On a criterion for hypoellipticity (Q1078751)

Consider \(\Omega \subset R^ n\) an open set and \(P=p(x,D_ x)\) a second order differential operator with real valued coefficients in \(C^{\infty}(\Omega)\). Assume that for any \(\epsilon >0\) and any compact \(K\subset \Omega\) there is a constant C(\(\epsilon\),K) such that \[ \| (\log (| D_ x|^ 2+1))^ 2u\| \leq \| \epsilon p(x,D_ x)u\| +C\| u\|,\quad u\in C_ 0^{\infty}(K). \] If \[ \sum^{n}_{j=1}\| P^{(j)}u\|^ 2+\| P_{(j)}u\|^ 2_{-1}\leq C_ 1(Re(Pu,\quad u)+\| u\|^ 2) \] holds for some \(C_ 1\), where \(P^{(j)}=(\partial /\partial \xi_ j)p(x,\xi)\), \(P_{(j)}=D_{xj}p(x,\xi)\) \(\| \|\) is the \(L^ 2\) norm and \(\| \|_{-1}\) is the \(H^{-1}\)-norm, then P is hypoelliptic in \(\Omega\). The result is extended to operators of higher order. An \(R^ 3\) application is given.