Abstract:
It is proved that the operator
$$
P\equiv-\frac{\partial^2}{\partial x_1^2}-\sum_{k=2}^n\frac{\partial}{\partial x_k}\varphi^2(x)\frac\partial{\partial x_k},
$$
where $\varphi(x)\in C^\infty(\Omega)$ ($\Omega$ is a domain in $\mathbf{R}^n$),
$\{x: \varphi(x)=0\}$ is a compactum in $\Omega$ which is the closure of its internal points,
has the property of global hypoellipticity in $\Omega$, i.e.,
$$
v\in D'(\Omega),\qquad Pv\in C^\infty(\Omega)\Longrightarrow v\in C^\infty(\Omega).
$$
This operator is not hypoelliptic.