Surjectivity of Euler operators on temperate distributions

Abstract

Euler operators are partial differential operators of the form $P(\theta )$ where $P$ is a polynomial and ${\theta }_j=x_j\partial /\partial x_j$. 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.

Cited by in corpus (1)

• ${\mathcal{E}}^{\prime }$ as an algebra by multiplicative convolution