Euler partial differential equations and Schwartz distributions

Abstract

Euler operators are partial differential operators of the form $P(\theta )$ where $P$ is a polynomial and ${\theta }_j=x_j\partial /\partial x_j$. They are surjective on the space of temperate distributions on $R^d$. We show that this is, in general, not true for the space of Schwartz distributions on $R^d$, $d\ge 3$, for $d=1$, however, it is true. It is also true for the space of distributions of finite order on $R^d$ and on certain open sets $\Omega \subset R^d$, like the euclidian unit ball.