# zbMATH — the first resource for mathematics

Remarks about global analytic hypoellipticity. (English) Zbl 0932.35046
In the paper under consideration a necessary and sufficient condition for the global analytic hypoellipticity (GAH) on the torus $$T^2$$ of the first-order operator $$L= \partial_t+ (a(t)+ ib(t))\partial_x$$ is proved. The coefficients $$a$$, $$b$$ of $$L$$ are real-valued, real-analytic functions on the unit circle.
In Section 3 a necessary and sufficient condition for GAH of the involutive system of vector fields $$L_j= \partial_j+ c_j(t_j)\partial_x$$, $$j= 1,\dots, n$$ on $$T^{n+1}$$ is shown. The author proposes several examples illustrating his main results.

##### MSC:
 35H10 Hypoelliptic equations
Full Text:
##### References:
 [1] M. S. Baouendi and F. Trèves, A local constancy principle for the solutions of certain overdetermined systems of first-order linear partial differential equations, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 245 – 262. [2] Adalberto P. Bergamasco, Perturbations of globally hypoelliptic operators, J. Differential Equations 114 (1994), no. 2, 513 – 526. · Zbl 0815.35009 · doi:10.1006/jdeq.1994.1158 · doi.org [3] Adalberto P. Bergamasco, Paulo D. Cordaro, and Pedro A. Malagutti, Globally hypoelliptic systems of vector fields, J. Funct. Anal. 114 (1993), no. 2, 267 – 285. · Zbl 0777.58041 · doi:10.1006/jfan.1993.1068 · doi.org [4] N. G. de Bruijn, Asymptotic methods in analysis, 3rd ed., Dover Publications, Inc., New York, 1981. · Zbl 0556.41021 [5] Fernando Cardoso and Jorge Hounie, Globally hypoanalytic first order evolution equations, An. Acad. Brasil. Ci. 49 (1977), no. 4, 533 – 535. · Zbl 0371.35007 [6] Paulo D. Cordaro and A. Alexandrou Himonas, Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1 (1994), no. 4, 501 – 510. · Zbl 0836.35036 · doi:10.4310/MRL.1994.v1.n4.a10 · doi.org [7] T. Gramchev, P. Popivanov, and M. Yoshino, Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients, Rend. Sem. Mat. Univ. Politec. Torino 51 (1993), no. 2, 145 – 172 (1994). · Zbl 0824.35027 [8] Stephen J. Greenfield, Hypoelliptic vector fields and continued fractions, Proc. Amer. Math. Soc. 31 (1972), 115 – 118. · Zbl 0229.35024 [9] François Trèves, Analytic-hypoelliptic partial differential equations of principal type, Comm. Pure Appl. Math. 24 (1971), 537 – 570. · Zbl 0222.35014 · doi:10.1002/cpa.3160240407 · doi.org [10] François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. · Zbl 0565.35079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.