A Generalization of a Levitin and Parnovski Universal Inequality for Eigenvalues

TL;DR

This paper extends universal eigenvalue inequalities to the Hodge de Rham Laplacian on Euclidean submanifolds and the Kohn Laplacian on the Heisenberg group, broadening the scope of previous inequalities.

Contribution

It generalizes the Levitin-Parnovski inequality to new settings involving geometric Laplacians on submanifolds and the Heisenberg group.

Findings

Derived universal inequalities for eigenvalues of the Hodge de Rham Laplacian.

Established inequalities for eigenvalues of the Kohn Laplacian on the Heisenberg group.

Obtained new Reilly and Asada inequalities.

Abstract

In this paper, we derive "universal" inequalities for the sums of eigenvalues of the Hodge de Rham Laplacian on Euclidean closed Submanifolds and of eigenvalues of the Kohn Laplacian on the Heisenberg group. These inequalities generalize the Levitin-Parnovski inequality obtained for the sums of eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain. New Reilly and Asada inequalities are also obtained.