On quasimöbius maps in real Banach spaces

Abstract

Suppose that E and E ′ denote real Banach spaces with dimension at least 2, that D E and D ′ E ′ are domains, that f : D → D ′ is an (M, C)-CQH homeomorphism, and that D is uniform. The main aim of this paper is to prove that D ′ is a uniform domain if and only if f extends to a homeomorphism f : D → D ′ and f is η-QM relative to ∂D. This result shows that the answer to one of the open problems raised by Väisälä from 1991 is affirmative.2000 Mathematics Subject Classification. Primary: 30C65, 30F45; Secondary: 30C20.

Quasimöbius maps, weakly quasimöbius maps and uniform perfectness in quasi-metric spaces

“…The introduction of quasimöbius maps has provided a handy tool when studying the quasisymmetric maps and the quasiconformal maps. Many references related to the relationships among quasimöbius maps, quasisymmetric maps and quasiconformal maps have been in literature; see [1,5,6,7,13,15,16,17,18,19,20,21,25,29,30] etc. The precise definition for quasimöbius maps is as follows.…”

Quasim

“…Then we show that all of these conditions are equivalent if the domain is ψ-natural. As applications, we partially answer an open question proposed by Väisälä, and provide a new method to prove a recent result of Huang et al (2013), which also gives an answer to another question raised by Väisälä.…”

“…It is worth mentioning that Huang, Li, Vuorinen, and Wang [4] answered Question 1.1 affirmatively. Their proofs are based on several concepts and results in the free quasiworld [22], such as coarse quasihyperbolic length and solid arcs, the equivalence of uniform and ϕ-uniform domains, the diameter cigar theorems for uniform domains.…”

“…Moreover, we show that Theorem 1.3 gives a way to affirmatively answer Question 1.1. Our method is quite different from the one used in [4]. Theorem 1.4.…”

“…Recently, this area is a subject of several investigations (see e.g. [5,4,6,9,11,15,26] and references therein).…”