On the stability of $$\varphi $$ φ -uniform domains

Abstract

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain G ⊂ R n . In the sequel, we investigate a class of domains, so called ϕ-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism ϕ from [0, ∞) to itself. Finally, we discuss a number of stability properties of ϕ-uniform domains. In particular, we show that the class of ϕ-uniform domains is stable in the sense that removal of a geometric sequence of points from a ϕ-uniform domain yields a ϕ 1 -uniform domain.2010 Mathematics Subject Classification. Primary 30F45; Secondary 30C65.

On Removability Properties of $$\psi $$ ψ -Uniform Domains in Banach Spaces

“…In these two papers, the removability questions were studied for the classes of uniform domains and John domains. For latest results in the case of ψ-uniform domains in R n see [8]. Our main result extends these results to Banach spaces.…”

Geometric properties of $$\varphi $$ φ -uniform domains

“…Similar results have been established for various classes of functions such as quasiconformal mappings and mappings with finite distortion. In such applications, uniform domains and their generalizations occur [Ge99, GO79, GH12, Ko09, Va71, Va88, Va91, Va98, Vu88]; ϕ-uniform domains have been recently studied in [KLSV14].…”

On the Removability Property of φ‐Natural Domains in Real Banach Spaces

“…To investigate the relationships between conformal invariants and quasi hyperbolic metrics, in [14], Vuorinen introduced the so-called ψ-uniform domains, which are a generalization of uniform domains. See [1,[15][16][17][18][19] for the development of ψ-uniform domains in recent years.…”