Some results on surjectivity of augmented semi-elliptic differential operators

We show that for a semi-elliptic polynomial P on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^2}$$\end{document} surjectivity of P(D) on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript{D}'(\Omega)}$$\end{document} implies surjectivity of the augmented operator P+(D) on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\fancyscript{D}'(\Omega\times\mathbb{R})}$$\end{document}, where P+(x1, x2, x3) := P(x1, x2). For arbitrary dimension n we give a sufficient geometrical condition on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega\subset\mathbb{R}^n}$$\end{document} 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).