Brascamp-Lieb Inequality Research Articles

Adjoint Brascamp–Lieb inequalities

AbstractThe Brascamp–Lieb inequalities are a generalization of the Hölder, Loomis–Whitney, Young, and Finner inequalities that have found many applications in harmonic analysis and elsewhere. In this paper, we introduce an “adjoint” version of these inequalities, which can be viewed as an version of the entropic Brascamp–Lieb inequalities of Carlen and Cordero–Erausquin. As applications, we reprove a log‐convexity property of the Gowers uniformity norms, and establish some reverse inequalities for various tomographic transforms. We conclude with some open questions.

$L^{p}$ improving properties and maximal estimates for certain multilinear averaging operators

In this article we focus on L^{p} estimates for two types of multilinear lacunary maximal averages over hypersurfaces with curvature conditions. Moreover, we give a different proof for the bilinear lacunary spherical maximal functions. To obtain our results, we make use of the L^{1} -improving estimates of multilinear averaging operators. We also obtain L^{p} -improving estimates for certain multilinear averages by means of the nonlinear Brascamp–Lieb inequality.

Loomis–Whitney inequalities on corank 1 Carnot groups

In this paper we provide another way to deduce the Loomis–Whitney inequality on higher dimensional Heisenberg groups \(\mathbb{H}^n\) based on the one on the first Heisenberg group \(\mathbb{H}^1\) and the known nonlinear Loomis–Whitney inequality (which has more projections than ours). Moreover, we generalize the result to the case of corank 1 Carnot groups and products of such groups. Our main tool is the modified equivalence between the Brascamp–Lieb inequality and the subadditivity of the entropy developed in Carlen and Cordero-Erausquin (2009).

Stability of the Borell–Brascamp–Lieb Inequality for Multiple Power Concave Functions

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry.

Stability of the Poincaré constant

We study stability of the sharp Poincaré constant of the invariant probability measure of a reversible diffusion process satisfying some natural conditions. The proof is based on the spectral interpretation of Poincaré inequalities and Stein’s method. In particular, these results are applied to gamma distributions, to the Brownian motion on spheres and to the Brascamp-Lieb inequality for one-dimensional log-concave measures.

Quantum Brascamp–Lieb Dualities

Brascamp–Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory—including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of ‘geometric’ Brascamp–Lieb inequalities.

Unifying the Brascamp-Lieb Inequality and the Entropy Power Inequality

The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential entropy of linear transformations of random vectors with independent components. The BLI, on the other hand, provides upper bounds on the differential entropy of a random vector in terms of the differential entropies of some of its linear transformations. In this paper, we define a family of entropy functionals, which we show are subadditive. We then establish that Gaussians are extremal for these functionals by adapting a proof technique from Geng and Nair (2014). As a consequence, we obtain a new entropy inequality that generalizes both the BLI and EPI. By considering a variety of independence relations among the components of the random vectors appearing in these functionals, we also obtain families of inequalities that lie between the EPI and the BLI.

Five-linear singular integral estimates of Brascamp–Lieb-type

We prove the full range of estimates for a five-linear singular integral of Brascamp-Lieb type. The study is methodology-oriented with the goal to develop a sufficiently general technique to estimate singular integral variants of Brascamp-Lieb inequalities that do not obey H\"older scaling. The invented methodology constructs localized analysis on the entire space from local information on its subspaces of lower dimensions and combines such tensor-type arguments with the generic localized analysis. A direct consequence of the boundedness of the five-linear singular integral is a Leibniz rule which captures nonlinear interactions of waves from transversal directions.

Multilinear duality and factorisation for Brascamp–Lieb-type inequalities

We initiate the study of a duality theory which applies to norm inequalities for pointwise weighted geometric means of positive operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the norm inequality under consideration. We relate our theory to the Maurey–Nikishin–Stein theory of factorisation of operators, and present a fully multilinear version of Maurey’s fundamental theorem on factorisation of operators through L^1 . The development of the theory involves convex optimisation and minimax theory, functional-analytic considerations concerning the dual of L^\infty , and the Yosida–Hewitt theory of finitely additive measures. We consider the connections of the theory with the theory of interpolation of operators. We discuss the ramifications of the theory in the context of concrete families of geometric inequalities, including Loomis–Whitney inequalities, Brascamp–Lieb inequalities and multilinear Kakeya inequalities.

Continuous Version of the Approximate Geometric Brascamp–Lieb Inequalities

Continuous Version of the Approximate Geometric Brascamp–Lieb Inequalities

Concentration inequalities for log-concave distributions with applications to random surface fluctuations

We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp–Lieb concentration inequality and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds. These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the ∇φ type. The classical Brascamp–Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of convex potentials whose second derivative vanishes only on a zero measure set, when the underlying graph is a d-dimensional discrete torus. The result applies, in particular, to potentials of the form U(x)=|x|p with p>1 and answers a question discussed by Brascamp–Lieb–Lebowitz (In Statistical Mechanics (1975) 379–390, Springer). Additionally, new tail probability bounds are obtained for the family of potentials U(x)=|x|p+x2, p>2. This result answers a question mentioned by Deuschel and Giacomin (Stochastic Process. Appl. 89 (2000) 333–354).

Positive Gaussian Kernels also Have Gaussian Minimizers

We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb’s results on maximizers of multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends several inverse inequalities.

Fourier duality in the Brascamp–Lieb inequality

AbstractIt was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

Regularized Brascamp–lieb Inequalities and an Application

AbstractWe present a certain regularized version of Brascamp–Lieb inequalities studied by Bennett, Carbery, Christ and Tao. Using the induction-on-scales method of Guth, these regularized inequalities lead to a generalization of the multilinear Kakeya inequality.

A family of sharp inequalities on real spheres

We prove a family of sharp multilinear integral inequalities on real spheres involving functions that possess some symmetries that can be described by annihilation by certain sets of vector fields. The Lebesgue exponents involved are seen to be related to the combinatorics of such sets of vector fields. Moreover, we derive some Euclidean Brascamp–Lieb inequalities localized to a ball of radius R, with a blow-up factor of type , where the exponent is related to the aforementioned Lebesgue exponents, and prove that in some cases δ is optimal.

An Algebraic Brascamp\u2013Lieb Inequality

The Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions f_jin L^{q_j}(mathbb {R}^{n_j}), for j=1,ldots ,m, under some corresponding linear maps B_j. This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where B_j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

Intertwinings, second-order Brascamp–Lieb inequalities and spectral estimates

We explore the consequences of the so-called intertwinings between gradients and Markov diffusion operators on $R^d$ in terms of second-order Brascamp-Lieb inequalities for log-concave distributions and beyond, extending our inequalities established in a previous paper. As a result, we derive some convenient lower bounds on the $(d+1)^{th}$ positive eigenvalue depending on the spectral gap of the dual Markov diffusion operator given by the intertwining. To see the relevance of our approach, we apply our spectral results in the case of perturbed product measures, freeing us from Helffer's classical method based on uniform spectral estimates for the one-dimensional conditional distributions.

On affine invariant and local Loomis–Whitney type inequalities

We prove various extensions of the Loomis–Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors w i of a not necessarily orthonormal basis of R n , or their orthogonal complements. In order to prove such inequalities, we estimate the constant in the Brascamp–Lieb inequality in terms of the vectors w i . Restricted and functional versions of the inequality will also be considered.

On the nonlinear Brascamp–Lieb inequality

We prove a nonlinear variant of the general Brascamp–Lieb inequality. Our proof consists of running an efficient, or “tight,” induction-on-scales argument, which uses the existence of Gaussian near-extremizers to the underlying linear Brascamp–Lieb inequality (Lieb’s theorem) in a fundamental way. A key ingredient is an effective version of Lieb’s theorem, which we establish via a careful analysis of near-minimizers of weighted sums of exponential functions. Instances of this inequality are quite prevalent in mathematics, and we illustrate this with some applications in harmonic analysis.

A variation on Hölder–Brascamp–Lieb inequalities

The class of Hölder–Brascamp–Lieb inequalities is a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis–Whitney inequalities. The full range of exponents was classified in a paper of Bennett, Carbery, Christ, and Tao [Math. Res. Lett. 17 (2010), no. 4, 647–666]. In a setting similar to that of Ivanisvili and Volberg [J. Lond. Mat. Sci (2) (2015), no. 3, 657–674], we introduce a notion of size for these inequalities which generalizes L p L^p norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Hölder–Brascamp–Lieb-type inequality to hold and establish sufficient conditions for maximizers to exist when the underlying linear maps match those of the convolution inequality of Young.