Sharp microlocal Kakeya--Nikodym estimates for eigenfunctions with applications

2025/09/01 by Gao, Chuanwei, Wu, Shukun, Xi, Yakun · 1 citation
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2509.01116

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 → L^p estimates for Hörmander operators in odd dimensions, improved L^q → L^p Fourier extension bounds, and improved bounds for the Bochner--Riesz conjecture in \mathbb R³.

Citations

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Cited by

• On Oscillatory Integral Operators Satisfying the cinematic curvature condition

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