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.

N$$ N $$‐dimensional wave packet transform and associated uncertainty principles in the free metaplectic transform domain

The free metaplectic transformation (FMT) is an ‐dimensional linear canonical transform. This transform is much useful, especially in multidimensional signal processing and applications. In this paper, our aim is to achieve an efficient time‐frequency representation of higher‐dimensional nonstationary signals by introducing the novel free metaplectic wave packet transform (FM‐WPT) in , based on the elegant convolution structure associated with the free metaplectic transforms. The FM‐WPT preserves the properties of classical wave packet transform (WPT) in and has better mathematical properties. Further, the validity of the proposed transform is demonstrated via a lucid example. The preliminary analysis encompasses the derivation of fundamental properties of the novel FM‐WPT, including boundedness, reconstruction formula, Moyal's formula, and the reproducing kernel. To extend the scope of the study, we formulate several uncertainty inequalities, including Lieb's inequality, Pitt's inequality, logarithmic inequality, Heisenberg's uncertainty inequality, and Nazarov's uncertainty inequality for the proposed transform.

Lieb's Inequality for Continuous Modulated Shearlet Transform on LCA Groups

We have proved a version of Lieb's inequality for continuous modulated shearlet transform on locally compact Abelian (LCA) groups. An alternative proof for the inequality has also been provided using Riesz-Thorin interpolation theorem.

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.

Decoupling inequalities for quadratic forms

We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.

Short-time free metaplectic transform: Its relation to short-time Fourier transform in $ L^2(\mathbb R^n) $ and uncertainty principles

<abstract><p>The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various applications in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short-time free metaplectic transform (ST-FMT). In this paper, we investigate the ST-FMT. First we propose the definition of the ST-FMT and provide the time-frequency analysis of the proposed transform in the FMT domain. Second we establish the relationship between the ST-FMT and short-time Fourier transform (STFT) in $ L^2(\mathbb R^n) $ and investigate the basic properties of the proposed transform including the reconstruction formula, Moyal's formula. The emergence of the ST-FMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. We extend some different uncertainty principles (UPs) from quantum mechanics including Lieb's inequality, Pitt's inequality, Hausdorff-Young inequality, Heisenberg's UP, Hardy's UP, Beurling's UP, Logarithmic UP and Nazarov's UP. Finally, we give a numerical example and a possible applications of the proposed ST-FMT.</p></abstract>

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).

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.

One dimensional weighted Ricci curvature and displacement convexity of entropies

AbstractIn the present paper, we prove that a lower bound on the 1‐weighted Ricci curvature is equivalent to a convexity of entropies on the Wasserstein space. Based on such characterization, we provide some interpolation inequalities such as the Prékopa–Leindler inequality, the Borel–Branscamp–Lieb inequality, and the Brunn–Minkowski inequality under the curvature bound.

Uncertainty Principle for the Two-Sided Quaternion Windowed Linear Canonical Transform

In this paper, we investigate the (two-sided) quaternion windowed linear canonical transform (QWLCT) and study the uncertainty principles associated with the QWLCT. First, several important properties of the QWLCT such as bounded, shift, modulation and orthogonality relations are presented based on the spectral representation of the quaternionic linear canonical transform (QLCT). Second, Pitt’s inequality and the Lieb inequality for the QWLCT are explored. Moreover, we study different kinds of uncertainty principles for the QWLCT, such as the logarithmic uncertainty principle, the entropic uncertainty principle, the Lieb uncertainty principle and Donoho–Stark’s uncertainty principle. Finally, we provide a numerical example and a potential application to signal recovery by using Donoho–Stark’s uncertainty principle associated with the QWLCT.

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.

Continuous Spherical Gabor Transform for Gelfand Pair

Using the basic properties of the Gelfand pairs and its spherical Fourier transform, we introduce the spherical Gabor transform. We show its fundamental properties, such as Plancherel, Parseval and inversion formula. Finally, we establish a number of uncertainty principles (such as Donoho–Stark’s uncertainty principle , Lieb inequality and Beckner’s uncertainty principle ) associated with this transformation.

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.