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.
Gao et al. (Mon,) studied this question.