Restriction Estimates via the Derivatives of the Heat Semigroup and Connexion with Dispersive Estimates

We consider an abstract non-negative self-adjoint operator H on an L-space. We derive a characterization for the restriction estimate ‖dEH(λ)‖Lp→Lp′ ≤ Cλ d 2 ( 1 p − 1 p )−1 in terms of higher order derivatives of the semigroup e . We provide an alternative proof of a result in [1] which asserts that dispersive estimates imply restriction estimates. We also prove L − L ′ estimates for the derivatives of the spectral resolution of H .