Analytic and Gevrey Hypoellipticity for Perturbed Sums of Squares Operators

TL;DR

This paper investigates how pseudodifferential perturbations affect the Gevrey and analytic hypoellipticity of sums of squares operators satisfying Hörmander's condition, establishing conditions for minimal regularity and hypoellipticity preservation.

Contribution

It demonstrates that certain pseudodifferential perturbations preserve Gevrey regularity and analytic hypoellipticity for sums of squares operators under specific conditions.

Findings

Perturbations of order less than the subelliptic index preserve Gevrey regularity.

Under additional conditions, pseudodifferential perturbations maintain analytic hypoellipticity.

Results extend understanding of regularity properties under perturbations for sums of squares operators.

Abstract

We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying H\"ormander's condition. The first is on the minimal Gevrey regularity: if a sum of squares with analytic coefficients is perturbed with a pseudodifferential operator of order strictly less than its subelliptic index it still has the Gevrey minimal regularity. We also prove a statement concerning real analytic hypoellipticity for the same type of pseudodifferential perturbations, provided the operator satisfies to some extra conditions (see Theorem 1.2 below) that ensure the analytic hypoellipticity.