A Bourgain–Brezis–Mironescu–Dávila theorem in Carnot groups of step two

IF 0.7 4区 数学 Q2 MATHEMATICS
Nicola Garofalo, Giulio Tralli
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引用次数: 12

Abstract

In this note we prove the following theorem in any Carnot group of step two $\mathbb{G}$:\[\lim_{s \nearrow 1/2} (1 - 2s) \mathfrak{P}_{H,s} (E) = \frac{4}{\sqrt{\pi}} \mathfrak{P}_H (E).\]Here, $\mathfrak{P}_H (E)$ represents the horizontal perimeter of a measurable set $E \subset \mathbb{G}$, whereas the nonlocal horizontal perimeter $\mathfrak{P}_{H,s} (E)$ is a heat based Besov seminorm. This result represents a dimensionless sub-Riemannian counterpart of a famous characterisation of Bourgain–Brezis–Mironescu and Dávila.
二阶卡诺群中的布尔干-布雷齐斯-米罗内斯库-达维拉定理
在本说明中,我们将在任何卡诺群的第二步$\mathbb{G}$中证明以下定理:\[\lim_{s \nearrow 1/2} (1 - 2s) \mathfrak{P}_{H,s} (E) = \frac{4}\{sqrt{pi}}.\这里,$mathfrak{P}_H (E)$ 表示可测量集合 $E (子集)\mathbb{G}$ 的水平周长,而非局部水平周长 $\mathfrak{P}_{H,s} (E)$ 是基于热的贝索夫半矩阵。这一结果代表了布尔甘-布雷齐斯-米罗内斯库和达维拉的著名特征的无量纲次黎曼对应。
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
CiteScore
1.60
自引率
0.00%
发文量
4
审稿时长
>12 weeks
期刊介绍: Publishes high-quality papers on subjects related to classical analysis, partial differential equations, algebraic geometry, differential geometry, and topology.
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