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

We establish sharp microlocal Kakeya--Nikodym estimates for Hörmander operators with positive-definite Carleson--Sjölin phases and for spectral projectors on smooth, compact Riemannian manifolds. As an application, we obtain sharp Lq→ Lᵖ estimates for the aforementioned Hörmander operators in odd dimensions, thereby completing the analysis in the odd-dimensional case. Further applications include Lq→ Lᵖ estimates for the Fourier extension operator, Lᵖ estimates for the Bochner--Riesz operator, microlocal Kakeya--Nikodym estimates for Laplace eigenfunctions, and Lᵖ estimates for Hecke--Maass forms on compact $3$-dimensional arithmetic hyperbolic manifolds.