Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings

$L^1$-Poincaré and Sobolev inequalities for differential forms in Euclidean spaces

In this paper, we prove Poincaré and Sobolev inequalities for differential forms in $L^1(\mathbb R^n)$. The singular integral estimates that it is possible to use for $L^p$, $p>1$, are replaced here with inequalities which go back to Bourgain-Brezis.

Riesz transform on manifolds and Poincaré inequalities

We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $p<2$ on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.

$L^1$-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups

In this paper, we prove interior Poincar{é} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis in Euclidean spaces, and to Chanillo-van Schaftingen in Heisenberg groups.

Higher order Poincare inequalities and Minkowski-type inequalities

We observe some higher order Poincare-type inequalities on a closed manifold, which is inspired by Hurwitz's proof of the Wirtinger's inequality using Fourier theory. We then give some geometric implication of these inequalities by applying them on the sphere. More specifically, by applying them to the support function of a convex hypersurface in the Euclidean space, we obtain some sharp Minkowski-type inequalities, such as a stability inequality for the classical Minkowski inequality and the Alexandrov-Fenchel inequality.

Poincar{é} and Sobolev inequalities for differential forms in Heisenberg groups

Poincar{é} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.

Poincaré and mean value inequalities for hypersurfaces in Riemannian manifolds and applications

In the first part of this paper we prove some new Poincaré inequalities, with explicit constants, for domains of any hypersurface of a Riemannian manifold with sectional curvatures bounded from above. This inequalities involve the first and the second symmetric functions of the eigenvalues of the second fundamental form of such hypersurface. We apply these inequalities to derive some isoperimetric inequalities and to estimate the volume of domains enclosed by compact self-shrinkers in terms of its scalar curvature. In the second part of the paper we prove some mean value inequalities and as co…

Improved L$^p$-Poincaré inequalities on the hyperbolic space

We investigate the possibility of improving the $p$-Poincaré inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge Λ_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $Λ_p:=[(N-1)/p]^{p}$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincaré-Hardy inequality, namely an improvement of the best $p$-Poincaré inequality in terms of the Hardy weight $r^{-p}$, $r$ being geodesic distance from a given pole. Certain Hardy-Maz'ya-type inequalities in the Euclidean half-space are also obtained.

Reverse Poincare inequalities, Isoperimetry, and Riesz transforms in Carnot groups

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

关于紧致流形上Poincaré-Sobolev型不等式的注记

A note on Poincaré-Sobolev type inequalities on compact manifolds

本文提出了一种在紧致流形上基于密度定义的偏差平均值的Poincaré-Sobolev型不等式，并探讨了其与密度Lebesgue范数的定量关系。

From Super Poincaré to Weighted Log-Sobolev and Entropy-Cost Inequalities

We derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\dd)}$-transportation cost inequality $$W^\rr_{2/(2-\dd)}(fμ,μ)^{2/(2-\dd)}\le Cμ(f\log f), μ(f)=1, f\ge 0$$ for some constant $C>0$, and they are equivalent if the curvature of the corresponding generator is bounded below. Weighted log-Sobolev and entropy-cost inequalities are als…

共形协变边界算子与Poincaré-Einstein流形上的尖锐高阶Sobolev迹不等式

Conformally Covariant Boundary Operators and Sharp Higher Order Sobolev Trace Inequalities on Poincaré-Einstein Manifolds

本文为Poincaré-Einstein流形引入共形协变边界算子，建立高阶Dirichlet问题以恢复分数阶GJMS算子，并推导出相关高阶迹Sobolev不等式，最终结合Beckner不等式得到球上的尖锐高阶迹不等式。

双曲空间上的尺度依赖Poincaré不等式、log-Sobolev不等式及海森堡不确定性原理的稳定性

Scale-Dependent Poincaré inequalities, log-Sobolev inequality and the stability of the Heisenberg Uncertainty Principle on the hyperbolic space

本文在双曲空间上建立尺度依赖的Poincaré-Hardy型恒等式，推导并改进多类不等式，获得海森堡不确定性原理的L2稳定性，并填补高斯测度对数Sobolev不等式的空白。

对双曲空间和黎曼流形上的对数Sobolev、泊松和Beckner不等式的研究

Logarithmic Sobolev, Poincaré and Beckner Inequalities on Hyperbolic Spaces and Riemannian Manifolds

本文研究了双曲空间和黎曼流形上的对数Sobolev、泊松和Beckner不等式，通过改进的Pólya-Szegö不等式和半群方法，揭示了这些不等式对几何和等周性质的依赖性。