Gromov hyperbolic John is quasihyperbolic John I

TL;DR

This paper introduces quasihyperbolic John spaces, provides a criterion for their characterization, and proves that Gromov hyperbolic quasihyperbolic John spaces satisfy certain conditions, answering an open question in geometric analysis.

Contribution

It establishes a necessary and sufficient condition for quasihyperbolic John spaces and proves a key property linking Gromov hyperbolicity with quasihyperbolic John spaces.

Findings

A criterion for quasihyperbolic John spaces is provided.

Gromov hyperbolic quasihyperbolic John spaces are shown to be quasihyperbolic John.

The paper answers an open question by Heinonen from 1989.

Abstract

In this paper, we introduce a concept of quasihyperbolic John spaces and provide a necessary and sufficient condition for a space to be quasihyperbolic John. Using this criteria, we exhibit a simple proof to show that a John space with a Gromov hyperbolic quasihyperbolization is quasihyperbolic John, quantitatively. This answers affirmatively to an open question proposed by Heinonen (Rev.~Math.~Iber, 1989), which was studied by Gehring et al. (Math.~Scand, 1989). As a tool, we study its connection between several geometric conditions.