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