Sharp microlocal Kakeya--Nikodym estimates for eigenfunctions with applications
Chuanwei Gao, Shukun Wu, Yakun Xi
TL;DR
This work advances spectral and oscillatory analysis by proving sharp microlocal Kakeya–Nikodym estimates for eigenfunctions, extending the admissible $L^p$ range and introducing an anisotropic variant of the microlocal norm on constant-curvature manifolds. The authors develop a unified wave-packet framework for Hörmander oscillatory integral operators and spectral projectors, coupled with refined decoupling and incidence geometry to implement an induction-on-scales strategy. They obtain optimal or near-optimal $(q,p)$ bounds for Hörmander operators in odd dimensions, improved $L^p$ bounds for Hecke–Maass forms on compact hyperbolic $3$-manifolds, and sharpened Fourier extension and Bochner–Riesz estimates, with concrete corollaries in hyperbolic and Euclidean settings. The anisotropic MKN norm plays a central role in capturing anisotropic concentration along tubular structures and leads to new bounds in the constant-curvature case, highlighting deep connections between geometric optics, spectral theory, and harmonic analysis.
Abstract
We extend the microlocal Kakeya--Nikodym bounds for eigenfunctions of Blair--Sogge to a larger range of exponents, which is optimal in odd dimensions. On manifolds of constant sectional curvature we introduce a new anisotropic variant of the microlocal Kakeya--Nikodym norm that further enlarges the admissible $p$-range. As a corollary, we obtain improved $L^p$ bounds for Hecke--Maass forms on compact hyperbolic $3$-manifolds by combining with a recent result of Hou. Further applications include sharp $L^q\!\to\!L^p$ estimates for Hörmander operators in odd dimensions, improved $L^q\!\to\!L^p$ Fourier extension bounds, and improved bounds for the Bochner--Riesz conjecture in $\mathbb R^3$.