On H∞ on the complement of C1+α curves

Let ρ be a quasiconformal mapping on the plane with complex dilatation μ. We show that if μ satisfies a certain Carleson measure condition, then one can transfer H∞ on the upper half plane onto the corresponding space in the complement of the quasicircle Γ=ρ(R), and that this condition on μ characterizes C1+α curves.