Refined $L^p$ restriction estimate for eigenfunctions on Riemannian surfaces

TL;DR

This paper refines $L^p$ restriction estimates for Laplace eigenfunctions on Riemannian surfaces, providing sharper bounds for restrictions to sets and neighborhoods, unifying previous results, and exploring eigenfunction concentration.

Contribution

It introduces new sharp $L^p$ restriction estimates for eigenfunctions on surfaces, including arbitrary Borel sets and tubular neighborhoods, extending and unifying prior bounds.

Findings

Sharp $L^p$ estimates for eigenfunction restrictions to Borel sets.

Unified framework combining $L^p(M)$ and $L^p( ext{curve})$ bounds.

Insights into eigenfunction concentration phenomena and variable coefficient Fourier estimates.

Abstract

We refine the $L^p$ restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, G\'erard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel sets on the surface, following the formulation of Eswarathasan and Pramanik. We achieve this by proving a variable coefficient version of a weighted Fourier extension estimate of Du and Zhang. Our results naturally unify the $L^p(M)$ estimates of Sogge and the $L^p(\gamma)$ restriction bounds of Burq, G\'erard, and Tzvetkov, and are sharp for all $p \geq 2$, up to a $\lambda^\varepsilon$ loss. Second, we derive sharp estimates for the restriction of eigenfunctions to tubular neighborhoods of a curve with non-vanishing geodesic curvature. These estimates are closely related to a variable coefficient version of the Mizohata--Takeuchi conjecture, providing new insights into eigenfunction concentration phenomena.