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

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

1994-11-30

arxiv.org/abs/math/9411202v1

Mathematics

Complex Variables

Scientific paper

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.

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