Analytic hypoellipticity for sums of squares of vector fields

A. Alexandrou Himonas

Annales Polonici Mathematici (1998)

  • Volume: 70, Issue: 1, page 117-129
  • ISSN: 0066-2216

Abstract

top
We discuss the open problem of analytic hypoellipticity for sums of squares of vector fields, including some recent partial results and a conjecture of Treves.

How to cite

top

A. Alexandrou Himonas. "Analytic hypoellipticity for sums of squares of vector fields." Annales Polonici Mathematici 70.1 (1998): 117-129. <http://eudml.org/doc/262561>.

@article{A1998,
abstract = {We discuss the open problem of analytic hypoellipticity for sums of squares of vector fields, including some recent partial results and a conjecture of Treves.},
author = {A. Alexandrou Himonas},
journal = {Annales Polonici Mathematici},
keywords = {analytic hypoellipticity; sum of squares of vector fields; eigenvalue; bracket condition; characteristic set; symplectic; torus; up to date review of the status of the theory},
language = {eng},
number = {1},
pages = {117-129},
title = {Analytic hypoellipticity for sums of squares of vector fields},
url = {http://eudml.org/doc/262561},
volume = {70},
year = {1998},
}

TY - JOUR
AU - A. Alexandrou Himonas
TI - Analytic hypoellipticity for sums of squares of vector fields
JO - Annales Polonici Mathematici
PY - 1998
VL - 70
IS - 1
SP - 117
EP - 129
AB - We discuss the open problem of analytic hypoellipticity for sums of squares of vector fields, including some recent partial results and a conjecture of Treves.
LA - eng
KW - analytic hypoellipticity; sum of squares of vector fields; eigenvalue; bracket condition; characteristic set; symplectic; torus; up to date review of the status of the theory
UR - http://eudml.org/doc/262561
ER -

References

top
  1. [BG] M. S. Baouendi and C. Goulaouic, Nonanalytic-hypoellipticity for some degenerate elliptic operators, Bull. Amer. Math. Soc. 78 (1972), 483-486. Zbl0276.35023
  2. [BM] D. R. Bell and S. A. Mohammed, An extension of Hörmander's theorem for infinitely degenerate second-order operators, Duke Math. J. 78 (1995), 453-475. Zbl0840.60053
  3. [BT] L. Boutet de Monvel and F. Treves, On a class of pseudo-differential operators with double characteristics, Invent. Math. 24 (1974), 1-34. Zbl0281.35083
  4. [BTa1] A. Bove and D. S. Tartakoff, Optimal non-isotropic Gevrey exponents for sums of squares of vector fields, Comm. Partial Differential Equations 22 (1997), 1263-1282. Zbl0921.35043
  5. [BTa2] A. Bove and D. S. Tartakoff, On a conjecture of Treves: Analytic hypoellipticity and Poisson strata, preprint, 1997. 
  6. [C] S. C. Chen, Global analytic hypoellipticity of the ∂̅-Neumann problem on circular domains, Invent. Math. 92 (1988), 173-185. Zbl0621.35067
  7. [Ch1] M. Christ, A class of hypoelliptic PDE admitting nonanalytic solutions, in: Contemp. Math. 137, Amer. Math. Soc., 1992, 155-167. 
  8. [Ch2] M. Christ, A necessary condition for analytic hypoellipticity, Math. Res. Lett. 1 (1994), 241-248. Zbl0841.35026
  9. [Ch3] M. Christ, Global analytic hypoellipticity in the presence of symmetry, ibid., 559-563. Zbl0841.35027
  10. [Ch4] M. Christ, A progress report on analytic hypoellipticity, in: Geometric Complex Analysis, J. Noguchi (ed.), World Sci., 1996, 123-146. Zbl0924.47029
  11. [Ch5] M. Christ, Intermediate optimal Gevrey exponents occur, Comm. Partial Differential Equations 22 (1997), 359-379. Zbl0893.35021
  12. [CH1] P. D. Cordaro and A. A. Himonas, Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1 (1994), 501-510. Zbl0836.35036
  13. [CH2] P. D. Cordaro and A. A. Himonas, Global analytic hypoellipticity for sums of squares of vector fields, Trans. Amer. Math. Soc., to appear. Zbl0914.35087
  14. [D] M. Derridj, Un problème aux limites pour une classe d'opérateurs du second ordre hypoelliptiques, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 4, 99-148. Zbl0215.45405
  15. [DT] M. Derridj and D. S. Tartakoff, Global analyticity for b on three dimensional CR manifolds, Comm. Partial Differential Equations 18 (1993), 1847-1868. Zbl0791.35087
  16. [DZ] M. Derridj et C. Zuily, Régularité analytique et Gevrey d'opérateurs elliptiques dégénérés, J. Math. Pures Appl. 52 (1973), 65-80. Zbl0263.35020
  17. [F] V. S. Fediĭ, Estimates in H ( s ) norms and hypoellipticity, Dokl. Akad. Nauk SSSR 193 (1970), 301-303 (in Russian). 
  18. [G] A. Gilioli, A class of second-order evolution equations with double characteristics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 187-229. Zbl0327.35018
  19. [GT] A. Gilioli and F. Treves, An example in the solvability theory of linear PDE's, Amer. J. Math. 96 (1974), 367-385. Zbl0308.35022
  20. [GR] A. Grigis and L. P. Rothschild, A criterion for analytic hypoellipticity of a class of differential operators with polynomial coefficients, Ann. of Math. 118 (1983), 443-460. Zbl0541.35017
  21. [GS] A. Grigis et J. Sjöstrand, Front d'onde analytique et sommes de carrés de champs de vecteurs, Duke Math. J. 52 (1985), 35-51. Zbl0581.35009
  22. [GW] S. J. Greenfield and N. R. Wallach, Global hypoelliptic and Liouville numbers, Proc. Amer. Math. Soc. 31 (1972), 112-114. 
  23. [Gr] V. V. Grushin, On a class of hypoelliptic operators, Mat. Sb. 83 (1970), 456-473 (in Russian). 
  24. [HH1] N. Hanges and A. A. Himonas, Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), 1503-1511. Zbl0745.35011
  25. [HH2] N. Hanges and A. A. Himonas, Analytic hypoellipticity for generalized Baouendi Goulaouic operators, J. Funct. Anal. 125 (1994), 309-325. Zbl0812.35026
  26. [HH3] N. Hanges and A. A. Himonas, Singular solutions for a class of Grusin type operators, Proc. Amer. Math. Soc. 124 (1996), 1549-1557. Zbl0858.35025
  27. [HH4] N. Hanges and A. A. Himonas, Non-analytic hypoellipticity in the presence of symplecticity, ibid. 126 (1998), 405-409. 
  28. [He] B. Helffer, Conditions nécessaires d'hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J. Differential Equations 44 (1982), 460-481. Zbl0458.35019
  29. [Ho] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. Zbl0156.10701
  30. [K] J. J. Kohn, Pseudo-differential operators and hypoellipticity, in: Proc. Sympos. Pure Math. 23, Amer. Math. Soc., 1973, 61-70. Zbl0262.35007
  31. [Ko] G. Komatsu, Global analytic hypoellipticity of the ∂̅-Neumann problem, Tôhoku Math. J. 28 (1976), 145-156. Zbl0319.35056
  32. [KS] S. Kusuoka and D. Strook, Applications of the Malliavin calculus, Part II, J. Fac. Sci. Tokyo Sect. IA Math. 32 (1985), 1-76. 
  33. [Kw] K. H. Kwon, Concatenations applied to analytic hypoellipticity of operators with double characteristics, Trans. Amer. Math. Soc., 283 (1984), 753-763. Zbl0542.35024
  34. [M] T. Matsuzawa, Sur les équations u t t + t α u x x = f ( α 0 ) , Nagoya Math. J. 42 (1971), 43-55. 
  35. [Me] A. Menikoff, Some examples of hypoelliptic partial differential equations, Math. Ann. 221 (1976), 167-181. Zbl0323.35019
  36. [Met1] G. Métivier, Analytic hypoellipticity for operators with multiple characteristics, Comm. Partial Differential Equations 1 (1981), 1-90. Zbl0455.35040
  37. [Met2] G. Métivier, Une classe d'opérateurs non hypoelliptiques analytiques, Indiana Univ. Math. J. 29 (1980), 823-860. Zbl0455.35041
  38. [OR1] O. A. Oleĭnik and E. V. Radkevich, On the analyticity of solutions of linear differential equations and systems, Dokl. Akad. Nauk SSSR 207 (1972), 1614-1618 (in Russian). Zbl0266.35001
  39. [OR2] O. A. Oleĭnik and E. V. Radkevich, Second Order Equations with Nonnegative Characteristic Form, Amer. Math. Soc. and Plenum Press, 1973. 
  40. [PR] Pham The Lai et D. Robert, Sur un problème aux valeurs propres non linéaire, Israel J. Math. 36 (1980), 169-186. Zbl0444.35065
  41. [RS] L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1977), 247-320. Zbl0346.35030
  42. [S] J. Sjöstrand, Analytic wavefront sets and operators with multiple characteristics, Hokkaido Math. J. 12 (1983), 392-433. Zbl0531.35022
  43. [St] E. M. Stein, An example on the Heisenberg group related to the Lewy operator, Invent. Math. 69 (1982), 209-216. Zbl0515.58032
  44. [Ta1] D. S. Tartakoff, On the local real analyticity of solutions to b and the ∂̅-Neumann problem, Acta Math. 145 (1980), 117-204. 
  45. [Ta2] D. S. Tartakoff, On the global real analyticity of solutions to b on compact manifolds, Comm. Partial Differential Equations 1 (1976), 283-311. 
  46. [Ta3] D. S. Tartakoff, Global (and local) analyticity for second order orerators constructed from rigid vector fields on products of tori, Trans. Amer. Math. Soc. 348 (1996), 2577-2583. Zbl0863.58063
  47. [Tr1] F. Treves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the ∂̅-Neumann problem, Comm. Partial Differential Equations 3 (1978), 475-642. Zbl0384.35055
  48. [Tr2] F. Treves, Concatenations of second-order evolution equations applied to local solvability and hypoellipticity, Comm. Pure Appl. Math. 26 (1973), 201-250. Zbl0266.35060
  49. [Tr3] F. Treves, Symplectic geometry and analytic hypo-ellipticity, preprint, 1996. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.