  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 $J$. "
