Euler partial differential equations and Schwartz distributions

TL;DR

This paper investigates the surjectivity of Euler differential operators on various distribution spaces, establishing their surjectivity on temperate and finite order distributions but not on Schwartz distributions in higher dimensions.

Contribution

It demonstrates that Euler operators are surjective on temperate and finite order distributions but generally not on Schwartz distributions in dimensions three and higher.

Findings

Euler operators are surjective on temperate distributions.

Surjectivity fails for Schwartz distributions in dimensions ≥ 3.

Surjectivity holds for distributions of finite order and certain open sets.

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.