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 $\Phi: D\rightarrow \Omega$ 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 $J$. "