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

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

Keywords

restriction estimates, semigroup, spectral multipliers, dispersive estimates

2010 Mathematics Subject Classification

35P05, 42B15, 47D03

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Published 13 June 2014