Characterizations of quasihyperbolic John domains and uniform domains in metric spaces

TL;DR

This paper provides a new elementary characterization of uniform domains in metric spaces, linking LLC, ball separation, and quasihyperbolic John conditions, thereby extending and improving previous results.

Contribution

It introduces a simplified approach to characterize uniform domains via LLC and quasihyperbolic John conditions, improving prior complex proofs and extending the scope of existing theorems.

Findings

Characterization of uniform domains via LLC and ball separation.

Equivalence between LLC-1 and quasihyperbolic John conditions.

Alternative proof of inner uniformity without Gromov hyperbolicity.

Abstract

In a recent work of Zhou and Ponnusamy [Ann. Sc. Norm. Super. Pisa Ci. Sci. 2025], the authors studied the following natural question: find sufficient and necessary conditions for a domain $\Omega$ in a metric space $X$ to be quasihyperbolic John. It was proved that Gromov hyperbolic John domains are quasihyperbolic John, quantitatively. As an application, they obtained a characterization of uniform domains in Ahlfors regular spaces. In a recent work, using a deep improved characterization of Gromov hyperbolicity, Guo, Huang and Wang [arXiv 2025] proved the quantitative equivalence bteween inner uniformity and the quasihyperbolic John condition in metric doubling spaces. However, the proof does not yield a similar characterization for uniform domains. In this article, we find a new elementary approach to successfully extend the above characterization to uniform domains: a domain $\Omega$ in a doubling length space $X$ is uniform if and only if it is linearly locally connected (LLC) and satisfies the ball separation condition, if and only if it is LLC-1 and quasihyperbolic John, quantitatively. This substantially improved the corresponding results of Zhou and Ponnusamy. Our new approach also allows us to give an alternative proof of the inner uniformity result of Guo-Huang-Wang without using the improved characterization on Gromov hyperbolicity.