Analytic hypoellipticity for generalized Baouendi-Goulaouic operators (Q1340831)

The authors consider the linear differential operator \[ P= \biggl( {\partial\over {\partial x}} \biggr)^ 2+ \biggl( a(x) {\partial\over {\partial y}} \biggr)^ 2+ \biggl( {\partial \over {\partial t}} \biggr)^ 2, \] \(a(x)\) being real-analytic, real-valued function near 0 and prove that \(P\) is analytic hypoelliptic at the origin in \(\mathbb{R}^ 3\) if and only if \(a(0)\neq 0\). This theorem generalizes some previous results of Baouendi, Goulaouic, Derridj, Zuily and the authors.