Vol. 10, No. 7, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Analytic hypoellipticity for sums of squares and the Treves conjecture, II

Antonio Bove and Marco Mughetti

Vol. 10 (2017), No. 7, 1613–1635
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.

Keywords
sums of squares of vector fields, analytic hypoellipticity, Treves conjecture
Mathematical Subject Classification 2010
Primary: 35H10, 35H20
Secondary: 35B65, 35A20, 35A27
Milestones
Received: 1 June 2016
Revised: 24 February 2017
Accepted: 17 June 2017
Published: 1 August 2017
Authors
Antonio Bove
Dipartimento di Matematica
Università di Bologna
Bologna
Italy
Marco Mughetti
Dipartimento di Matematica
Università di Bologna
Bologna
Italy