An unsolvable hypoelliptic differential operator

It is proved that the differential operator D ₁ +ix ₁ D ₂ ² is hypoelliptic everywhere, but is not locally solvable in any open set which intersects the line x ₁=0. Thus, this operator is not contained in the usual classes of hypoelliptic differential operators. The proofs involve certain properties of the characteristic Cauchy problem for the backward heat operator.