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

Frederic Bernicot (1) , El Maati Ouhabaz (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Équipe Analyse

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 $H$.

Mots clés: Restriction estimates; semigroup; spectral multipliers; dispersive estimates

Domaines: Analyse classique [math.CA]

Frederic Bernicot, El Maati Ouhabaz. Restriction estimates via the derivatives of the heat semigroup and connexion with dispersive estimates. Mathematical Research Letters, 2013, 20 (6), pp.1047-1058. ⟨hal-00812285⟩