Euler partial differential equations and Schwartz distributions

Abstract

Euler operators are partial differential operators of the form P ( θ ) P(\theta ) where P P is a polynomial and θ j = x j ∂ / ∂ x j \theta _j = x_j \partial /\partial x_j . They are surjective on the space of temperate distributions on R d \mathbb {R}^d . We show that this is, in general, not true for the space of Schwartz distributions on R d \mathbb {R}^d , d ≥ 3 d\ge 3 , for d = 1 d=1 ; however, it is true. It is also true for the space of distributions of finite order on R d \mathbb {R}^d and on certain open sets Ω ⊂ R d \Omega \subset \mathbb {R}^d , like the Euclidean unit ball.