Some results on surjectivity of augmented differential operators

Considering a problem of Bonet and Domanski (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 x R), where P(+)(x(1), x(2), x(3)) := P(xi, x2). 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. (C) 2011 Elsevier Inc. All rights reserved.