On the sharp Gevrey regularity for a generalization of the M\'etivier operator

TL;DR

This paper establishes the sharp Gevrey regularity for a generalized Métivier operator, extending previous hypoellipticity results to a broader class involving higher powers and mixed terms in the operator.

Contribution

It provides the first sharp Gevrey hypoellipticity results for a generalized class of Métivier operators with polynomially weighted derivatives.

Findings

Proves sharp Gevrey hypoellipticity for the generalized operator

Extends previous hypoellipticity results to more complex operators

Identifies conditions under which the operator exhibits Gevrey regularity

Abstract

The sharp Gevrey hypoellipticity is provided for the following generalization of the M\'etivier operator, "Non-hypoellipticit\'e analytique pour $D_{x}^{2}+\left( x^{2} + y^{2}\right)D_{y}^{2}$" by G. M\'etivier, \begin{align*} D_{x}^{2}+\left(x^{2n+1}D_{y}\right)^{2}+\left(x^{n}y^{m}D_{y}\right)^{2}, \end{align*} in $\Omega$ open neighborhood of the origin in $\mathbb{R}^{2}$, where $n$ and $m$ are positive integers.