Quasisymmetry of freely quasiconformal mappings in Banach spaces

In this paper, we establish several characterizations of freely quasiconformal mappings in Banach spaces in terms of quasisymmetry in the quasihyperbolic metric. This provides a partial solution to an open problem posed by Väisälä. Also, we consider the special case when the control function satisfies η(t)≤Mt for some M≥1.