Sharp microlocal Kakeya--Nikodym estimates for Hörmander operators and spectral projectors

Abstract: We establish sharp microlocal Kakeya--Nikodym estimates for H\"ormander operators with positive-definite Carleson--Sj\"olin phases and for spectral projectors on smooth, compact Riemannian manifolds. As an application, we obtain sharp $L^q\to L^p$ estimates for the aforementioned H\"ormander operators in odd dimensions, thereby completing the analysis in the odd-dimensional case. Further applications include $L^q\to L^p$ estimates for the Fourier extension operator, $L^p$ estimates for the Bochner--Riesz operator, microlocal Kakeya--Nikodym estimates for Laplace eigenfunctions, and $L^p$ estimates for Hecke--Maass forms on compact $3$-dimensional arithmetic hyperbolic manifolds.