Eigenfunctions Restriction Estimates on Riemannian Manifolds and Bilinear Kakeya–Nikodym Averages with Principal Eigenvalue of Nonlocal Operators

Abstract

The 𝐿𝑝 norm estimates and an improvement of eigenfunctions restricted to submanifolds, for compact boundaryless Riemannian manifolds with nonpositive sectional curvature and constant negative curvature are studied. We show the refined, microlocal and bilinear Kakeya-Nikodym averages bounds for eigenfunctions in two dimensions, on compact Riemannian surfaces and lower bounded for nodal sets of eigenfunctions in higher dimensions with 𝐿𝑝 –norms. Simple criterion for the existence and properties of principal eigenvalue of the elliptic operators in Euclidean space and principal eigenfunctions and spectrum points of some nonlocal dispersal operators, and applications are considered.