Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 7 (2017), 1613-1635.

Analytic hypoellipticity for sums of squares and the Treves conjecture, II

Antonio Bove and Marco Mughetti

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/apde.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. The Treves conjecture defines a stratification and states that an operator of this type is analytic hypoelliptic if and only if all the strata in the stratification are symplectic manifolds.

Albano, Bove, and Mughetti (2016) produced an example where the operator has a single symplectic stratum, according to the conjecture, but is not analytic hypoelliptic.

If the characteristic manifold has codimension 2 and if it consists of a single symplectic stratum, defined again according to the conjecture, it has been shown that the operator is analytic hypoelliptic.

We show here that the above assertion is true only if the stratum is single, by producing an example with two symplectic strata which is not analytic hypoelliptic.

Article information

Source
Anal. PDE, Volume 10, Number 7 (2017), 1613-1635.

Dates
Received: 1 June 2016
Revised: 24 February 2017
Accepted: 17 June 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843557

Digital Object Identifier
doi:10.2140/apde.2017.10.1613

Mathematical Reviews number (MathSciNet)
MR3683923

Zentralblatt MATH identifier
1375.35094

Subjects
Primary: 35H10: Hypoelliptic equations 35H20: Subelliptic equations
Secondary: 35B65: Smoothness and regularity of solutions 35A20: Analytic methods, singularities 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE [See also 32C38, 58J15]

Keywords
sums of squares of vector fields analytic hypoellipticity Treves conjecture

Citation

Bove, Antonio; Mughetti, Marco. Analytic hypoellipticity for sums of squares and the Treves conjecture, II. Anal. PDE 10 (2017), no. 7, 1613--1635. doi:10.2140/apde.2017.10.1613. https://projecteuclid.org/euclid.apde/1510843557


Export citation

References

  • P. Albano and A. Bove, Wave front set of solutions to sums of squares of vector fields, Mem. Amer. Math. Soc. 1039, Amer. Math. Soc., Providence, RI, 2013.
  • P. Albano, A. Bove, and M. Mughetti, “Analytic hypoellipticity for sums of squares and the Treves conjecture”, preprint, 2016.
  • F. A. Berezin and M. A. Shubin, The Schrödinger equation, Mathematics and its Applications (Soviet Series) 66, Kluwer Academic Publishers Group, Dordrecht, 1991.
  • P. Bolley, J. Camus, and J. Nourrigat, “La condition de Hörmander–Kohn pour les opérateurs pseudo-différentiels”, Comm. Partial Differential Equations 7:2 (1982), 197–221.
  • A. Bove and M. Mughetti, “On a new method of proving Gevrey hypoellipticity for certain sums of squares”, Adv. Math. 293 (2016), 146–220.
  • A. Bove and F. Treves, “On the Gevrey hypo-ellipticity of sums of squares of vector fields”, Ann. Inst. Fourier $($Grenoble$)$ 54:5 (2004), 1443–1475.
  • A. Bove, M. Mughetti, and D. S. Tartakoff, “Hypoellipticity and nonhypoellipticity for sums of squares of complex vector fields”, Anal. PDE 6:2 (2013), 371–445.
  • P. D. Cordaro and N. Hanges, “A new proof of Okaji's theorem for a class of sum of squares operators”, Ann. Inst. Fourier $($Grenoble$)$ 59:2 (2009), 595–619.
  • B. Helffer, Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics 1336, Springer, 1988.
  • B. Helffer and J. Sjöstrand, “Multiple wells in the semiclassical limit, I”, Comm. Partial Differential Equations 9:4 (1984), 337–408.
  • L. Hörmander, “Hypoelliptic second order differential equations”, Acta Math. 119 (1967), 147–171.
  • L. Hörmander, “Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients”, Comm. Pure Appl. Math. 24 (1971), 671–704.
  • D. Jerison, “The Poincaré inequality for vector fields satisfying Hörmander's condition”, Duke Math. J. 53:2 (1986), 503–523.
  • A. Martinez, “Estimations de l'effet tunnel pour le double puits, I”, J. Math. Pures Appl. $(9)$ 66:2 (1987), 195–215.
  • G. Métivier, “Non-hypoellipticité analytique pour $D\sp{2}\sb{x}+(x\sp{2}+y\sp{2})D\sp{2}\sb{y}$”, C. R. Acad. Sci. Paris Sér. I Math. 292:7 (1981), 401–404.
  • M. Mughetti, “Hypoellipticity and higher order Levi conditions”, J. Differential Equations 257:4 (2014), 1246–1287.
  • M. Mughetti, “On the spectrum of an anharmonic oscillator”, Trans. Amer. Math. Soc. 367:2 (2015), 835–865.
  • T. Ōkaji, “Analytic hypoellipticity for operators with symplectic characteristics”, J. Math. Kyoto Univ. 25:3 (1985), 489–514.
  • B. Simon, “Semiclassical analysis of low lying eigenvalues, I: Nondegenerate minima: asymptotic expansions”, Ann. Inst. H. Poincaré Sect. A $($N.S.$)$ 38:3 (1983), 295–308. Correction in 40:2 (1984), 224.
  • F. Treves, “Symplectic geometry and analytic hypo-ellipticity”, pp. 201–219 in Differential equations: La Pietra 1996 (Florence), edited by M. Giaquinta et al., Proc. Sympos. Pure Math. 65, Amer. Math. Soc., Providence, RI, 1999.
  • F. Treves, “On the analyticity of solutions of sums of squares of vector fields”, pp. 315–329 in Phase space analysis of partial differential equations, edited by A. Bove et al., Progr. Nonlinear Differential Equations Appl. 69, Birkhäuser, Boston, 2006.
  • F. Treves, “Aspects of analytic PDE”, book in preparation.
  • M. Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138, Amer. Math. Soc., Providence, RI, 2012.