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

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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

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

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

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


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.

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