## Analysis & PDE

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

### Hypoellipticity and nonhypoellipticity for sums of squares of complex vector fields

#### 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,

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

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 $k<ℓq$, for every $s≥ℓq∕(ℓq−k)=1+k∕(ℓq−k)$. We show that in the Gevrey spaces below this index, the operator is not hypoelliptic. Moreover, if $k≥ℓq$, 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
Revised: 16 January 2012
Accepted: 13 February 2012
First available in Project Euclid: 20 December 2017

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

#### 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

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