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 ⁺(x ₁, x ₂, x ₃) := P(x ₁, x ₂). 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).