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.

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Abstract. In this paper we construct a Sobolev extension domain which, together with its complement, is topologically as nice as possible and yet not uniform. This shows that the well known implication that Uniform ⇒ Sobolev extension can not be reversed under strongest possible topological conditions.