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 $n\ge 3$ 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 Hörmander operators, and we characterize the $L^q\to L^p$ boundedness of Hörmander operators with positive-definite phase in all dimensions $n\ge 3$, thereby fully resolving a question going back to Hörmander. Further applications include improved $L^q\to L^p$ Fourier extension bounds, and improved bounds related to the Bochner--Riesz conjecture in ${\mathbb{R}}^3$.

36 pages. Added new sharpness examples, which show that our $(q,p)$ bounds for Hörmander operators with positive-definite phase, as well as the corresponding microlocal Kakeya--Nikodym estimates, are sharp in all dimensions. Similar examples also show that interpolation between the Tomas--Stein and Bourgain--Guth bounds yields the complete picture in the absence of the positivity assumption