Abstract
We first prove Kato's inequalities for the Laplacian and a Schrodinger-type operator on smooth functions on closed Riemannian manifolds. We then apply the result to establish some new domination inequalities for spectral zeta functions and their related spectral zeta ker-nels on n-dimensional unit spheres using Kato's inequalities and majorisation techniques. Our results are the generalisations of Kato's comparison inequalities for Riemannian surfaces to n -dimensional closed Riemannian manifolds.