Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori
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- by David S. Tartakoff PDF
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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 \[ P=\left ( \frac \partial {\partial x_1}\right ) ^2+\left ( \frac \partial { \partial x_2}\right ) ^2+\left ( a(x_1,x_2)\frac \partial {\partial t}\right )^2 \] (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 [Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1 (1994), 501–510] and Himonas in [On degenerate elliptic operators of infinite type, Math. Z. (to appear)] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [Analyticity for degenerate elliptic equations and applications, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1971, pp. 79–84], Hanges and Himonas [Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), 1503–1511], and Christ [Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 10 (1991), 1695–1707]) were globally analytic hypoelliptic on products of tori.References
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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
- Received by editor(s): November 21, 1994
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 2577-2583
- MSC (1991): Primary 32F10, 35N15, 35B65
- DOI: https://doi.org/10.1090/S0002-9947-96-01573-5
- MathSciNet review: 1344213