Improved spectral projection estimates

TL;DR

This paper presents improved spectral projection estimates on manifolds with non-positive curvature, achieving sharper results and simplified proofs by leveraging pointwise estimates and microlocal analysis techniques.

Contribution

It introduces stronger spectral projection bounds, simplifies previous methods, and extends results to manifolds of negative curvature and tori with improved techniques.

Findings

Sharper spectral projection estimates for non-positive curvature manifolds

Simplified proofs using pointwise estimates and microlocal techniques

New results for manifolds of negative curvature and tori

Abstract

We obtain new improved spectral projection estimates on manifolds of non-positive curvature, including sharp ones for relatively large spectral windows for general tori. Our results are stronger than those in an earlier work of the first and third authors [6], and the arguments have been greatly simplified. We more directly make use of pointwise estimates that are implicit in the work of Berard [2] and avoid the use of weak-type spaces that were used in the previous works [6] and [22]. We also simplify and strengthen the bilinear arguments by exploiting the use of microlocal $L^2\to L^{q_c}$ Kakeya-Nikodym estimates and avoiding the of $L^2\to L^2$ ones as in earlier results. This allows us to prove new results for manifolds of negative curvature and some new sharp estimates for tori. We also have new and improved techniques in two dimensions for general manifolds of non-positive curvature.