Brascamp-Lieb Inequality Research Articles

Generating monotone quantities for the heat equation

Abstract The purpose of this article is to expose and further develop a simple yet surprisingly far-reaching framework for generating monotone quantities for positive solutions to linear heat equations in euclidean space. This framework is intimately connected to the existence of a rich variety of algebraic closure properties of families of sub/super-solutions, and more generally solutions of systems of differential inequalities capturing log-convexity properties such as the Li–Yau gradient estimate. Various applications are discussed, including connections with the general Brascamp–Lieb inequality and the Ornstein–Uhlenbeck semigroup.

Sharpness of the Brascamp--Lieb inequality in Lorentz spaces

We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.

Behaviour of the Brascamp–Lieb constant

Recent progress in multilinear harmonic analysis naturally raises questions about the local behaviour of the best constant (or bound) in the general Brascamp–Lieb inequality as a function of the underlying linear transformations. In this paper we prove that this constant is continuous, but is not in general differentiable.

Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities

In this work we consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. For this we use optimal transport methods and the Borell-Brascamp-Lieb inequality. These refinements can be written as a deficit in the classical inequalities. They have the right scale with respect to the dimension. They lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behaviour for the laws of solutions to stochastic differential equations.

Isotropic measures and stronger forms of the reverse isoperimetric inequality

The reverse isoperimetric inequality, due to Keith Ball, states that if K K is an n n -dimensional convex body, then there is an affine image K ~ \tilde {K} of K K for which S ( K ~ ) n / V ( K ~ ) n − 1 S(\tilde {K})^n/V(\tilde {K})^{n-1} is bounded from above by the corresponding expression for a regular n n -dimensional simplex, where S S and V V denote the surface area and volume functional. It was shown by Franck Barthe that the upper bound is attained only if K K is a simplex. The discussion of the equality case is based on the equality case in the geometric form of the Brascamp-Lieb inequality. The present paper establishes stability versions of the reverse isoperimetric inequality and of the corresponding inequality for isotropic measures.

The Borell–Ehrhard game

A precise description of the convexity of Gaussian measures is provided by sharp Brunn-Minkowski type inequalities due to Ehrhard and Borell. We show that these are manifestations of a game-theoretic mechanism: a minimax variational principle for Brownian motion. As an application, we obtain a Gaussian improvement of Barthe's reverse Brascamp-Lieb inequality.

BRASCAMP–LIEB INEQUALITY AND QUANTITATIVE VERSIONS OF HELLY'S THEOREM

We provide new quantitative versions of Helly's theorem. For example, we show that for every family of closed half-spaces in such that has positive volume, there exist and such that where are absolute constants. These results complement and improve previous work of Bárány et al and Naszódi. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp–Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality.

Riemannian metrics on ℝ n and Sobolev-type Inequalities

Poincare-type estimates for a logarithmically concave measure μ on a convex set Ω are obtained. For this purpose, Ω is endowed with a Riemannian metric g in which the Riemannian manifold with measure (Ω, g, μ) has nonnegative Bakry–Emery tensor and, as a corollary, satisfies the Brascamp–Lieb inequality. Several natural classes of metrics (such as Hessian and conformal metrics) are considered; each of these metrics gives new weighted Poincare, Hardy, or log-Sobolev type inequalities and other results.

Geometric stability via information theory

Geometric stability via information theory, Discrete Analysis, 2016:10, 28pp. Let $A$ be a subset of $\mathbb R^3$. Then we can project $A$ onto the $xy$-plane, the $yz$-plane and the $xz$-plane. If we are given the areas of these projections, how large can the volume of $A$ be? A natural example to consider is when $A$ is a cuboid aligned with the coordinate axes. In that case the volume of $A$ is the square root of the product of the areas of the projections. That turns out to be the best we can do, by a famous result of Loomis and Whitney. The Loomis-Whitney inequality is the following more general statement. Let $g_1,\dots,g_d$ be non-negative functions defined on $\mathbb R^{d-1}$ and for $1\leq i\leq d$ let $\tilde{\pi}_i:\mathbb R^d\to\mathbb R^{d-1}$ be the projection that removes the $i$th coordinate from a vector. Then $$\int_{\mathbb R^d}\prod_{i=1}^dg_i(\tilde{\pi}_i(x))\,dx\leq\prod_{i=1}^d\|g_i\|_{d-1}.$$ To see how this relates to the statement about volumes, consider the case where each $g_i$ takes values 0 and 1. Then we can think of $g_i$ as the characteristic function of a set $A_i$, and $\prod (g_i \circ \tilde{\pi}_i)$ is then the characteristic function of the largest set $A$ such that $\tilde{\pi}_i(A)\subset A_i$ for each $i$. Also, $\|g_i\|_{d-1}=|A_i|^{1/(d-1)}$, so the right-hand side is equal to the product of the $(d-1)$th roots of the volumes of the $A_i$. Setting $d=3$ we obtain the result stated in the first paragraph. The Loomis-Whitney inequality has been further generalized. For example, the uniform-cover inequality of Chung, Frankl, Graham and Shearer (which is equivalent to the Box Theorem of Bollobas and Thomason) relates the volume of a set to the volumes of lower-dimensional projections (where these dimensions may differ), and the Brascamp-Lieb inequality can be used to relate the volume of a set to the volumes of projections that are not necessarily orthogonal to each other. Given any inequality, it is natural to try to characterize the equality cases, which for the Loomis-Whitney and uniform-cover inequalities are given by products of subsets of $\mathbb{R}$. It is also interesting, and often very useful, to understand the _near_-equality cases. The aim here tends to be to prove _stability_ results, which state that if equality holds, then the example for which it holds is close (in some appropriate sense) to one for which it holds exactly. This paper proves stability versions of the Loomis-Whitney and uniform-cover inequalities, and obtains as a consequence a stability version of the edge-isoperimetric inequality in the grid $\mathbb Z^d$. This last inequality states that if $A$ is a subset of $\mathbb{Z}^d$ of cardinality $m$, then the number of edges incident with points in $A$ is minimized when $A$ is a cuboid (in the obvious sense). Returning to the result mentioned at the beginning, it follows from the main result of this paper that if $A\subset\mathbb R^3$ has volume as large as the product of the square roots of the areas of its three projections, then there exist three subsets $K_1,K_2,K_3 \subset \mathbb{R}$ such that the measure of the symmetric difference $A\bigtriangleup (K_1 \times K_2 \times K_3)$ is small. To prove a stability version of an inequality, it is sometimes enough to examine carefully the proof that gives the equality cases and replace each step by its almost version. For example, it is an easy exercise to prove a stability version of the Cauchy-Schwarz inequality, which will say that if $u$ and $v$ are vectors in an inner-product space and the real part of the inner product $\langle u,v\rangle$ is at least $(1-\epsilon)\|u\|_2\|v\|_2$, then the unit vectors $u/\|u\|_2$ and $v/\|v\|_2$ are close. (More precisely, their distance is at most $\sqrt{2\epsilon}$, and this cannot be improved.) A naive argument of this kind can be used for the Loomis-Whitney inequality as well (based on the proof of the Loomis-Whitney inequality which uses Holder's inequality and induction on the dimension), but it does not give an optimal dependence on the closeness parameter $\epsilon$. In general for stability results, having an optimal dependence is highly desirable, because it means that in a certain sense one has a complete structural characterization of the near-extremal cases. That is, the fact that near-equality holds implies structural information, and that structural information implies that near-equality holds, and if one applies both implications, one ends up back where one started without any appreciable loss of information. The authors base their arguments on one of the well-known approaches to proving the Loomis-Whitney inequality: entropy methods. The main difficulty they face is that entropy methods yield probability distributions, whereas they need to end up with a set (in fact, a Cartesian product of 1-dimensional sets). Considerable care is needed to prove that in moving back from distributions to sets, they do not throw too much away. This requires them to introduce a new measure of distance, which they call _hole weight_.

Convolution estimates for singular measures and some global nonlinear Brascamp—Lieb inequalities

We give an L2 x L2 → L2 convolution estimate for singular measures supported on transversal hypersurfaces in ℝn, which improves earlier results of Bejenaru et al. as well as Bejenaru and Herr. The quantities arising are relevant to the study of the validity of bilinear estimates for dispersive partial differential equations. We also prove a class of global, nonlinear Brascamp–Lieb inequalities with explicit constants in the same spirit.

Hessian of Bellman functions and uniqueness of the Brascamp–Lieb inequality

Under some assumptions on the vectors $a_{1},..,a_{n} \in\mathbb{R}^{k}$ and the function $B : \mathbb{R}^{n} \to \mathbb{R}$ we find the sharp estimate of the expression $\int_{\mathbb{R}^{k}} B(u_{1}(a_{1}\cdot x),..., u_{n}(a_{n}\cdot x))dx$ in terms of $\int_{\mathbb{R}}u_{j}(y)dy, j=1,...,n.$ In some particular case we will show that these assumptions on $B$ imply that there is only one Brascamp--Lieb inequality.

Improved Hölder and reverse Hölder inequalities for Gaussian random vectors

We propose algebraic criteria that yield sharp Hölder types of inequalities for the product of functions of Gaussian random vectors with arbitrary covariance structure. While our lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the geometric Brascamp–Lieb inequality for Gaussian measures. As an application, we retrieve the Gaussian hypercontractivity as well as its reverse and we present a generalization of the sharp Young and reverse Young inequalities. From the latter, we recover several known inequalities in the literature including the Prékopa–Leindler and Barthe inequalities.

Limit theorems for additive functionals of stationary fields, under integrability assumptions on the higher order spectral densities

We obtain central limit theorems for additive functionals of stationary fields under integrability conditions on the higher-order spectral densities. The proofs are based on the Hölder–Young–Brascamp–Lieb inequality.

Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems

AbstractWe give a short proof of the Brascamp–Lieb theorem, which asserts that a certain general formof Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prèkopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.

A Brascamp-Lieb type covariance estimate

In this article, we derive a new covariance estimate. The estimate has a similar structure as the Brascamp-Lieb inequality and is optimal for ferromagnetic Gaussian measures. It can be naturally applied to deduce decay of correlations of lattice systems of continuous spins. We also discuss the relation of the new estimate with known estimates like a weighted estimate due to Helffer & Ledoux. The main ingredi.ent of the proof of the new estimate is a directional Poincare inequality which seems to be unknown.

Log-Concavity and Strong Log-Concavity: a review.

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on ℝ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

A connection of the Brascamp-Lieb inequality with Skorokhod embedding

We reveal a connection of the Brascamp-Lieb inequality with Skorokhod embedding. Error bounds for the inequality in terms of variance are also provided.

A refinement of the Brascamp–Lieb–Poincaré inequality in one dimension

In this short note, we give a refinement of the Brascamp–Lieb inequality in the style of the Houdré–Kagan extension for the Poincaré inequality in one dimension. This is inspired by works by Helffer and by Ledoux.

Dimensional variance inequalities of Brascamp–Lieb type and a local approach to dimensional Prékopaʼs theorem

We give a new approach, inspired by Hörmanderʼs L2-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 Brunn–Minkowski inequalities. We also present several applications of these variance inequalities, including reverse Hölder inequalities for convex functions, weighted Brascamp–Lieb inequalities and sharp weighted Poincaré inequalities for generalized Cauchy measures.

On small deviations of stationary Gaussian processes and related analytic inequalities

Let \( \{ X_j, j\in {\mathbb Z}\}\) be a Gaussian stationary sequence having a spectral function F of infinite type. Then for all n and z ≥ 0, $$ {\mathbb P}\left\{ \sup_{j=1}^n |X_j|\le z \right\}\le \Bigg( \int_{-z/\sqrt{G(f)}}^{z/\sqrt{G(f)}} e^{-x^2/2}\frac{{\rm d} x}{\sqrt{2\pi}} \Bigg)^n, $$ where G(f) is the geometric mean of the Radon Nykodim derivative of the absolutely continuous part f of F. The proof uses properties of finite Toeplitz forms. Let \( \{X(t), t\in {\mathbb R}\}\) be a sample continuous stationary Gaussian process with covariance function γ > 0. We also show that there exists an absolute constant K such that for all T > 0, a > 0 with T ≥ ε(a), $$ {\mathbb P} \bigg\{ \sup_{0\le s,t\le T} |X(s)-X(t)|\le a\bigg\} \le \exp \Bigg\{-{ KT \over {\varepsilon}(a) p({\varepsilon}(a))}\Bigg\}, $$ where \({\varepsilon} (a)= \min\big\{ b>0: \delta (b)\ge a\big\}\), \(\delta (b)=\min_{u\ge 1}\{ \sqrt{2(1-{\gamma}((ub))}, u\ge 1\}\), and \(p(b) = 1+\sum_{j=2}^\infty {|2{\gamma} (jb )-{\gamma} ((j-1)b )-{\gamma} ((j+1)b )|}/{2(1-{\gamma}(b))}\). The proof is based on some decoupling inequalities arising from Brascamp-Lieb inequality, from which we also derive a general upper bound for the small values of stationary sample continuous Gaussian processes. A two-sided inequality for correlated suprema in the case of the Ornstein-Uhlenbeck process. We also establish an unexpected link between the Littlewood hypothesis and small values of cyclic Gaussian processes. In the discrete case, we obtain a general bound by combining Anderson’s inequality with a weighted inequality for quadratic forms. In doing so, we also clarify link between matrices with dominant principal diagonal and Geršgorin’s disks. Both approaches are developed and compared on examples.