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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: 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.


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Additional Information

David S. Tartakoff
Affiliation: Department of Mathematics, University of Illinois at Chicago, 851 S. Morgan St., m/c 349, Chicago, Illinois 60607-7045
Email: dst@uic.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01573-5
PII: S 0002-9947(96)01573-5
Received by editor(s): November 21, 1994
Article copyright: © Copyright 1996 American Mathematical Society