Local properties of quasihyperbolic mappings in metric spaces

Abstract

Abstract. In this paper, we consider Väisälä's problem and obtain that a homeomorphism which is both semi-local M -QH and semi-local η-QS between two suitable metric spaces is an M 1 -QH map.

Quasihyperbolic mappings in Banach spaces

“…The main advantage of this approach avoids to make use of volume integrals and conformal modulus, which allows one to study the quasiconformality of mappings in Banach spaces with dimension infinity and metric spaces without volume measures. This research has recently attracted substantial interest in the research community (see e.g., [3,7,8,10,17] and reference therein).…”

“…We remark that the authors in [7] recently considered Problem 1.1 in metric spaces. However, the target spaces in [7] are always assumed to be proper. Here a metric space being proper means that every closed ball in this space is compact.…”

Quasihyperbolic mappings in length metric spaces

“…Thus the local conditions for the mappings both in Theorem A and Theorem 2 are welldefined. Note that B(x, d G (x)) may not lie in G even if X is quasi-convex, for more discussions see [3,4]. Definition 7.…”

“…Definition 7. In [4], (X , d ) is said to be dense if for any two points x, y ∈ X and two positive constants r 1 , r 2 with d (x, y) < r 1 + r 2 , we have B(x, r 1 ) ∩ B(y, r 2 ) = .…”

“…For the proof of the necessity part, we assume that f is L-QH. Since the assumption that (X 1 , d 1 ) is length implies that (X 1 , d 1 ) is 2-quasi-convex, it follows from [4,Theorem 3.7]…”