# Global Hypoellipticity for Strongly Invariant Operators

## TL;DR

This paper establishes a necessary and sufficient condition for the global hypoellipticity of invariant operators by analyzing their matrix symbols at infinity, linking it to subelliptic estimates.

## Contribution

It introduces a new criterion based on the behavior at infinity of matrix symbols for determining global hypoellipticity of invariant operators.

## Key findings

- Derived a necessary and sufficient condition for global hypoellipticity.
- Connected global hypoellipticity with subelliptic estimates.
- Provided analysis of matrix symbols at infinity for invariant operators.

## Abstract

In this note, by analyzing the behavior at infinity of the matrix symbol of an invariant operator $P$ with respect to a fixed elliptic operator, we obtain a necessary and sufficient condition to guarantee that $P$ is globally hypoelliptic. We also investigate relations between the global hypoellipticity of $P$ and global subelliptic estimates.