On a problem of hypoellipticity (Q1116016)

The author presents a proof of the following result. Let P(x,D) be an m- th order partial differential operator of principal type with analytic coefficients in an open subset \(\Omega\) of \({\mathbb{R}}^ n\). If P is not hypoelliptic in \(\Omega\), there exists an open subset \(\omega_ 0\) and \(\Omega\) such that, for any \(\ell \geq m\), one can find \(u\in C^{\ell}(\omega_ 0)\setminus C^{\ell +1}(\omega_ 0)\) with \(P(x,D)u=0\) in \(\omega_ 0\).