Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions
Matthew D. Blair, Christopher D. Sogge
TL;DR
This paper improves bounds on eigenfunctions in two dimensions by developing sharper microlocal estimates that leverage phase space decomposition aligned with geodesic flow.
Contribution
It introduces new microlocal estimates and refined Kakeya-Nikodym bounds for eigenfunctions, enhancing understanding of their behavior in two-dimensional settings.
Findings
Sharper Kakeya-Nikodym bounds achieved
Development of phase space decomposition adapted to geodesic flow
Enhanced microlocal estimates for eigenfunctions
Abstract
We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.
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