Examples of irregular domains for some hypoelliptic differential operators (Q1079267)

Consider smooth vectorfields \(X_ 1,...,X_ n\) on a k-dimensional manifold M and assume that their commutators up to order r span the tangent space at each \(x\in M\). Rothschild and Stein opened a canonical way to study properties of the hypoelliptic operator \(L=X^ 2_ 1+...+X^ 2_ n\). They introduced a lifting procedure which connects L with an operator \(\tilde L=\tilde X^ 2_ 1+...+\tilde X^ 2_ n\) on a certain m-dimensional manifold N which contains M as a submanifold. The \(\tilde X_ i\) are liftings of the \(X_ i\), their commutators up to order r span the tangent space at each \(x\in N\), and the advantage is, that now the \(\tilde X_ i\) are free up to order r. For such situations there are models: \(\tilde L\) behaves like a sub-Laplacian on a stratified algebra of type I and this is especially true for the corresponding Green functions. So it is interesting to see examples of irregular domains in these models.