Some results on surjectivity of augmented semi-elliptic differential operators

Abstract

We show that for a semi-elliptic polynomial P on \({\mathbb{R}^2}\) surjectivity of P(D) on \({\fancyscript{D}'(\Omega)}\) implies surjectivity of the augmented operator P+(D) on \({\fancyscript{D}'(\Omega\times\mathbb{R})}\), where P+(x1, x2, x3) := P(x1, x2). For arbitrary dimension n we give a sufficient geometrical condition on \({\Omega\subset\mathbb{R}^n}\) such that an analogous implication is true for semi-elliptic P. Moreover, we give an alternative proof of a result due to Vogt which says that for elliptic P the operator P+(D) is surjective if this is true for P(D).