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

Abstract

This paper is devoted 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. In particular, we introduce the concept of weak $(L,M)$-quasisymmetry, serving as a bridge between local quasiconformality and global quasisymmetry. We show that in general metric spaces local regularity and some connectivity together with the Loewner condition are sufficient for a quasiconformal map to be weakly $(L,M)$-quasisymmetric, and subsequently, quasisymmetric with respect to the internal metrics.