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

DOI:
https://doi.org/10.1090/S0002-9947-96-01573-5

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 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

(with analytic , 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.

**1.**M. S. Baouendi and C. Goulaouic,*Analyticity for degenerate elliptic equations and applications*, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1971, pp. 79--84. MR**50:5167****2.**A. Bove and D. S. Tartakoff,*Microlocal Gevrey hypoellipticity for subelliptic operators*, (to appear).**3.**M. Christ,*Certain sums of squares of vector fields fail to be analytic hypoelliptic*, Comm. Partial Differential Equations**10**(1991), 1695--1707. MR**92k:35056****4.**P. Cordaro and A. Himonas,*Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus*, Math. Res. Lett.**1**(1994), 501--510. MR**95j:05048****5.**M. Derridj and D. S. Tartakoff,*Global analyticity for on three dimensional pseudoconvex CR manifolds*, Comm. Partial Differential Equations**18**(11) 1993, 1847--1868. MR**94i:32021****6.**N. Hanges and A. Himonas,*Singular solutions for sums of squares of vector fields*, Comm. Partial Differential Equations**16**(1991), 1503--1511. MR**92i:35031****7.**B. Helffer and C. Mattera,*Analyticité de itérés réduits d'un système de champs de vecteurs*, Comm. Partial Differential Equations**5**(1980), 1065--1072. MR**81m:35034****8.**A. Alexandrou Himonas,*On degenerate elliptic operators of infinite type*, Math. Z. (to appear).**9.**L. Hormander,*Linear partial differential operators*, Springer-Verlag, New York, 1969. MR**40:1687****10.**P. Popivanov and D. S. Tartakoff,*Gevrey hypoellipticity for fourth order differential operators*, Comm. Partial Differential Equations**20**(1995), 309--314. MR**95i:35056****11.**D. S. Tartakoff,*Gevrey hypoellipticity for subelliptic boundary value problems*, Comm. Pure Appl. Math.**26**(1973), 251--312. MR**49:7586****12.**D. S. Tartakoff,*On the global real analyticity of solutions to on compact manifolds*, Comm. Partial Differential Equations**1**(1976), 283--311. MR**53:14552****13.**D. S. Tartakoff,*Local analytic hypoellipticity for on nondegenerate Cauchy Riemann manifolds*, Proc. Nat. Acad. Sci. U.S.A.**75**(1978), 3027--3028. MR**80g:58045**

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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:
https://doi.org/10.1090/S0002-9947-96-01573-5

Received by editor(s):
November 21, 1994

Article copyright:
© Copyright 1996
American Mathematical Society