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

Abstract

This paper contributes to the study of a fundamental problem in the theory of quasiconformal analysis: under what conditions local quasiconformality of a homeomorphism implies its global quasisymmetry. We show that in general metric spaces local regularity and some connectivity together with the Loewner condition are necessary and sufficient for a quasiconformal map to be globally quasisymmetric with respect to the internal metrics. In this endeavor, two major new ingredients are used. One is the recently introduced concept of weakly $(L,M)$-quasisymmetry, serving as a bridge between local quasiconformality and global quasisymmetry. Another is the quasi-invariance of conformal modulus under weakly $(L,M)$-quasisymmetric maps, which is developed in this paper.