On global hypoellipticity of degenerate elliptic operators (Q1297990)

This paper deals with the global \(C^\infty\) hypoellipticity on the torus of some classes of degenerate second order elliptic operators. In Theorem 1.1 an operator \(P\) having the form of sum of squares of real-valued smooth vector fields is considered, and it is proved that \(P\) is \(C^\infty\) hypoelliptic if and only if an algebraic condition involving Liouville vectors and simultaneous approximability holds. The operator \(P\) may not be locally hypoelliptic. Theorem 1.2 shows that the zero order term may influence the global hypoellipticity of a class of second order differential operators on the two-dimensional torus.