Global hypoellipticity property of a differential operator (Q805885)

A differential operator P in a domain \(Q\subset {\mathbb{R}}^ n\) is called globally hypoelliptic if for any distribution \(u\in D'(Q)\) with \(Pu\in C^{\infty}(Q)\) we have \(u\in C^{\infty}(Q)\). It is known that there are non-hypoelliptic operators with the property of global hypoellipticity (Fedij, 1972). In this paper the following theorem is proved. Theorem. Let \(X_ j(j=1,...,m)\) be real vector fields in a neighbourhood of the closure of Q. If the differential operator \(P=\sum^{n}_{j=1}X_ j^ m\) is elliptic in some neighbourhood of the boundary of Q and the Lie algebra, generated by the fields \(X_ 1,...,X_ m\), coincides with the whole space \({\mathbb{R}}^ n\) (that is, it has rank n), then the operator P is globally hypoelliptic.