Abstract

A new range of uniform L p resolvent estimates is obtained in the setting of the flat torus, improving previous results of Bourgain, Shao, Sogge and Yao. The arguments rely on the ℓ 2 -decoupling theorem and multidimensional Weyl sum estimates.

Highlights

  • This article continues a line of investigation pursued by Dos Santos Ferreira, Kenig and Salo [DSFKS14] and Bourgain, Shao, Sogge and Yao [BSSY15] concerning uniform Lp estimates for resolvents of Laplace–Beltrami operators on compact manifolds

  • It was shown in [BSSY15] that the desired resolvent estimates are equivalent to certain spectral projection bounds

  • By Theorem 4, the uniform resolvent estimates in Theorem 1 are equivalent to the following spectral projection bounds

Read more

Summary

Introduction

This article continues a line of investigation pursued by Dos Santos Ferreira, Kenig and Salo [DSFKS14] and Bourgain, Shao, Sogge and Yao [BSSY15] concerning uniform Lp estimates for resolvents of Laplace–Beltrami operators on compact manifolds. New bounds are obtained only in the special case of the flat n-dimensional torus Tn := Rn \ Zn but, in order to contextualise the results, it is useful to recall the general setup from [DSFKS14, BSSY15] To this end, let (M , g ) be a smooth, compact manifold of dimension n ≥ 3 without boundary and ∆g be the associated Laplace–Beltrami operator. Bourgain, Shao, Sogge and Yao [BSSY15] showed that the region RDKSS is, optimal in the case of Zoll manifolds (one example being the standard euclidean sphere Sn), in the sense that here it is not possible to relax |μ| ≥ 1 to |μ| ≥ λ−α for any α > 0 in RDKSS.

Spectral projections
The proof of Theorem 1
Improvements via multidimensional Weyl sum estimates

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.