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

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 (Formula presented.) -quasisymmetry, serving as a bridge between local quasiconformality and global quasisymmetry. Another is the quasi-invariance of conformal modulus under weakly (Formula presented.) -quasisymmetric maps, which is developed in this paper.