Sharp Spectral Projection Estimates for the Torus at $p_c=\frac{2(n+1)}{n-1}$

TL;DR

This paper establishes sharp spectral projection estimates for tori at a critical exponent, improving previous results by reducing window widths and eliminating sub-polynomial losses through a combination of advanced analytical techniques.

Contribution

It introduces a unified approach that combines bilinear decomposition, microlocal analysis, and decoupling theory to achieve optimal spectral projection estimates at the critical exponent.

Findings

Sharp spectral projection estimates at the critical exponent for all dimensions.

Extension of window widths down to mbda^{-1+\u03ba} without loss.

Improved bounds over previous works by removing sub-polynomial losses.

Abstract

We prove sharp spectral projection estimates for tori in all dimensions at the exponent $p_c=\frac{2(n+1)}{n-1}$ for shrinking windows of width $1$ down to windows of length $\lambda^{-1+\kappa}$ for fixed $\kappa>0$. This improves and slightly generalizes the work of Blair-Huang-Sogge who proved sharp results for windows of width $\lambda^{-\frac{1}{n+3}}$, and the work of Hickman, Germain-Myerson, and Demeter-Germain who proved results for windows of all widths but incurred a sub-polynomial loss. Our work uses the approaches of these two groups of authors, combining the bilinear decomposition and microlocal techniques of Blair-Huang-Sogge with the decoupling theory and explicit lattice point lemmas used by Hickman, Germain-Myerson, and Demeter-Germain to remove these losses.