Perturbations of globally hypoelliptic pseudo-differential operators on $\mathbb{R}^n$

TL;DR

This paper proves that the global regularity of certain hypoelliptic pseudo-differential operators on ^n is stable under lower-order perturbations, especially for Shubin and SG classes, but not in Hf6rmander classes.

Contribution

It establishes the stability of global hypoellipticity under lower-order perturbations for a broad class of pseudo-differential operators, including Shubin and SG classes.

Findings

Global regularity is preserved under lower-order perturbations.

Stability does not extend to Hf6rmander classes.

Results apply to operators on ^n with hypoelliptic symbols.

Abstract

This paper demonstrates the stability of the global regularity for a class of pseudo-differential operators under lower-order perturbations. We establish that if an operator has a globally hypoelliptic symbol, its global regularity (in the sense of Schwartz functions and tempered distributions) is preserved when perturbed by operators of sufficiently lower order. This result applies in particular to operators within the Shubin and SG classes. Furthermore, we discuss why this stability result does not hold in the standard H\"ormander classes.