Quantitative Borell-Brascamp-Lieb inequalities for compactly supported power concave functions (and some applications)

On Witten Laplacians and Brascamp-Lieb's inequality on manifolds with boundary

In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the Neumann and Dirichlet self-adjoint realizations of the Witten Laplacian. They moreover imply the usual Brascamp-Lieb's inequality and its generalization to compact Riemannian manifolds without boundary.

带有限多个输出函数的Borell–Brascamp–Lieb不等式

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。

Sharp trace Gagliardo-Nirenberg-Sobolev inequalities for convex cones, and convex domains

We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, aswell as a weighted sharp trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality, coming from the Brunn-Minkowski theory.

Hessian of Bellman functions and uniqueness of 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.

The Brascamp-Lieb inequality and its influence on Fourier analysis

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.

具有完全单项式权重的尖锐Poincaré不等式和Brascamp--Lieb不等式的稳定性

The stability of Sharp Poincaré and Brascamp--Lieb inequalities with full monomial weights

本文证明完全单项式权重高斯测度满足ICD(2,∞)条件，从而得到最优常数2的尖锐Poincaré不等式及Brascamp--Lieb稳定性，并借助Laguerre谱分解导出精确亏损恒等式和梯度稳定性。

Ball不等式的非线性变体

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。

Rearrangement methods in the work of Elliott Lieb

We review some topics in the theory of symmetric decreasing rearrangements with a particular focus on Lieb's fundamental contributions. Topics covered include the Brascamp--Lieb--Luttinger theorem, the sharp Young and Hardy--Littlewood--Sobolev inequalities, as well as the continuity of the rearrangement map on Sobolev spaces.

To logconcavity and beyond

In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity. In this paper, introducing a variation of concavity, we show that it preserves in fact a stronger property than logconcavity and we identify the strongest concavity preserved by the heat flow.