Quasiconformal mappings and Ahlfors-David curves

Abstract:

We show that if $\rho$ is a quasiconformal mapping of the plane whose dilatation satisfies a certain quadratic Carleson measure condition relative to an Ahlfors-David curve $\Gamma$ then $\rho$ is differentiable almost everywhere on $\Gamma$ and $\log |\rho \prime | \in {\text {BMO}}$. When $\Gamma$ is chord-arc we show that its image is a Bishop-Jones curve. If the Carleson norm is small then we show that $\rho$ is absolutely continuous on $\Gamma$, the image of $\Gamma$ is an Ahlfors-David curve, and $\rho \prime = {e^a}$, where $a \in {\text {BMO}}$ with a small norm.