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