A Sobolev estimate for the adjoint restriction operator

TL;DR

This paper establishes optimal Sobolev space estimates for the adjoint restriction operator on hypersurfaces, leading to new weighted Strichartz estimates for certain dispersive propagators.

Contribution

It provides the first optimal $H^s$-$L^q$ estimates for the adjoint restriction operator with additional regularity assumptions, including mixed norm generalizations.

Findings

Optimal $H^s$-$L^q$ estimate derived

Weighted Strichartz estimates proved for dispersive propagators

Generalization to mixed norm estimates achieved

Abstract

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 $e^{it(-\Delta)^{\alpha/2}}\varphi$, $\alpha>0$.