Uniform $L^p$ Resolvent Estimates on the Torus

TL;DR

This paper establishes new uniform $L^p$ resolvent estimates for the flat torus, enhancing previous bounds by leveraging advanced harmonic analysis techniques like $ ext{l}^2$-decoupling and Weyl sum estimates.

Contribution

It introduces improved uniform $L^p$ resolvent estimates on the flat torus, utilizing modern harmonic analysis tools to extend prior results.

Findings

Enhanced $L^p$ resolvent bounds for the flat torus

Application of $ ext{l}^2$-decoupling theorem in this context

Use of multidimensional Weyl sum estimates

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 $\ell^2$-decoupling theorem and multidimensional Weyl sum estimates.