Improved Spectral Cluster Bounds for Orthonormal Systems

TL;DR

This paper refines spectral cluster bounds for orthonormal systems on flat tori and nonpositive curvature spaces by narrowing spectral bands, utilizing advanced harmonic analysis techniques to improve previous results.

Contribution

It introduces a method to tighten spectral band estimates for orthonormal systems, extending prior bounds to narrower spectral intervals as the frequency parameter grows.

Findings

Spectral bands are reduced from fixed width to shrinking intervals as ()  0.

Improved bounds are established for spectral cluster estimates at p=.

Method leverages Bourgain-Shao-Sogge-Yao approach for sharper spectral analysis.

Abstract

We improve Frank-Sabin's work concerning the spectral cluster bounds for orthonormal systems at $p=\infty$, on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from $[\lambda^{2}, (\lambda+1)^{2})$ to $[\lambda^{2}, (\lambda+\epsilon(\lambda))^{2})$, where $\epsilon(\lambda)$ is a function of $\lambda$ that goes to $0$ as $\lambda$ goes to $\infty$. In achieving this, we invoke the method developed by Bourgain-Shao-Sogge-Yao.