On quasisymmetry of quasiconformal mappings

Abstract

Suppose that f : D → D ′ is a quasiconformal mapping, where D and D ′ are domains in R n , and that D is a broad domain. Then for every arcwise connected subset A in D, the weak quasisymmetry of the restriction f | A : A → f (A) implies its quasisymmetry, and as a consequence, we see that the answer to one of the open problems raised by Heinonen from 1989 is affirmative under the additional condition that A is arcwise connected. As an application, we establish nine equivalent conditions for a bounded domain, which is quasiconformally equivalent to a bounded and simply connected uniform domain, to be John. This result is a generalization of the main result of Heinonen from [15].

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.…”

A note on Väisälä's problem concerning free quasiconformal mappings

“…The main advantage of this approach is that it avoids using volume integrals and conformal modulus, allowing one to study the quasiconformality of mappings in infinite dimensional Banach spaces and metric spaces without doubling volume measure. This line of research has recently attracted substantial interest, and some recent results can be found from [4,5,8,10] and references therein.…”

“…This class of domains has numerous geometric and function theoretic properties that are useful in many fields of mathematical analysis (see e.g. [2,5,13,16,18]).…”

Weakly Quasisymmetric Maps and Uniform Spaces

“…Also, Väisälä proved the quantitative equivalence between free quasiconformality and quasisymmetry of homeomorphisms between two Banach spaces, see [48,Theorem 7.15]. Against this background, it is not surprising that the study of quasisymmetry in metric spaces has recently attracted significant attention [4,24,23,39,52].…”