Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 2 (2013), 371-445.

Hypoellipticity and nonhypoellipticity for sums of squares of complex vector fields

Antonio Bove, Marco Mughetti, and David Tartakoff

Full-text: Open access

Abstract

In this paper we consider a model sum of squares of complex vector fields in the plane, close to Kohn’s operator but with a point singularity,

P = B B + B ( t 2 + x 2 k ) B , B = D x + i x q 1 D t .

The characteristic variety of P is the symplectic real analytic manifold x=ξ=0. We show that this operator is C-hypoelliptic and Gevrey hypoelliptic in Gs, the Gevrey space of index s, provided k<q, for every sq(qk)=1+k(qk). We show that in the Gevrey spaces below this index, the operator is not hypoelliptic. Moreover, if kq, the operator is not even hypoelliptic in C. This fact leads to a general negative statement on the hypoellipticity properties of sums of squares of complex vector fields, even when the complex Hörmander condition is satisfied.

Article information

Source
Anal. PDE, Volume 6, Number 2 (2013), 371-445.

Dates
Received: 28 August 2011
Revised: 16 January 2012
Accepted: 13 February 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731313

Digital Object Identifier
doi:10.2140/apde.2013.6.371

Mathematical Reviews number (MathSciNet)
MR3071394

Zentralblatt MATH identifier
1335.35024

Subjects
Primary: 35H10: Hypoelliptic equations 35H20: Subelliptic equations
Secondary: 35B65: Smoothness and regularity of solutions

Keywords
sums of squares of complex vector fields hypoellipticity Gevrey hypoellipticity pseudodifferential operators

Citation

Bove, Antonio; Mughetti, Marco; Tartakoff, David. Hypoellipticity and nonhypoellipticity for sums of squares of complex vector fields. Anal. PDE 6 (2013), no. 2, 371--445. doi:10.2140/apde.2013.6.371. https://projecteuclid.org/euclid.apde/1513731313


Export citation

References

  • L. Boutet de Monvel, “Opérateurs pseudo-différentiels analytiques et opérateurs d'ordre infini”, Ann. Inst. Fourier $($Grenoble$)$ 22:3 (1972), 229–268.
  • L. Boutet de Monvel, “Hypoelliptic operators with double characteristics and related pseudo-differential operators”, Comm. Pure Appl. Math. 27 (1974), 585–639.
  • L. Boutet de Monvel and P. Krée, “Pseudo-differential operators and Gevrey classes”, Ann. Inst. Fourier $($Grenoble$)$ 17:1 (1967), 295–323.
  • A. Bove and M. Mughetti, “Analytic and Gevrey hypoellipticity for a class of pseudodifferential operators in one variable”, J. Differential Equations 255:4 (2013), 728–758.
  • A. Bove and D. Tartakoff, “Optimal non-isotropic Gevrey exponents for sums of squares of vector fields”, Comm. Partial Differential Equations 22:7-8 (1997), 1263–1282.
  • A. Bove and D. S. Tartakoff, “Gevrey hypoellipticity for non-subelliptic operators”, Pure Appl. Math. Q. 6:3 (2010), 663–675.
  • 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. Derridj, J. J. Kohn, and D. S. Tartakoff, “Sums of squares of complex vector fields and (analytic-) hypoellipticity”, Math. Res. Lett. 13:5 (2006), 683–701.
  • A. Bove, M. Mughetti, and D. S. Tartakoff, “Gevrey hypoellipticity for an interesting variant of Kohn's operator”, pp. 51–73 in Complex analysis, edited by P. Ebenfelt et al., Birkhäuser, Basel, 2010.
  • M. Christ, “A remark on sums of squares of complex vector fields”, preprint, 2005.
  • W. Feller, An introduction to probability theory and its applications, vol. 1, 2nd ed., Wiley, New York, 1957.
  • A. Grigis and L. P. Rothschild, “A criterion for analytic hypoellipticity of a class of differential operators with polynomial coefficients”, Ann. of Math. $(2)$ 118:3 (1983), 443–460.
  • A. Grothendieck, Topological vector spaces, Gordon and Breach, New York, 1973.
  • B. Helffer, “Sur l'hypoellipticité des opérateurs pseudodifferentiels à caractéristiques multiples (perte de $3/2$ dérivées)”, Bull. Soc. Math. France Suppl. Mém. 51-52 (1977), 13–61.
  • L. H örmander, The analysis of linear partial differential operators, III: Pseudodifferential operators, Grundlehren Math. Wiss. 274, Springer, Berlin, 1985.
  • J. J. Kohn, “Hypoellipticity and loss of derivatives”, Ann. of Math. $(2)$ 162:2 (2005), 943–986.
  • H. Kumano-go, Pseudodifferential operators, MIT Press, Cambridge, MA, 1982. http://msp.org/idx/zbl/489:35003MR 489:35003
  • G. Métivier, “Une classe d'opérateurs non hypoelliptiques analytiques”, Indiana Univ. Math. J. 29:6 (1980), 823–860.
  • G. Métivier, “Analytic hypoellipticity for operators with multiple characteristics”, Comm. Partial Differential Equations 6:1 (1981), 1–90.
  • L. Boutet de Monvel and F. Trèves, “On a class of pseudodifferential operators with double characteristics”, Invent. Math. 24 (1974), 1–34.
  • J. Sj östrand and M. Zworski, “Elementary linear algebra for advanced spectral problems”, Ann. Inst. Fourier $($Grenoble$)$ 57:7 (2007), 2095–2141.
  • F. Treves, “Symplectic geometry and analytic hypo-ellipticity”, pp. 201–219 in Differential equations: La Pietra 1996 (Florence, 1996), edited by M. Giaquinta et al., Proc. Sympos. Pure Math. 65, American Mathematical Society, Providence, RI, 1999.