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

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

sums of squares of vector fields analytic hypoellipticity Treves conjecture


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.

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