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

Abstract

Abstract. In this paper, first, we define the weakly quasimöbius maps in quasi-metric spaces and obtain a series of elementary properties of these maps. Then we find conditions under which a weakly quasimöbius map is quasimöbius in quasi-metric spaces. With the aid of uniform perfectness, three related results are proved, and some applications are also given.

On uniform perfectness in quasimetric spaces

“…In this paper, we study the equivalence conditions for uniform perfectness of quasimetric spaces. The first version of this paper was first finished in 2016 (an early version of this paper see [23]), there were several applications in the area of geometric function theory based on that version [1,[19][20][21]. We start with the definition of quasimetric spaces.…”

“…For example, it is worth to mention that Buyalo and Schroeder established the quasisymmetric and quasimöbius extension theorems for visual geodesic hyperbolic spaces which possess uniformly perfect boundaries at infinity [4,Chapter 7]. In [20], the first author and Wang found several conditions under which a weakly quasimöbius map is quasimöbius in uniformly perfect quasimetric spaces. The authors in [22] investigated the invariance of doubling property under sphericalization and flattening transformations in uniformly perfect spaces.…”

“…Moreover, as we mentioned in the beginning of this paper, there were already several applications based on Theorems 1.4 and 1.6. By using Theorems 1.4 and 1.6, the first author and Wang investigated the relations between quasimöbius mappings and weakly quasimöbius mappings in uniform perfect quasimetric spaces [20]. By using the equivalence of weakly quasisymmetric mappings and quasisymmetric mappings on uniform perfect quasimetric spaces as established in [20], Vellis further studied the quasisymmetric and bilipschitz extension properties of planer uniform domains with uniform perfect boundaries [19].…”

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].…”

An extension property of quasimöbius mappings in metric spaces

“…Also, it has been well known that the class of QM mappings has played an important role in the study of QC mappings (which is the abbreviation of quasiconformal mappings), QS mappings and their relationships (cf. [2,9,11,14] etc. ).…”