Smoothing effects of Schrödinger evolution groups on Riemannian manifolds

The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds

“…This property of gain of regularity has been first observed in the case of R d in the works of Constantin-Saut [11], Sjölin [24] and Vega [26] and it was later generalized to different perturbations of the flat Laplacian (see Ben Artzi-Klainerman [6], Ben Artzi-Devinatz [5], Constantin-Saut [10] and Doï ( [17,15])).…”

Strichartz estimates on asymptotically hyperbolic manifolds

“…To be more complete, let us point out that the analysis contained in this paper and a classical argument due to [Staffilani and Tataru 2002] (see also [Bouclet and Tzvetkov 2007, Section 5]), using the local smoothing effect [Doi 1996], would give the following global in space estimates. Theorem 1.3.…”

Untitled

“…Second, we adapt a result of Burq [2002], which provides Strichartz estimates without loss for a nontrapping problem, with a metric that equals the identity outside a compact set. The proof relies on a local smoothing effect for the free evolution exp (it D ), first observed independently by Constantin and Saut [1989], Sjölin [1987], and Vega [1988] in the flat case, and then by Doi [1996] on nontrapping manifolds and by Burq et al [2004a] on exterior domains. Following a strategy suggested by Staffilani and Tataru [2002], we prove that away from the obstacle the free evolution enjoys the Strichartz estimates exactly as for the free space.…”