Analytic regularity for solutions to sums of squares: an assessment

Abstract

We present a brief survey on the state of the theory of the real analytic regularity (real analytic hypoellipticity) for the solutions to sums of squares of vector fields satisfying the Hörmander condition.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    Albano, P., Bove, A.: Wave front set of solutions to sums of squares of vector fields. Mem. Am. Math. Soc 221, 1039 (2013)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Albano, P., Bove, A., Chinni, G.: Minimal microlocal gevrey regularity for “sums of squares”. Int. Math. Res. Not. 12, 2275–2302 (2009)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Albano, P., Bove, A., Mughetti, M.: Analytic hypoellipticity for sums of squares and the treves conjecture. J. Funct. Anal. 274(10), 2725–2753 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Baouendi, M.S., Goulaouic, C.: Nonanalytic-hypoellipticity for some degenerate operators. Bull. Am. Math. Soc. 78, 483–486 (1972)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Bender, C.M., Wang, Q.: A class of exactly solvable eigenvalue problems. J. Phys. A 34, 9835–9847 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Berezin, F.A., Shubin, M.A.: The Schrödinger Equation, Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1991)

    Google Scholar 

  7. 7.

    Bolley, P., Camus, J., Nourrigat, J.: La condition de Hörmander-Kohn pour les opérateurs pseudo-différentiels. Commun. Partial Differ. Equ. 7(2), 197–221 (1982)

    MATH  Article  Google Scholar 

  8. 8.

    Bove, A., Mughetti, M.: Analytic and Gevrey hypoellipticity for a class of pseudodifferential operators in one variable. J. Differ. Equ. 255(4), 728–758 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Bove, A., Mughetti, M.: On a new method of proving Gevrey hypoellipticity for certain sums of squares. Adv. Math. 293, 146–220 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Bove, A., Mughetti, M.: Analytic hypoellipticity for sums of squares and the treves conjecture II. Anal. PDE 10(7), 1613–1635 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Bove, A., Mughetti, M.: Analytic hypoellipticity for sums of squares in the presence of symplectic non treves strata. J. Inst. Math. Jussieu. https://doi.org/10.1017/S1474748018000580

  12. 12.

    Bove, A., Mughetti, M.: Gevrey Regularity for a Class of Sums of Squares of Monomial Vector Fields, preprint, (2019)

  13. 13.

    Bove, A., Tartakoff, D.S.: Optimal non-isotropic Gevrey exponents for sums of squares of vector fields. Commun. Partial Differ. Equ. 22(7–8), 1263–1282 (1997)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Bove, A., Tartakoff, D.S.: Gevrey hypoellipticity for sums of squares of vector fields in \(\mathbb{R}^{2}\) with quasi-homogeneous polynomial vanishing. Indiana Univ. Math. J. 64, 613–633 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Bove, A., Treves, F.: On the Gevrey hypo-ellipticity of sums of squares of vector fields. Ann. Inst. Fourier (Grenoble) 54, 1443–1475 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Chanillo, S., Helffer, B., Laptev, A.: Nonlinear eigenvalues and analytic hypoellipticity. J. Funct. Anal. 209(2), 425–443 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Chinni, G.: (Semi)-global analytic hypoellipticity for a class of “sums of squares” which fail to be locally analytic hypoelliptic, to appear in Proc. Amer. Math. Soc., https://doi.org/10.1090/proc/14464

  18. 18.

    Chinni, G.: On the sharp Gevrey regularity for a generalization of the Métivier operator, preprint (2019)

  19. 19.

    Christ, M.: Certain sums of squares of vector fields fail to be analytic hypoelliptic. Commun. Partial Differ. Equ. 16, 1695–1707 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Christ, M.: Intermediate optimal Gevrey exponents occur. Commun. Partial Differ. Equ. 22(3–4), 359–379 (1997)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Cordaro, P.D., Hanges, N.: A new proof of Okaji’s theorem for a class of sum of squares operators. Ann. Inst. Fourier (Grenoble) 59(2), 595–619 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Derridj, M.: Un problème aux limites pour une classe d’opérateurs du second ordre hypoelliptiques. Ann. Inst. Fourier (Grenoble) 21(4), 99–148 (1971)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Ehrenpreis, L.: Solutions of some problems of division IV. Am. J. Math 82, 522–588 (1960)

    MATH  Article  Google Scholar 

  24. 24.

    Fefferman, C., Phong, D.H.: The uncertainty principle and sharp Gårding inequalities. Commun. Pure Appl. Math. 34, 285–331 (1981)

    MATH  Article  Google Scholar 

  25. 25.

    Grigis, A., Sjöstrand, J.: Front d’onde analytique et sommes de carrés de champs de vecteurs. Duke Math. J. 52(1), 35–51 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Grušin, V.V.: On a class of Hypoelliptic operators. Math. USSR Sb. 12, 458–476 (1970)

    Article  Google Scholar 

  27. 27.

    Grušin, V.V.: On a class of elliptic pseudodifferential operators degenerating at a submanifold. Mat. Sbornik 84(2), 163–195 (1971)

    Google Scholar 

  28. 28.

    Gundersen, G.G.: A class of anharmonic oscillators whose eigenfunctions have no recurrence relations. Proc. Am. Math. Soc. 58, 109–113 (1976)

    MATH  Article  Google Scholar 

  29. 29.

    Hanges, N., Himonas, A.A.: Singular solutions for sums of squares of vector fields. Commun. Partial Differ. Equ. 16, 1503–1511 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Hanges, N., Himonas, A.A.: Singular solutions for a class of Grusin type operators. Proc. Am. Math. Soc. 124(5), 1549–1557 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Hanges, N., Himonas, A.A.: Non-analytic hypoellipticity in the presence of symplecticity. Proc. Am. Math. Soc. 126(2), 405–409 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Helffer, B.: Sur l’hypoellipticité des opérateurs pseudodifférentiels à caractéristiques multiples (perte de 3/2 dérivées). Mémoires de la S. M. F. 51–52, 13–61 (1977)

    MATH  Google Scholar 

  33. 33.

    Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques. Astérisque 112, 9–197 (1984)

    MATH  Google Scholar 

  34. 34.

    Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Hörmander, L.: Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients. Commun. Pure Appl. Math. 24, 671–704 (1971)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Hörmander, L.: The Analysis of Partial Differential Operators, I. Springer, New York (1985)

    Google Scholar 

  37. 37.

    Hörmander, L.: The Analysis of Partial Differential Operators. Springer Verlag III, New York (1985)

    Google Scholar 

  38. 38.

    Hoshiro, T.: Failure of analytic hypoellipticity for some operators of \(X^{2}+Y^{2}\) type. J. Math. Kyoto Univ. 35(4), 569–581 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Kohn, J.J.: Pseudo-differential operators and non-elliptic problems, 1969 Pseudo-Diff. Operators (C.I.M.E., Stresa, pp. 158–165. Edizioni Cremonese, Roma (1968)

  40. 40.

    Kohn, J.J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18, 443–492 (1965)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Łojasiewicz, S.: Ensembles semi-analytiques, cours à Orsay, saisie en LaTeX par M. Coste, Juillet (1965)

    Google Scholar 

  42. 42.

    Métivier, G.: Une Classe d’Opérateurs Non Hypoelliptiques Analytiques. Indiana Univ. Math. J. 29, 823–860 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Métivier, G.: Hypoellipticité analitique sur des groupes nilpotents de rang 2. Duke Math. J. 47, 195–221 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Métivier, G.: Non-hypoellipticité Analytique pour \(D_{x}^{2} + (x^{2} + y^{2}) D_{y}^{2}\), C. R. Acad. Sci. Paris Sér. I Math. 292(7), 401–404 (1981)

  45. 45.

    Métivier, G.: Analytic hypoellipticity for operators with multiple characteristics. Comm. Partial Differ. Equ. 6(1), 1–90 (1981)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Mughetti, M.: Regularity properties of a double characteristics differential operator with complex lower order terms. J. Pseudo-Differ. Oper. Appl. 5(3), 343–358 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Mughetti, M.: On the spectrum of an anharmonic oscillator. Trans. Am. Math. Soc. 367, 835–865 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Ōkaji, T.: Analytic hypoellipticity for operators with symplectic characteristics. J. Math. Kyoto Univ. 25(3), 489–514 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Oleĭnik, O. A.: On the analyticity of solutions of partial differential equations and systems, Colloque International CNRS sur les Équations aux Dérivées Partielles Linéaires (Univ. Paris- Sud, Orsay, 1972), pp. 272–285. Astérisque, 2 et 3. Societé Mathématique de France, Paris (1973)

  50. 50.

    Oleĭnik, O.A., Radkevič, E.V.: The analyticity of the solutions of linear partial differential equations , (Russian) Mat. Sb. (N.S.), 90(132), 592–606 (1973)

  51. 51.

    Rothschild, L., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(3–4), 247–320 (1976)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Sjöstrand, J.: Analytic wavefront set and operators with multiple characteristics. Hokkaido Math. J. 12, 392–433 (1983)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Sjöstrand, J.: Singularités analytiques microlocales. Astérisque 95 (1982)

  54. 54.

    Tartakoff, D.S.: On the local real analyticity of solutions to \(\Box _{b}\) and the \(\bar{\partial }\)-Neumann problem. Acta Math. 145, 117–204 (1980)

    MathSciNet  Article  Google Scholar 

  55. 55.

    Tartakoff, D.S.: Elementary proofs of analytic hypoellipticity for \(\Box _{b}\) and the \(\overline{\partial }\)-Neumann problem, Analytic solutions of partial differential equations (Trento, 1981), 85–116; Astérisque, 89–90. Soc. Math. France, Paris (1981)

    Google Scholar 

  56. 56.

    Trèves, F.: Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the \(\bar{\partial }\)-Neumann problem. Commun. Partial Differ. Equ. 3(6–7), 475–642 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Treves, F.: Introduction to Pseudodifferential and Fourier Integral Operators, vol. 1. Plenum Press, New York (1980)

    Google Scholar 

  58. 58.

    Treves, F., Symplectic geometry and analytic hypo-ellipticity, in Differential equations, La Pietra,: (Florence). In: Proc. Sympos. Pure Math. 65, Amer. Math. Soc. Providence, vol. 1999, pp. 201–219 (1996)

  59. 59.

    Treves, F.: On the analyticity of solutions of sums of squares of vector fields, Phase space analysis of partial differential equations, Bove, Colombini, Del Santo ed.’s, pp. 315–329, Progr. Nonlinear Differential Equations Appl., 69, Birkhäuser Boston, Boston, MA (2006)

  60. 60.

    Treves, F.: Aspects of Analytic PDE, book in preparation

  61. 61.

    Zworski, M.: Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence (2012)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Antonio Bove.

Additional information

In memory of Nick Hanges.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bove, A., Mughetti, M. Analytic regularity for solutions to sums of squares: an assessment. Complex Anal Synerg 6, 18 (2020). https://doi.org/10.1007/s40627-020-00055-8

Download citation

Keywords

  • Sums of squares of vector fields
  • Analytic hypoellipticity
  • Treves conjecture

Mathematics Subject Classification

  • 35H10
  • 35H20 (primary)
  • 35B65
  • 35A20
  • 35A27 (secondary)