Some results on surjectivity of augmented differential operators

Considering a problem of Bonet and Domański (2006) [1, Problem 9.1], we prove that for a polynomial P on R2 surjectivity of the differential operator P(D) on D′(X) implies surjectivity of the augmented operator P+(D) on D′(X×R), where P+(x1,x2,x3):=P(x1,x2). Moreover we give a sufficient geometrical condition on an open subset X of Rd such that an analogous implication is true for arbitrary dimension d in case of P being homogeneous, semi-elliptic, or of principal type.