Related papers: Improved log-Sobolev inequalities, hypercontractiv…
We are interested in Sobolev type inequalities and their relationship with concentration properties on higher dimensions. We consider unbounded spin systems on the d-dimensional lattice with interactions that increase slower than a…
For a function $f$ on the hypercube $\{0,1\}^n$ with Fourier expansion $f=\sum_{S\subseteq[n]}\hat f(S)\chi_S$, the hypercontractive inequality allows bounding norms of $T_\rho f=\sum_S\rho^{|S|} \hat f(S)\chi_S$ in terms of norms of $f$.…
We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted)…
According to the theory of functional inequalities, a non-symmetric Markov semigroup has better properties than the corresponding symmetric one. For instance, there exist non-symmetric Markov semigroups which are hypercontractive (and thus…
This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity…
The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $\Sigma$ of the Euclidean space $\mathbb R^{n+m}$. Our first result, stated for…
The continuum $\varphi^4_2$ and $\varphi^4_3$ measures are shown to satisfy a log-Sobolev inequality uniformly in the lattice regularisation under the optimal assumption that their susceptibility is bounded. In particular, this applies to…
We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality: \begin{equation}\int |S_t u_0-S_t v_0|…
We establish a family of uncertainty principles for finite linear combinations of Hermite functions. More precisely, we give a geometric criterion on a subset $S\subset \RR^d$ ensuring that the $L^2$-seminorm associated to $S$ is equivalent…
In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and…
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…
We begin by studying semigroup estimates that are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for showing such log--Sobolev…
In a 2013 paper, the author showed that the convolution of a compactly supported measure on the real line with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal…
In the nice recent work [48], S. Wang established uniform log-Sobolev inequalities for mean field particles when the energy is flat convex. In this note we comment how to extend his proof to some semi-convex energies provided the curvature…
The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard…
By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of this proof is…
Recently, an entropic uncertainty relation for multiple measurements has been proposed by Liu et al. in [Phys. Rev. A 91, 042133 (2015)]. However, the lower bound of the relation is not always tight with respect to different measurements.…
We study Boolean functions on the $p$-biased hypercube $(\{0,1\}^n,\mu_p^n)$ through the lens of Fourier (spectral) entropy, i.e. the Shannon entropy of the squared $p$-biased Fourier coefficients. Motivated by recent restriction-based…
We present new proofs of two theorems of E.B. Davies and B. Simon about ultracontractivity property for of semigroups of operators and logarithmic Sobolev inequalities with parameter (LSIWP for short) satisfied by the generator of the…
For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log-Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very…