Some domination inequalities for spectral zeta kernels on closed Riemannian manifolds

We first prove Kato's inequalities for the Laplacian and a Schr$\ddot{o}$dinger-type operator on smooth functions closed Riemannian manifolds. then apply result to establish some new domination spectral zeta their related kernels $n$-dimensional unit spheres using majorisation techniques. Our results are generalisations of comparison surfaces