On the inverse problem for free quasiconformality in Banach spaces

It is known that the inverse of a quasiconformal homeomorphism of domains in R n is also quasiconformal. This paper focuses on the inverse problem for free quasiconformality in Banach spaces. We first show that the inverse of a fully semisolid homeomorphism is fully semisolid under an additional coarsely Lipschitz condition in the quasihyperbolic metric. This gives several partial answers to two open problems posed by V & auml;is & auml;l & auml;. Next, we prove that the inverse of a locally quasisymmetric homeomorphism is also locally quasisymmetric. As applications, we obtain new characterizations of freely quasiconformal mappings in Banach spaces, and study the relation between freely quasiconformal mappings and quasisymmetric mappings between uniform domains.