Jonathan Bennett, Anthony Carbery, Michael Christ, Terence Tao
A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalities of Holder, Young, and Loomis-Whitney. This is a companion to a recent paper by the same authors, containing a different proof and discussing certain variants.
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We prove a singular Brascamp-Lieb inequality, stated in Theorem 1, with a large group of involutive symmetries.
We present a regularized version of Hölder-Brascamp-Lieb inequalities studied by Bennett, Carbery, Christ, and Tao. These inequalities lead to a generalization of the multilinear Kakeya inequality.
We prove a sharp common generalization of endpoint multilinear Kakeya and local discrete Brascamp-Lieb inequalities.
Certain rearrangement inequalities of a type considered by Hardy, Riesz, and Brascamp-Lieb-Luttinger are studied. Subsets of the real line that extremize these inequalities are characterized. Our results apply only to special cases, and complement earlier results by Burchard treating other cases. This draft incorporates minor corrections and clarifications.
Quiver Brascamp-Lieb inequalities
该研究针对二部箭图表示提出广义Brascamp-Lieb不等式，给出其成立的充要条件，且证实高斯函数无法让部分此类不等式达到饱和，修正了相关认知。
It 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-…
We present an overview of results on multi-linear singular integrals in the broader context of Brascamp-Lieb inequalities. This elaborates a lecture given at the inspiring conference on Geometric Aspects of Harmonic Analysis at Cortona 2018 in honor of Fulvio Ricci.
Discrete nonlinear Hölder--Brascamp--Lieb inequalities: a local approach
研究\(\mathbb{Z}^n\)上非线性赫尔德 - 布拉斯坎普 - 利布不等式，通过嵌入整数到紧致化空间应用局部理论，得出\(\mathbb{Z}^n\)中合适子格上多线性泛函的界。
We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis.
Remarks on Gaussian-stability for Brascamp--Lieb Inequalities
本文研究了Brascamp-Lieb不等式在勒贝格指数严格介于1和2之间的高斯稳定性问题，证明了近极值情况接近高斯分布。
We use the method of induction-on-scales to prove certain diffeomorphism invariant nonlinear Brascamp--Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru--Herr--Tataru and Bennett--Carbery--Wright.
A note on ubiquity of geometric Brascamp-Lieb data
本文借助 Garg 等人的工作证明了几何 Brascamp-Lieb 数据具有普遍性，从而回答了 Bennett 和 Tao 在伴随 Brascamp-Lieb 不等式研究中提出的问题。
We establish a set of necessary conditions and a set of sufficient conditions for boundedness of a family of Brascamp-Lieb forms in Lorentz spaces and $L^p$-spaces with power weights. The conditions are close to optimal.
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.
A symmetrization inequality of Rogers and of Brascamp-Lieb-Luttinger states that for a certain class of multilinear integral expressions, among tuples of sets of prescribed Lebesgue measures, tuples of balls centered at the origin are among the maximizers. Under natural hypotheses, we characterize all maximizing tuples for these inequalities for dimensions strictly greater than 1. We establish a sharpened form of the inequality.
Estimates for more Brascamp-Lieb forms in $L^p$-spaces with power weights
本文研究一类 Brascamp-Lieb 形式在带幂权重的 $L^p$ 空间上的有界性，给出了接近充要的条件。
We prove a nonlinear variant of the general Brascamp-Lieb inequality. Instances of this inequality are quite prevalent in analysis, and we illustrate this with substantial applications in harmonic analysis and partial differential equations. Our proof consists of running an efficient, or "tight", induction on scales argument, which uses the existence of gaussian near-extremisers 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-minimisers of weigh…
The Hölder-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ 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 extremizers to exist when the underlyi…
On the family of singular Brascamp-Lieb inequalities with dimension datum $(1, 2, 2, 1)$
本文分类了维度数据为$(1,2,2,1)$的奇异Brascamp-Lieb不等式族，确定了Lebesgue指数的精确范围，并给出了关键估计的简单证明及有界性反例。
We establish a nonlinear generalisation of the classical Brascamp-Lieb inequality in the case where the Lebesgue exponents lie in the interior of the finiteness polytope. As a corollary we show that the best constant in Young's convolution inequality in a small neighbourhood of the identity of a general Lie group, approaches the euclidean constant as the size of the neighbourhood approaches zero, answering a question of Cowling, Martini, Müller and Parcet. Our proof consists of running an efficient, or "tight", induction on scales argument which uses the existence of gaussian extremisers to th…
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 quantities are relevant in 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.
We prove a range of $L^p$ bounds for singular Brascamp-Lieb forms with cubical structure. We pass through sparse and local bounds, the latter proved by an iteration of Fourier expansion, telescoping, and the Cauchy-Schwarz inequality. We allow $2^{m-1}<p\le \infty$ with $m$ the dimension of the cube, extending an earlier result that required $p=2^m$. The threshold $2^{m-1}$ is sharp in our theorems.
This paper establishes a necessary and sufficient condition for $L^p$-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence relations. The testing condition involves bounding the average of an inverse power of certain Jacobian-type quantities along fibers of associated projections and covers many widely-studied special cases, including convolution with measures on nondegenerate hypersurfaces or on nondegenerate curves. The heart of the proof is based on Guth's visibility lemma in o…
An inequality of Brascamp-Lieb-Luttinger generalizes the Riesz-Sobolev inequality, stating that certain multilinear functionals, acting on nonnegative functions of one real variable with prescribed distribution functions, are maximized when these functions are symmetrized. It is shown that under certain hypotheses, when the functions are indicator functions of sets of prescribed measures, then up to the natural translation symmetries of the inequality, the maximum is attained only by intervals centered at the origin. Moreover, a quantitative form of this uniqueness is established, sharpening t…
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 Hölder 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 whi…
We prove the folklore endpoint multilinear $k_j$-plane conjecture originated from the paper \cite{bennett2006multilinear} of Bennett, Carbery and Tao. Along the way we prove a more general result, namely the endpoint multilinear $k_j$-variety theorem. Finally, we generalize our results to the endpoint perturbed Beascamp-Lieb inequality using techniques in this paper.
Inequalities of Brascamp-Lieb type on the Heisenberg group
本文研究Heisenberg群上的Brascamp–Lieb型不等式，刻画相关多重线性广义Radon变换的$L^p$有界性，并指出其充要条件在部分重要情形一致、其余情形存在差异。
An Algebraic Brascamp-Lieb Inequality
本文利用 Zhang 和 Zorin-Kranich 的多线性 Kakeya 型不等式，对多项式与有理映射证明整体非线性 Brascamp-Lieb 不等式，并通过仿射不变权得到仅依赖映射次数的一致常数。
Finiteness of the Hölder-Brascamp-Lieb Constant Revisited
本研究针对Hölder-Brascamp-Lieb常数有限性的刻画问题，提出了一种基于子空间有向图的全新互补性刻画方式，可通过分析有限子空间列表判定常数有限性，其证明是对已有半显式因式分解算法的显式化。
An analytic approach to the ubiquity of geometric Brascamp-Lieb data
通过解析方法证明几何Brascamp-Lieb数据在所有可行数据中的稠密性，并将结果推广到箭图Brascamp-Lieb数据。
Mixed-norm Brascamp-Lieb inequalities
本文建立混合范数Brascamp-Lieb不等式框架，确定经典情形未覆盖的新区域，用张量积技巧改进Christ的论证并获端点结果，还探讨其与Kakeya猜想的联系及独特难点。
Effective Brascamp-Lieb inequalities
本文研究了加权线性映射相关Brascamp-Lieb常数的有效上界问题，提出了一种新的方法来确定该上界。
This paper considers the problem of establishing $L^p$-improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension $n$ of the ambient space and the dimension $k$ of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang and…
Brascamp-Lieb inequality is an important mathematical tool in analysis, geometry and information theory. There are various ways to prove Brascamp-Lieb inequality such as heat flow method, Brownian motion and subadditivity of the entropy. While Brascamp-Lieb inequality is originally stated in Euclidean Space, discussed Brascamp-Lieb inequality for discrete Abelian group and discussed Brascamp-Lieb inequality for Markov semigroups. Many mathematical inequalities can be formulated as algebraic inequalities which asserts some given polynomial is nonnegative. In 1927, Artin proved that any non- neg…
On trilinear singular Brascamp-Lieb integrals
本研究运用有限维代数表示论结果完成三线性奇异Brascamp-Lieb形式的全分类，提出旋转方法分解高维奇异积分核，并证明相关形式的新界与条件界，为高维双线性Hilbert变换的界研究提供路线图。
The optimal constants are found for Lebesgue norm multilinear inequalities of Holder-Brascamp-Lieb type for arbitrary discrete Abelian groups. Previously a criterion for finiteness of the constants had been established for finitely generated Abelian groups, and the optimal constant had been found in the torsion-free case. The main step here is the analysis of finite groups.
Localised variants of multilinear restriction
本文研究多重线性限制定理的局部化变体，利用 Kakeya-Brascamp-Lieb 不等式理论给出 Bejenaru 定理的新证明，并据此获得实质性推广。
A new proof is given for the fact that centered gaussian functions saturate the Euclidean forward-reverse Brascamp-Lieb inequalities, extending the Brascamp-Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp-Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure and gaussian-extremizability for the Brascamp-Lieb inequality due to Bennett, Carbery, Christ and Tao are generalized to the setting of the forward-reverse Brascamp-Lieb ine…
Regularized Brascamp--Lieb inequalities
本文采用热流方法研究正则化输入函数下的逆向Brascamp–Lieb不等式，将Barthe–Wolff定理推广到该情形，并在正-逆向Brascamp–Lieb不等式框架下推导出正则化不等式，重新得到相关高斯饱和性质。
Brascamp--Lieb inequalities for fractal dimensions
本文利用泛函分析中的Brascamp--Lieb不等式，证明了关于分形集的上盒维数、包络维数和Assouad维数的新不等式，这些不等式以某些投影的维数为条件。与Hausdorff和下盒维数不同，类似不等式不成立。作者将这些分形Brascamp--Lieb不等式应用于正交投影的例外集估计，并为某些受限和集提供精确的维数估计，同时通过非线性Brascamp--Lieb不等式建立了类似的非线性不等式。
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.
From the Sphere to the Grassmannian: Continuous Brascamp--Lieb Inequalities via Measure Approximation and Heat Flow
本文研究球面与格拉斯曼流形上各向同性测度的连续Brascamp--Lieb不等式，通过离散逼近与热半群保持两种方法给出证明，并概述混合秩情形。
We consider the Brascamp--Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse the inequalities from a structural perspective. We obtain two new proofs of Lieb's fundamental theorem concerning exhaustion by gaussians. Our techniques are partly based upon monotonicity formulas for po…
Adjoint Brascamp-Lieb inequalities
本文提出 Brascamp-Lieb 不等式的伴随形式，将其刻画为熵版本的 $L^p$ 对应物，并用于重证 Gowers 范数对数凸性及建立层析变换的反向不等式。
Algebraicity of the Brascamp-Lieb constants
研究布拉斯坎普 - 利布常数的代数性，通过证明其是可行数据集合上的半代数函数，得出它满足多项式关系，在更一般的二分箭图表示相关箭图布拉斯坎普 - 利布常数设置下也有类似结论。
We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum, and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work of Lieb and the second is inspired by the work of Bennett et al. Recent works of Garg et al.and Allen-Zhu et al. also imply a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the opera…
Brascamp-Lieb inequalities have been important in analysis, mathematical physics and neighboring areas. Recently, these inequalities have had a deep influence on Fourier analysis and, in particular, on Fourier restriction theory. In this article we motivate and explain this connection. A lot of our examples are taken from a rapidly developing subarea called "decoupling". It is the author's hope that this article will be accessible to graduate students in fields broadly related to analysis.
From Brunn-Minkowski to Prékopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities
本文通过Brunn-Minkowski不等式推导出Prékopa-Leindler和Borell-Brascamp-Lieb不等式，并证明了其在整数空间上的离散形式。
The Brascamp--Lieb inequality on compact Lie groups and its extinction on homogeneous Lie groups
研究了在紧致李群和齐性李群上Brascamp-Lieb不等式的性质，确定了其常数的表达式和有限性条件。
Operator capacity, the Brascamp--Lieb inequality and geometric programming
本文通过几何规划视角探讨算子容量与布拉斯坎普-利布常数，提出新的研究成果，包括算子容量的近似极小值和局部霍尔德正则性，以及推广到有向图数据容量的可能。
Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $σ$ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations, we introduce the Brascamp-Lieb operator $T_{V,σ}$ associated to $(V,σ)$ and study its capacity, denoted by $\mathbf{D}_Q(V, σ)$. When $Q$ is the $m$-subspace quiver, the capacity of quiver data is intimately related to the Brascamp-Lieb constants that occur in the $m$-multilinear Brascamp-Lieb inequality in analysis. We show that the positivity of $\mathbf{D}_Q(V, σ)$ is equivalent to the $σ$-semi-stab…
Hölder-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of $L^p$ norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suitable data, have characterized the set of all tuples of exponents for which such an inequality holds for specified data, as the convex polyhedron defined by a particular finite set of affine inequalities. In this paper we advance the theory of such inequ…
A Nonlinear Variant of Ball's Inequality
本文通过尺度归纳论证，在特定热流下建立非线性Brascamp-Lieb泛函的全局近单调性，并由此得到全局非线性Brascamp-Lieb不等式有限性的稳定性结果。
The celebrated Brascamp-Lieb (BL) inequalities (and their extensions) are an important mathematical tool, unifying and generalizing numerous inequalities in analysis, convex geometry and information theory. While their structural theory is very well understood, far less is known about computing their main parameters. We give polynomial time algorithms to compute feasibility of BL-datum, the optimal BL-constant and a weak separation oracle for the BL-polytope. The same result holds for the so-called Reverse BL inequalities of Barthe. The best known algorithms for any of these tasks required at …
Combinatorial Bounds for List Recovery via Discrete Brascamp--Lieb Inequalities
本文利用离散熵Brascamp--Lieb不等式，为多类线性码和折叠线性码的列表恢复问题建立了新的组合界，解决了列表大小能否被ℓ的多项式界定的长期开放问题。
Exceptional set estimate through Brascamp-Lieb inequality
本文利用 Brascamp-Lieb 不等式为例外集估计 $T(a,s)$ 建立新上界，改进 He 的结果，并构造例子确定特定参数下 $T(a,s)$ 的精确值。
Borell--Brascamp--Lieb inequality with finitely many output functions
本文将经典Borell–Brascamp–Lieb不等式扩展为每个输入函数对应不同输出函数的形式，建立了含加权幂平均、幂变换的新不等式，其特例可还原经典不等式或对应多输出Prékopa–Leindler不等式的特例。
We consider a special class of the multi-linear forms studied by Brascamp and Lieb. For these forms, we are able to characterize the L^p spaces for which the form is bounded. We use this characterization to study a non-linear map that arises in scattering theory.
This paper gives a complete geometric characterization in all dimensions and codimensions of those Radon-like transforms which, up to endpoints, satisfy the largest possible range of local $L^p \rightarrow L^q$ inequalities permitted by quadratic-type scaling. The necessary and sufficient curvature-type criterion is phrased in terms of an associated Newton-like diagram. In the case of averages over families of polynomial graphs, the curvature condition implies sharp endpoint estimates as well. The proof relies on the recently-developed multilinear Radon-Brascamp-Lieb testing criterion and a re…
This work studies the application of the discrete Holder-Brascamp-Lieb (HBL) inequalities to the design of communication optimal algorithms. In particular, it describes optimal tiling (blocking) strategies for nested loops that lack data dependencies and exhibit linear memory access patterns. We attain known lower bounds for communication costs by unraveling the relationship between the HBL linear program, its dual, and tile selection. The methods used are constructive and algorithmic. The case when all arrays have one index is explored in depth, as a useful example in which a particularly eff…
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