Surjectivity of Euler operators on temperate distributions

TL;DR

This paper proves that all non-trivial Euler operators are surjective on the space of temperate distributions, highlighting a unique property that differs from their behavior on other function spaces.

Contribution

It establishes the surjectivity of Euler operators on temperate distributions, a result contrasting their known behavior on differentiable and analytic function spaces.

Findings

All non-trivial Euler operators are surjective on temperate distributions.

Surjectivity does not hold on spaces of differentiable or analytic functions.

Highlights a fundamental difference in operator behavior across function spaces.

Abstract

Euler operators are partial differential operators of the form $P(\theta)$ where $P$ is a polynomial and $\theta_j = x_j \partial/\partial x_j$. We show that every non-trivial Euler operator is surjective on the space of temperate distributions on $R^d$. This is in sharp contrast to the behaviour of such operators when acting on spaces of differentiable or analytic functions.