Bilinear Kakeya-Nikodym averages of eigenfunctions on compact Riemannian surfaces

TL;DR

This paper improves bilinear estimates for eigenfunctions on compact Riemannian surfaces by employing microlocal analysis and advanced oscillatory integral techniques, enhancing understanding of eigenfunction behavior.

Contribution

It introduces refined bilinear Kakeya-Nikodym estimates using microlocal methods and a bilinear Hörmander theorem, advancing prior eigenfunction analysis.

Findings

Enhanced bilinear estimates for eigenfunctions

Application of microlocal techniques to eigenfunction analysis

Improved understanding of eigenfunction concentration patterns

Abstract

We obtain an improvement of the bilinear estimates of Burq, G\'erard and Tzvetkov in the spirit of the refined Kakeya-Nikodym estimates of Blair and the second author. We do this by using microlocal techniques and a bilinear version of H\"ormander's oscillatory integral theorem.