An example of a second-order nonhypoelliptic operator with the property of global hypoellipticity

V. S. Fedii

Matematičeskie zametki, Tome 12 (1972) no. 3, pp. 269-274.

Résumé

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.