A new characterization of Sobolev spaces on Lipschitz differentiability spaces
Bang-Xian Han, Zhe-Feng Xu, Zhuonan Zhu
TL;DR
This work develops a sharp characterization of metric Sobolev spaces on Lipschitz differentiability spaces by studying the asymptotics of $(\mathfrak{m}\times\mathfrak{m})(E_{\lambda,u})$ with $E_{\lambda,u}=\{(x,y): x\neq y, |u(x)-u(y)| \ge \lambda |d(x,y)|^{N/p+1}\}$. It yields both a lower bound and an upper bound under doubling and density conditions, culminating in an asymptotic formula $\lim_{\lambda\to\infty} \lambda^p (\mathfrak{m}\times\mathfrak{m})(E_{\lambda,u}) = \| \nabla u \|^p_{K_{p,\mathfrak{C}}}$ that is expressed via a tangent-space norm and Cheeger differentiability. A key novelty is the removal of the Poincaré inequality from the analysis and the explicit link between Brezis–Van Schaftingen–Yung-type asymptotics and Cheeger differentiability, including the Lip–lip equality $Lip(u)=lip(u)=|Du|_*$ a.e. for Lipschitz differentiability spaces. The results also prove the optimality of the doubling-dimension parameter $N=\frac{\log\beta}{\log 2}$, enhancing understanding of Sobolev-norm structures in non-PI metric spaces and offering new tools for analysis on such spaces.
Abstract
We prove a new characterization of metric Sobolev spaces, in the spirit of Brezis--Van Schaftingen--Yung's asymptotic formula. A new feature of our work is that we do not need Poincaré inequality which is a common tool in the literature. Another new feature is that we find a direct link between Brezis--Van Schaftingen--Yung's asymptotic formula and Cheeger's Lipschitz differentiability.