Jordan domains with a rectifiable arc in their boundary

Abstract

We show that if an open arc J of the boundary of a Jordan domain $\Omega$ is rectifiable, then the derivative $\Phi$' of the Riemann map $\Phi :D\to \Omega$ from the open unit disk D onto $\Omega$ behaves as an $H^1$ function when we approach the arc ${\Phi }^{-1}(J^{\prime })$,where $J^{\prime }$ is any compact subarc of $J$. "