Sharp microlocal Kakeya--Nikodym estimates for eigenfunctions with applications

Abstract

We extend the microlocal Kakeya--Nikodym bounds for eigenfunctions of Blair--Sogge to a larger range of exponents, which is optimal in all dimensions $n\ge3$ on general manifolds. 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, by combining our results with a recent theorem of Hou, we obtain improved $L^p$ bounds for Hecke--Maass forms on compact hyperbolic $3$-manifolds. In particular, our method applies to general H\"ormander operators, and we characterize the $L^q \to L^p$ boundedness of H\"ormander operators with positive-definite phase in all dimensions $n\ge3$, thereby fully resolving a question going back to H\"ormander. Further applications include improved $L^q \to L^p$ Fourier extension bounds, and improved bounds related to the Bochner--Riesz conjecture in $\mathbb R^3$.