  We present and prove a version of the elliptic regularity theorem for partial
differential equations involving fractional Riemann-Liouville derivatives. In
this case, regularity is defined in terms of Sobolev spaces $H^s(X)$: if the
forcing of a linear elliptic fractional PDE is in one Sobolev space, then the
solution is in the Sobolev space of increased order corresponding to the order
of the derivatives. We also mention a few applications and potential extensions
of this result.
