On $H^{\mathrm{\infty }}$ on the complement of C^{1+\alpha} curves

Let $\rho$ be a quasiconformal mapping on the plane with complex dilatation $\mu$. We show that if $\mu$ satisfies a certain Carleson measure condition, then one can transfer $H^{\mathrm{\infty }}$ on the upper half plane onto the corresponding space in the complement of the quasicircle $\mathrm{\Gamma }=\rho (\mathbb{R})$, and that this condition on $\mu$ characterizes $C^{1+\alpha }$ curves.