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Lower order perturbation and global analytic vectors for a class of globally analytic hypoelliptic operators


Authors: N. Braun Rodrigues, G. Chinni, P. D. Cordaro and M. R. Jahnke
Journal: Proc. Amer. Math. Soc. 144 (2016), 5159-5170
MSC (2010): Primary 35H10, 35H05, 35N15
DOI: https://doi.org/10.1090/proc/13178
Published electronically: May 31, 2016
MathSciNet review: 3556261
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Abstract: In this work we return to the class of globally analytic hypoelliptic Hörmander's operators defined on the $ N$-dimensional torus introduced by Cordaro and Himonas and prove that if $ P$ is any operator in this class, then a perturbation of $ P$ by an analytic pseudodifferential operator with degree smaller than the subelliptic index of $ P$ remains globally analytic hypoelliptic. We also study the Gevrey regularity of the Gevrey vectors for such a class and at the end we also show that Cordaro and Himonas's result can be extended to a similar class of operators now defined in a product of compact Lie group by a compact manifold.


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  • [1] Angela A. Albanese and David Jornet, Global regularity in ultradifferentiable classes, Ann. Mat. Pura Appl. (4) 193 (2014), no. 2, 369-387. MR 3180923, https://doi.org/10.1007/s10231-012-0279-5
  • [2] P. Bolley, J. Camus, and C. Mattera, Analyticité microlocale et itères d'opérateurs, Seminairie Goulaouic-Schwartz, Ecole Polytechnique France (1978-79).
  • [3] P. Bolley, J. Camus, and L. Rodino, Hypoellipticité analytique-Gevrey et itérés d'opérateurs, Rend. Sem. Mat. Univ. Politec. Torino 45 (1987), no. 3, 1-61 (1989) (French). MR 1037999
  • [4] Michael Christ, Global analytic hypoellipticity in the presence of symmetry, Math. Res. Lett. 1 (1994), no. 5, 559-563. MR 1295550, https://doi.org/10.4310/MRL.1994.v1.n5.a4
  • [5] 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. MR 1302393, https://doi.org/10.4310/MRL.1994.v1.n4.a10
  • [6] 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, https://doi.org/10.1090/S0002-9947-98-01987-4
  • [7] Aparajita Dasgupta and Michael Ruzhansky, Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces, Bull. Sci. Math. 138 (2014), no. 6, 756-782. MR 3251455, https://doi.org/10.1016/j.bulsci.2013.12.001
  • [8] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR 1738431
  • [9] A. Alexandrou Himonas and Gerson Petronilho, On $ C^\infty $ and Gevrey regularity of sublaplacians, Trans. Amer. Math. Soc. 358 (2006), no. 11, 4809-4820 (electronic). MR 2231873, https://doi.org/10.1090/S0002-9947-06-03819-0
  • [10] A. Alexandrou Himonas and Gerson Petronilho, On Gevrey regularity of globally $ C^\infty $ hypoelliptic operators, J. Differential Equations 207 (2004), no. 2, 267-284. MR 2102665, https://doi.org/10.1016/j.jde.2004.07.023
  • [11] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. MR 0222474
  • [12] Lars Hörmander, The analysis of linear partial differential operators. IV: Fourier integral operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. MR 781537
  • [13] Takeshi Kotake and Mudumbai S. Narasimhan, Regularity theorems for fractional powers of a linear elliptic operator, Bull. Soc. Math. France 90 (1962), 449-471. MR 0149329
  • [14] Kil Hyun Kwon, Concatenations applied to analytic hypoellipticity of operators with double characteristics, Trans. Amer. Math. Soc. 283 (1984), no. 2, 753-763. MR 737898, https://doi.org/10.2307/1999160
  • [15] Alberto Parmeggiani, A remark on the stability of $ C^\infty $-hypoellipticity under lower-order perturbations, J. Pseudo-Differ. Oper. Appl. 6 (2015), no. 2, 227-235. MR 3351885, https://doi.org/10.1007/s11868-015-0118-8
  • [16] Guy Métivier, Propriété des itérés et ellipticité, Comm. Partial Differential Equations 3 (1978), no. 9, 827-876 (French). MR 504629, https://doi.org/10.1080/03605307808820078
  • [17] Gerson Petronilho, On Gevrey solvability and regularity, Math. Nachr. 282 (2009), no. 3, 470-481. MR 2503164, https://doi.org/10.1002/mana.200810748
  • [18] Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247-320. MR 0436223
  • [19] E. M. Stein, An example on the Heisenberg group related to the Lewy operator, Invent. Math. 69 (1982), no. 2, 209-216. MR 674401, https://doi.org/10.1007/BF01399501

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Additional Information

N. Braun Rodrigues
Affiliation: Universidade de São Paulo, IME-USP, São Paulo, SP, Brazil
Email: braun@ime.usp.br

G. Chinni
Affiliation: Universidade de São Paulo, IME-USP, São Paulo, SP, Brazil
Email: gregorio.chinni@gmail.com

P. D. Cordaro
Affiliation: Universidade de São Paulo, IME-USP, São Paulo, SP, Brazil
Email: cordaro@ime.usp.br

M. R. Jahnke
Affiliation: Universidade de São Paulo, IME-USP, São Paulo, SP, Brazil
Email: jahnke@ime.usp.br

DOI: https://doi.org/10.1090/proc/13178
Keywords: Sums of squares, global analytic hypoellipticity, Gevrey vectors.
Received by editor(s): October 30, 2015
Received by editor(s) in revised form: January 29, 2016
Published electronically: May 31, 2016
Additional Notes: The first and fourth authors were supported by doctoral fellowships from CNPq
The second author was supported by a posdoctoral fellowship from Fapesp
The third author was partially supported by CNPq and Fapesp
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2016 American Mathematical Society

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