Some domination inequalities for spectral zeta kernels on closed Riemannian manifolds

TL;DR

This paper extends Kato's inequalities and domination inequalities for spectral zeta functions from Riemannian surfaces to higher-dimensional closed Riemannian manifolds, using majorisation techniques.

Contribution

It generalizes Kato's comparison inequalities to n-dimensional closed Riemannian manifolds and establishes new domination inequalities for spectral zeta kernels.

Findings

Proved Kato's inequalities for Laplacian and Schrödinger operators on closed manifolds.

Established new domination inequalities for spectral zeta functions on spheres.

Extended inequalities from surfaces to higher-dimensional manifolds.

Abstract

We first prove Kato's inequalities for the Laplacian and a Schr$\ddot{o}$dinger-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 kernels 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.