Ahlfors-regular curves and Carleson measures

We study the relation between the boundary of a simply connected domain Ω being Ahlfors-regular and the invariance of Carleson measures under the push-forward operator induced by a conformal mapping from the unit disk Δ onto the domain Ω. As an application, we characterize the chord-arc curves with small norms and the asymptotically smooth curves in terms of the complex dilatation of some quasiconformal reflection with respect to the curve.