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 $\mathbb {R}^d$. We show that this is, in general, not true for the space of Schwartz distributions on $\mathbb {R}^d$, $d\ge 3$, for $d=1$; however, it is true. It is also true for the space of distributions of finite order on $\mathbb {R}^d$ and on certain open sets $\Omega \subset \mathbb {R}^d$, like the Euclidean unit ball.