Quasiconformal mappings and chord-arc curves

Abstract

Given a quasiconformal mapping ρ \rho on the plane, what conditions on its dilatation μ \mu guarantee that ρ ( R ) \rho ({\mathbf {R}}) is rectifiable and ρ | R \rho {|_{\mathbf {R}}} is locally absolutely continuous? We show in this paper that if μ \mu satisfies certain quadratic Carleson measure conditions, with small norm, then ρ ( R ) \rho ({\mathbf {R}}) is a chord-arc curve with small constant, and ρ ( x ) = ρ ( 0 ) + ∫ 0 x e a ( t ) d t \rho (x) = \rho (0) + \int _0^x {{e^{a(t)}}dt} for x ∈ R x \in {\mathbf {R}} , with a ∈ BMO a \in \operatorname {BMO} having small norm. Conversely, given any such map from R → C {\mathbf {R}} \to {\mathbf {C}} , we show that it has an extension to C {\mathbf {C}} with the right kind of dilatation. Similar results hold with R {\mathbf {R}} replaced by a chord-arc curve. Examples are given that show that it would be hard to improve these results. Applications are given to conformal welding and the theorem of Coifman and Meyer on the real analyticity of the Riemann mapping on the manifold of chord-arc curves.