论文标题
Bourgain-Brezis-Mironescu-dávila定理在第二步
A Bourgain-Brezis-Mironescu-Dávila theorem in Carnot groups of step two
论文作者
论文摘要
在此注释中，我们在第两个$ \ mathbb {g} $：\ [\ underSet {s \ nearRow 1/2} {\ lim} {\ lim}（1-2S）\ Mathfrak p_ { p_h（e）。 \]在这里，$ \ mathfrak p_h（e）$表示可测量集的水平周围$ e \ subset \ subset \ mathbb {g} $，而非局部水平的$ \ mathfrak $ \ mathfrak p_ {h，s}（h，s}（e）$ is afte besov besov besov besov besov besov besov seminorm。该结果代表了无尺度的亚里曼尼亚人的著名表征，这是波尔加因 - 布雷西斯 - 米洛尼斯库和达维拉的著名特征。
In this note we prove the following theorem in any Carnot group of step two $\mathbb{G}$: \[ \underset{s\nearrow 1/2}{\lim} (1 - 2s) \mathfrak P_{H,s}(E) = \frac{4}{\sqrt π}\ \mathfrak P_H(E). \] 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.
