagrand's inequality,…
Hypercontractive inequalities are a useful tool in dealing with extremal questions in the geometry of high-dimensional discrete and continuous spaces. In this survey we trace a few connections between different manifestations of…
We establish dimension-free quantum Talagrand-type inequalities with explicit constants on the quantum Boolean cube, via a unified variance-decay perspective. For individual observables, short-time variance decay along the depolarizing…
We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace…
We show that the Ornstein-Uhlenbeck semigroup associated with a general Poisson random measure is hypercontractive, whenever it is restricted to non-increasing mappings on configuration spaces. We deduce from this result some versions of…
We give a $L^2\times L^2 \rightarrow L^2$ convolution estimate for singular measures supported on transversal hypersurfaces in $\mathbb{R}^n$, which improves earlier results of Bejenaru, Herr & Tataru as well as Bejenaru & Herr. The arising…
In this article we study generalization of the classical Talagrand transport-entropy inequality in which the Wasserstein distance is replaced by the entropic transportation cost. This class of inequalities has been introduced in the recent…
We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove…
We continue our investigation of the intertwining relations for Markov semigroups and extend the results of [9] to multi-dimensional diffusions. In particular these formulae entail new functional inequalities of Brascamp-Lieb type for…
This note is concerned with an extension, at second order, of an inequality on the discrete cube $C_n=\{-1,1\}$ (equipped with the uniform measure) due to Talagrand (\cite{TalL1L2}). As an application, the main result of this note is a…
In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new…
The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly…
The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or…
This paper builds upon several recent works, where semigroup proofs of Brascamp-Lieb inequalities are provided in various settings (Euclidean space, spheres and symmetric groups). Our aim is twofold. Firstly, we provide a general, unifying,…
Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $\sigma_n$. Talagrand's influence inequality (1994) asserts that there exists $C\in(0,\infty)$ such that for every $n\in\mathbb{N}$,…
We use Brascamp-Lieb's inequality to obtain new decoupling inequalities for general Gaussian vectors, and for stationary cyclic Gaussian processes. In the second case, we use a version by Bump and Diaconis of the strong Szego limit theorem.…
We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to…
This note reviews the studies of the last decades emphasizing a common principle based on entropy, logarithmic Sobolev inequality and hypercontractivity, behind four most celebrated inequalities by M. Talagrand: the convex distance…
We prove intertwining relations by twisted gradients for Markov semi-groups. These relations are applied to Brascamp-Lieb type inequalities and spectral gap results. It generalizes the results of [1] from the Euclidean space to Riemannian…
We give a new approach, inspired by H\"ormander's $L^2$-method, to weighted variance inequalities which extend results obtained by Bobkov and Ledoux. It provides in particular a local proof of the dimensional functional forms of the…
Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on…