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 R 2 surjectivity of the differential operator P ( D ) on D ′ ( X ) implies surjectivity of the augmented operator P + ( D ) on D ′ ( X × R ) , where P + ( x 1 , x 2 , x 3 ) : = P ( x 1 , x 2 ) . Moreover we give a sufficient geometrical condition on an open subset X of R d such that an analogous implication is true for arbitrary dimension d in case of P being homogeneous, semi-elliptic, or of principal type.