On some classes of non-hypoelliptic second order partial differential operators (Q1116017)

The author studies the non-hypoellipticity of second order linear partial differential operators of the form \[ L=\sum a_{ij}\partial^ 2_{ij}+\sum b_ k\partial_ k+c+\partial_ t, \] where \(\partial_ k=\partial /\partial x_ k\). Here \(x=(x_ 1,...,x_ n)\) runs over an open subset \(\omega\) of \({\mathbb{R}}^ n\), and \(t\in]-T, +T[.\) Assume that \(a_{ij}=a_{ji}\) and put \(a=\sum a_{ij}\xi_ i\xi_ j\). The author gives a sufficient condition for L not to be globally hypoelliptic (or hypoelliptic) near the origin mainly in terms of the behavior of Re a(x,t,\(\xi)\) along \(x=0\) or in terms of the behavior of a(x,t,\(\xi)\) along the integral curve of \(\sum b_ k\partial_ k+\partial_ t\) through the origin.