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

Abstract

We consider an abstract non-negative self-adjoint operator $H$ on an $L^2$-space. We derive a characterization for the restriction estimate $\Vert d{E}_H(\lambda ){\Vert }_{L^p\to L^{p^{\prime }}}\le C{\lambda }^{\frac{d}{2}(\frac{1}{p}-\frac{1}{p^{\prime }})-1}$ 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 $L^p-L^{p^{\prime }}$ estimates for the derivatives of the spectral resolution of $H$.

arXiv comment9 pages