  We study the mapping behavior of the Marchaud fractional derivative with
different extensions in the scale of fractional weighted Sobolev spaces. In
particular we show that the $\alpha$--order Riemann--Liouville fractional
derivative maps $W^{p,s}_0(\Omega)$ to $W^{p,s-\alpha}(\Omega)$, for all
$0<\alpha<s<1$ and the $\alpha$--order Marchaud fractional derivative with even
extension maps the fractional Sobolev space $W^{p,s}((0,\infty))$ to
$W^{p,s-\alpha}(\real)$ for all $0<\alpha<s<1$ and $ps\geq1$ . The proof is
based on the Calder\'{o}n--Lions interpolation theorem.
