From semiclassical Strichartz estimates to uniform $L^p$ resolvent estimates on compact manifolds

Nicolas Burq (1) , David dos Santos Ferreira (2, 3) , Katya Krupchyk (4)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
3 IECL - Institut Élie Cartan de Lorraine
4 Department of Mathematics [Irvine]

We prove uniform $L^p$ resolvent estimates for the stationary damped wave operator. The uniform $L^p$ resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Dos Santos Ferreira-Kenig-Salo and advanced further by Bourgain-Shao-Sogge-Yao. Here we provide an alternative proof relying on the techniques of semiclassical Strichartz estimates. This approach allows us also to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semiclassical spectral analysis framework.

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Burq, David dos Santos Ferreira, Katya Krupchyk. From semiclassical Strichartz estimates to uniform $L^p$ resolvent estimates on compact manifolds. International Mathematics Research Notices, 2018, 2018 (16), pp.5178-5218. ⟨10.1093/imrn/rnx042⟩. ⟨hal-01251701⟩