$L^2$ to $L^p$ bounds for spectral projectors on the Euclidean two-dimensional torus

TL;DR

This paper establishes bounds for spectral projectors on the 2D torus, extending Sogge's classical results by employing advanced decoupling techniques and analyzing the convolution kernel.

Contribution

It introduces new L2 to Lp bounds for spectral projectors on the 2D torus with narrow spectral windows, using modern harmonic analysis methods.

Findings

Derived L2 to Lp bounds extending Sogge's classical results

Introduced a new question on the convolution kernel of the projector

Applied decoupling and exponential sum estimates to spectral analysis

Abstract

We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include l2 decoupling, small cap decoupling, and estimates of exponential sums.