Global (and Local) Analyticity for Second Order Operators Constructed from Rigid Vector Fields on Products of Tori

We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the H\"ormander condition and where $P$ satisfies a `maximal' estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is $$P= \left({\partial \over {\partial x_1}}\right)^2 + \left({\partial \over {\partial x_2}}\right)^2 + \left(a(x_1,x_2){\partial \over {\partial t}}\right)^2.$$ (with analytic $a(x), a(0)=0,$ naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in \cite{Cordaro-Himonas 1994} and Himonas in \cite{Himonas 199X} which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic \cite{Baouendi-Goulaouic 1971}, Hanges and Himonas \cite{Hanges-Himonas 1991}, and Christ \cite{Christ 1991a}) were {\it globally} analytic hypoelliptic on products of tori.