Solvability of differential equations with linear coefficients of nilpotent type

Abstract:

Let $L$ be the vector field on ${{\mathbf {R}}^n}$ associated with a real nilpotent $(n \times n)$-matrix. It is shown that $L$ regarded as a differential operator defines a surjective mapping of the space $\mathcal {S}’$ of tempered distributions onto itself; i.e. $L\mathcal {S}’({{\mathbf {R}}^n}) = \mathcal {S}’({{\mathbf {R}}^n})$. Replacing $\mathcal {S}’$ by the space $\mathcal {D}’$ of ordinary distributions, this is not true in general.