A Sobolev estimate for the adjoint restriction operator

In this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption. We obtain the optimal $H^s$-$L^q$ estimate and its mixed norm generalization. As applications we prove some weighted Strichartz estimates for the propagator $e^{it(-\Delta)^{\alpha/2}}\varphi$, $\alpha>0$.