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.
Euler partial differential equations and Schwartz distributions
Right inverses for partial differential operators on spaces of Whitney functions
Surjectivity of the $\overline{\partial}$-operator between spaces of weighted smooth vector-valued functions
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