Vol. 6, No. 2, 2013

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Hypoellipticity and nonhypoellipticity for sums of squares of complex vector fields

Antonio Bove, Marco Mughetti and David S. Tartakoff

Vol. 6 (2013), No. 2, 371–445
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 = BB + B(t2 + x2k)B,B = D x + ixq1D 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 s q(q k) = 1 + k(q k). We show that in the Gevrey spaces below this index, the operator is not hypoelliptic. Moreover, if k q, 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.

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