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ISSN 1088-6850(e) ISSN 0002-9947(p)

Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori

Author(s): David S. Tartakoff
Journal: Trans. Amer. Math. Soc. 348 (1996), 2577-2583.
MSC (1991): Primary 32F10, 35N15, 35B65
MathSciNet review: 1344213
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Abstract: 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 Hörmander 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

$\begin{displaymath}P=\left ( \frac \partial {\partial x_1}\right ) ^2+\left ( \frac \partial { \partial x_2}\right ) ^2+\left ( a(x_1,x_2)\frac \partial {\partial t}\right )^2 \end{displaymath}$

(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 [4] and Himonas in [8] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [1], Hanges and Himonas [6], and Christ [3]) were globally analytic hypoelliptic on products of tori.

References:

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