We extend the microlocal Kakeya--Nikodym bounds for eigenfunctions of Blair--Sogge to a larger range of exponents, which is optimal in odd dimensions. 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, we obtain improved $L^p$ bounds for Hecke--Maass forms on compact hyperbolic $3$-manifolds by combining with a recent result of Hou. Further applications include sharp $L^q\!\to\!L^p$ estimates for H\"ormander operators in odd dimensions, improved $L^q\!\to\!L^p$ Fourier extension bounds, and improved bounds for the Bochner--Riesz conjecture in $\mathbb R^3$.
Categories: math.CA math.AP math.SP