On degenerate elliptic operators of infinite type (Q1909558)

Necessary and sufficient conditions are proved for global \(C^\infty\)-hypoellipticity of certain classes of second-order degenerate elliptic partial differential operators on the torus, which may be of finite type at most points. For one class it is shown that reachability is a necessary and sufficient condition for global hypoellipticity, while for another class it is only a sufficient condition. The necessity involves the notion of a Liouville number. These operators may not be locally hypoelliptic and do not satisfy local subelliptic estimates. The key point of the proof is the derivation of an appropriate \(L^2\)-estimate near characteristic points.