Restriction estimates via the derivatives of the heat semigroup and connection with dispersive estimates

Abstract

We consider an abstract non-negative self-adjoint operator H on an L2- space. We derive a characterization for the restriction estimate lldEH(λ)/dλll Lp→Lp ≤ Cλ d/2 (1/p − 1/p')−1 (involving the Radon-Nikodym derivative of the spectral measure) in terms of higher order derivatives of the semigroup e−tH. We provide an alternative proof of a result in [1] which asserts that dispersive estimates imply restriction estimates. We also prove Lp − Lp' estimates for the derivatives of the spectral resolution of H.