Perturbations of globally hypoelliptic operators on closed manifolds

TL;DR

This paper establishes criteria for when perturbations of globally hypoelliptic operators on closed manifolds remain hypoelliptic, analyzing eigenvalue behavior and providing examples that both preserve and destroy hypoellipticity.

Contribution

It provides necessary and sufficient conditions for the preservation of global hypoellipticity under perturbations on closed manifolds, extending previous results and constructing explicit examples.

Findings

Perturbations invariant under the Laplacian can preserve hypoellipticity.

Constructed examples show low order perturbations can destroy hypoellipticity.

Eigenvalue analysis at infinity is key to understanding hypoellipticity stability.

Abstract

Inspired by results of A. Bergamasco on perturbations of vector fields defined on the two-dimensional torus, and of J. Delgado and M. Ruzhansky on properties of invariant operators with respect to an elliptic operator defined on a closed manifold, we give necessary and sufficient conditions to ensure that perturbations of a globally hypoelliptic operator defined on $\T\times M$, continue to be globally hypoelliptic, where $\T$ is the flat torus and $M$ is a closed smooth manifold. For this, we analyze the behavior, at infinity, of the sequences of eigenvalues generated by the family of matrices given by the restrictions of this on the eigenspaces of a fixed elliptic operator. As an application, we construct perturbations, invariant with respect to the Laplacian, of the vector field $D_t + \omega D_x$ on $\T^2$. In the case where these perturbations commute with the operator $D_x$, our examples recover and extend some results of Bergamasco. Additionally, we construct examples of low order perturbations that destroy the global hypoellipticity, in the presence of diophantine phenomena.