Quasisymmetry of weakly quasisymmetric homeomorphisms

Quasi‐invariance of modulus and quasisymmetry of weakly (L,M)‐quasisymmetric maps in metric spaces

“…In this paper, we focus on the quasisymmetry in the internal metrics. This is because one important way to study the global distortion property of a quasiconformal map (in original metric) is to investigate under what conditions such a map is quasisymmetric in the internal metrics and a lot of work has been done in this line (see, for example, [6,7,11,17,30]). Corollary 1.2 is a generalization of corresponding results in [11,17] from Euclidean space to general metric spaces.…”

“…In this line of argument, the HTB condition is crucial. It is worth noting that, for maps between certain domains in $$\mathbb {R}^n$$, η‐quasisymmetry can be derived from H ‐quasisymmetry without the HTB property (see [6, 17]). However, the proofs in [6, 17] used some special properties and modulus estimates in the Euclidean space that may not be true in general metric spaces.…”

Quasiconformality to quasisymmetry via weak (L,M)-quasisymmetry

“…More recently, to complete the reversion of (1.4) in this setting, in [16] and [6] the authors derived quasisymmetry from weak H-quasisymmetry and, at the same time, eliminated the above LLC 2 condition on f (A). More specifically, the following theorem is established in [16].…”