  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 $H^{\infty}$ on the upper half plane onto the
corresponding space in the complement of the quasicircle
$\Gamma=\rho(\mathbb{R})$, and that this condition on $\mu$ characterizes
$C^{1+\alpha}$ curves.
