Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

高阶离散与光滑等周不等式的一个统一方法

A unified approach to higher order discrete and smooth isoperimetric inequalities

本文提出统一方法，通过离散或光滑傅里叶分析构造线性算子，获得高阶Wirtinger不等式，从而推导出离散多边形、光滑曲线及Chernoff型的高阶等周不等式，并给出等周亏量的上下界。

Isoperimetric functional inequalities via the maximum principle: the exterior differential systems approach

The goal of this note is to give the unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems studied by R.~Bryant and P.~Griffiths. In this note we list several classical by now isopereimetric inequalities which can be proved in a unified way. This unified approach reduces them to the so-called exterior differential systems studied by Robert Bryant and Phillip Griffiths. To the best of our knowledge, this is the first article where this connection is used. After reviewing a list of classical inequalities (log-Sobolev inequali…

Some inequalities related to isoperimetric inequalities with partial free boundary

The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free boundary, and the other is an integral inequality (due to Duff [9]) for any nonnegative function under Schwarz equimeasurable rearrangement. These ingredients also allow us to establish some Moser-Trudinger type inequalities, and obtain some estimates on the principal frequency of a membrane with partial free boundary, which extend early results of Nehari [15] and …

基于Meyers-Ziemer框架的加权Sobolev不等式：测度、等周不等式和端点估计

Weighted Sobolev Inequalities via the Meyers--Ziemer Framework: Measures, Isoperimetric Inequalities, and Endpoint Estimates

本文基于Meyers-Ziemer框架，建立新的加权Sobolev不等式，扩展到加权变分函数，并提供端点估计和等周不等式，同时揭示新的双权Sobolev不等式。

Isoperimetric inequalities for Bergman analytic content

The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $Ω$ measures the $L^{p}(Ω)$-distance between $\overline{z}$ and the Bergman space $A^{p}(Ω)$ of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman $p$-analytic content in terms of the St Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.

通过等周不等式得到的加权分数阶 Poincaré 不等式

Weighted fractional Poincaré inequalities via isoperimetric inequalities

本文利用分数阶等周不等式证明加权分数阶 Poincaré-Sobolev 不等式，改进 Bourgain-Brezis-Mironescu 估计与 Meyers-Ziemer 定理，并推广含 $A_p$ 权的结果。