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,

$$P=B{B}^{\ast }+B^{\ast }\left(t^{2\ell }+x^{2k}\right)B,B=D_x+i{x}^{q-1}{D}_t.$$

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.