Weighted fractional Poincaré inequalities via isoperimetric inequalities

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-08-22 DOI:10.1007/s00526-024-02813-6

Kim Myyryläinen, Carlos Pérez, Julian Weigt

引用次数: 0

Abstract

Our main result is a weighted fractional Poincaré–Sobolev inequality improving the celebrated estimate by Bourgain–Brezis–Mironescu. This also yields an improvement of the classical Meyers–Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincaré–Sobolev estimate with $A_p$ weights of Fabes–Kenig–Serapioni by means of a fractional type result in the spirit of Bourgain–Brezis–Mironescu. Examples are given to show that the corresponding $L^p$-versions of weighted Poincaré inequalities do not hold for $p>1$.

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通过等周不等式的加权分数波因卡雷不等式

我们的主要结果是一个加权分数 Poincaré-Sobolev 不等式，它改进了 Bourgain-Brezis-Mironescu 的著名估计。这也从几个方面改进了经典的 Meyers-Ziemer 定理。该证明基于分数等周不等式，即使在非加权设置中也是全新的。我们还通过布尔干-布雷齐斯-米罗内斯库（Bourgain-Brezis-Mironescu）精神中的分数型结果，扩展了法贝斯-凯尼格-塞拉皮奥尼（Fabes-Kenig-Serapioni）著名的具有（A_p\）权重的庞加莱-索博列夫估计（Poincaré-Sobolev estimate）。举例说明了加权 Poincaré 不等式的相应 $L^p$-versions 对于 $p>1$ 不成立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.

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