Hypoellipticity and nonhypoellipticity for sums of squares of complex vector fields

The characteristic variety of $P$ is the symplectic real analytic manifold $x = ξ = 0$ . We show that this operator is $C^{\infty}$ -hypoelliptic and Gevrey hypoelliptic in $G^{s}$ , the Gevrey space of index $s$ , provided $k < ℓ q$ , for every $s \geq ℓ q ∕ (ℓ q - k) = 1 + k ∕ (ℓ q - k)$ . We show that in the Gevrey spaces below this index, the operator is not hypoelliptic. Moreover, if $k \geq ℓ q$ , the operator is not even hypoelliptic in $C^{\infty}$ . 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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 35H10, 35H20

Secondary: 35B65

Milestones

Received: 28 August 2011

Revised: 16 January 2012

Accepted: 13 February 2012

Published: 24 June 2013

Authors

Antonio Bove
Department of Mathematics University of Bologna Piazza di Porta San Donato 5 I-40127 Bologna Italy

Marco Mughetti
Department of Mathematics University of Bologna Piazza di Porta San Donato 5 I-40127 Bologna Italy

David S. Tartakoff
Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago 322 Science and Engineering Offices (M/C 249) 851 South Morgan Street Chicago, IL 60607-7045 United States

Vol. 6, No. 2, 2013

Antonio Bove, Marco Mughetti and David S. Tartakoff

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors