On hypoelliptic operators with multiple characteristics of odd order (Q796730)

Denote by \(\bar S^ m\) the set of functions p(x,\(\xi)\in {\mathcal B}^{\infty}({\mathbb{R}}^ n_ x\times {\mathbb{R}}^ n_{\xi})\) such that p(x,\(\xi)\) is positively homogeneous of degree m with respect to \(\xi\) if \(| \xi | \geq 1/2\). The author studies the hypoellipticity of pseudodifferential operators \(L(x,D)=P(x,D)+A(x,D)\) where P and A have principal symbols \(p_ 1^{\ell}\) (with \(p_ 1\in \bar S^ 1)\) and \(a\in \bar S^{\ell -1}\) respectively. The author shows that if certain conditions (too detailed to state here) hold, L satisfies suitable a priori estimates which imply that it is hypoelliptic. These estimates are obtained by adapting the techniques developed in the proof of the Egorov- Hörmander subellipticity theorem. A counterexample is given which shows that if one of the conditions does not hold, L may fail to be hypoelliptic.