The mapping properties of fractional derivatives in weighted fractional Sobolev space

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 $W_0^{p,s}(\mathrm{\Omega })$ to $W^{p,s-\alpha }(\mathrm{\Omega })$, for all $0<\alpha <s<1$ and the $\alpha$–order Marchaud fractional derivative with even extension maps the fractional Sobolev space $W^{p,s}((0,\mathrm{\infty }))$ to $W^{p,s-\alpha }(\mathrm{\Re })$ for all $0<\alpha <s<1$ and $ps\ge 1$ . The proof is based on the Calderón–Lions interpolation theorem.