On on the complement of 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 C 1 + α curves.