#### Vol. 6, No. 2, 2013

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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,

$P=B{B}^{\ast }+{B}^{\ast }\left({t}^{2\ell }+{x}^{2k}\right)B,\phantom{\rule{1em}{0ex}}B={D}_{x}+i{x}^{q-1}{D}_{t}.$

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