Further Study on Domains and Quasihyperbolic Distances

TL;DR

This paper develops geometric tools to analyze $L^s$-averaging properties of domains and provides bounds for quasihyperbolic distances, enhancing understanding of domain classifications and their unions.

Contribution

It introduces constructive geometric methods for identifying $L^s$-averaging domains and explores their relationship with $p$-Poincaré domains, including union properties.

Findings

Established bounds for $L^s$-integrals of quasihyperbolic distance

Constructed examples illustrating geometric tools and domain relationships

Analyzed finite unions of $L^s$-averaging domains

Abstract

We establish constructive geometric tools for determining when a domain is $L^s$-averaging and obtain upper and lower bounds for the $L^s$-integrals of the quasihyperbolic distance. We also construct examples which are helpful to understand our geometric tools and the relationship between $p$-Poincar\'{e} domains and $L^s$-averaging domains. Finally, finite unions of $L^s(\mu)$-averaging domains are explored.