## On $C^\infty$ and Gevrey regularity of sublaplacians

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- by A. Alexandrou Himonas and Gerson Petronilho PDF
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**358**(2006), 4809-4820 Request permission

## Abstract:

In this paper we consider zero order perturbations of a class of sublaplacians on the two-dimensional torus and give sufficient conditions for global $C^\infty$ regularity to persist. In the case of analytic coefficients, we prove Gevrey regularity for a general class of sublaplacians when the finite type condition holds.## References

- Kazuo Amano,
*The global hypoellipticity of a class of degenerate elliptic-parabolic operators*, Proc. Japan Acad. Ser. A Math. Sci.**60**(1984), no. 9, 312–314. MR**778515** - M. S. Baouendi and C. Goulaouic,
*Nonanalytic-hypoellipticity for some degenerate elliptic operators*, Bull. Amer. Math. Soc.**78**(1972), 483–486. MR**296507**, DOI 10.1090/S0002-9904-1972-12955-0 - Denis R. Bell and Salah Eldin A. Mohammed,
*An extension of Hörmander’s theorem for infinitely degenerate second-order operators*, Duke Math. J.**78**(1995), no. 3, 453–475. MR**1334203**, DOI 10.1215/S0012-7094-95-07822-3 - E. Bernardi, A. Bove, and D. S. Tartakoff,
*On a conjecture of Treves: analytic hypoellipticity and Poisson strata*, Indiana Univ. Math. J.**47**(1998), no. 2, 401–417. MR**1647900**, DOI 10.1512/iumj.1998.47.1409 - Paulo D. Cordaro and A. Alexandrou Himonas,
*Global analytic regularity for sums of squares of vector fields*, Trans. Amer. Math. Soc.**350**(1998), no. 12, 4993–5001. MR**1433115**, DOI 10.1090/S0002-9947-98-01987-4 - Makhlouf Derridj,
*Un problème aux limites pour une classe d’opérateurs du second ordre hypoelliptiques*, Ann. Inst. Fourier (Grenoble)**21**(1971), no. 4, 99–148 (French, with English summary). MR**601055**, DOI 10.5802/aif.395 - V. S. Fediĭ,
*Estimates in the $H_{(s)}$ norms, and hypoellipticity*, Dokl. Akad. Nauk SSSR**193**(1970), 301–303 (Russian). MR**0271536** - Daisuke Fujiwara and Hideki Omori,
*An example of a globally hypo-elliptic operator*, Hokkaido Math. J.**12**(1983), no. 3, 293–297. MR**719969**, DOI 10.14492/hokmj/1470081007 - 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). MR**1289385** - Stephen J. Greenfield and Nolan R. Wallach,
*Global hypoellipticity and Liouville numbers*, Proc. Amer. Math. Soc.**31**(1972), 112–114. MR**296508**, DOI 10.1090/S0002-9939-1972-0296508-5 - A. Alexandrou Himonas,
*On degenerate elliptic operators of infinite type*, Math. Z.**220**(1995), no. 3, 449–460. MR**1362255**, DOI 10.1007/BF02572625 - A. Alexandrou Himonas,
*Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions*, Proc. Amer. Math. Soc.**129**(2001), no. 7, 2061–2067. MR**1825918**, DOI 10.1090/S0002-9939-00-05996-7 - A. Alexandrou Himonas and Gerson Petronilho,
*Global hypoellipticity and simultaneous approximability*, J. Funct. Anal.**170**(2000), no. 2, 356–365. MR**1740656**, DOI 10.1006/jfan.1999.3524 - Lars Hörmander,
*Hypoelliptic second order differential equations*, Acta Math.**119**(1967), 147–171. MR**222474**, DOI 10.1007/BF02392081 - Lars Hörmander,
*The analysis of linear partial differential operators. I*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR**717035**, DOI 10.1007/978-3-642-96750-4 - J. J. Kohn,
*Pseudo-differential operators and hypoellipticity*, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 61–69. MR**0338592** - O. A. Oleĭnik and E. V. Radkevič,
*Second order equations with nonnegative characteristic form*, Plenum Press, New York-London, 1973. Translated from the Russian by Paul C. Fife. MR**0457908**, DOI 10.1007/978-1-4684-8965-1 - Luigi Rodino,
*Linear partial differential operators in Gevrey spaces*, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR**1249275**, DOI 10.1142/9789814360036 - Linda Preiss Rothschild and E. M. Stein,
*Hypoelliptic differential operators and nilpotent groups*, Acta Math.**137**(1976), no. 3-4, 247–320. MR**436223**, DOI 10.1007/BF02392419 - E. M. Stein,
*An example on the Heisenberg group related to the Lewy operator*, Invent. Math.**69**(1982), no. 2, 209–216. MR**674401**, DOI 10.1007/BF01399501 - K. Taira,
*Le principe du maximum et l’hypoellipticité globale*, Bony-Sjöstrand-Meyer seminar, 1984–1985, École Polytech., Palaiseau, 1985, pp. Exp. No. 1, 11 (French). MR**819767** - David S. Tartakoff,
*Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori*, Trans. Amer. Math. Soc.**348**(1996), no. 7, 2577–2583. MR**1344213**, DOI 10.1090/S0002-9947-96-01573-5 - François Treves,
*Symplectic geometry and analytic hypo-ellipticity*, Differential equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, pp. 201–219. MR**1662756**, DOI 10.1090/pspum/065/1662756

## Additional Information

**A. Alexandrou Himonas**- Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
- MR Author ID: 86060
- Email: himonas.1@nd.edu
**Gerson Petronilho**- Affiliation: Department of Mathematics, Federal University of São Carlos, São Carlos, SP 13565-905, Brazil
- MR Author ID: 250320
- Email: gerson@dm.ufscar.br
- Received by editor(s): July 29, 2003
- Received by editor(s) in revised form: August 5, 2004
- Published electronically: January 24, 2006
- Additional Notes: The first author was partially supported by the NSF under grant number DMS-9970857, and the second author was partially supported by CNPq.
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**358**(2006), 4809-4820 - MSC (2000): Primary 35H10, 35B20
- DOI: https://doi.org/10.1090/S0002-9947-06-03819-0
- MathSciNet review: 2231873