Domains with Controlled Modulus and Quasi conformal Mappings

Abstract

In this paper we introduce the notion of controlled modulus condition as a generalization of the quasiextremal distance (or QED) condition introduced by Gehring and Martio. We show that domains satisfying the controlled modulus condition enjoy a lot of properties that QED domains have, such as the linear local connectivity and the extendability and Lipschitz continuity of quasiconformal maps defined on such domains. On the other hand, we also establish some properties for domains with controlled modulus, such as the quasimobius invariance, the controlled modulus condition. But the converse remains open. We also prove that the controlled modulus condition is equivalent to the Loewner condition recently introduced by Heinonen and Koskela in their study of quasiconformality in general metric spaces.