Fundamental solutions of multilinear differential operators with constant coefficients

This paper generalizes part of the author's previous results. Let $L$ be a multilinear differential operator with constant coefficients. The fundamental solution $\phi$ supported in a convex cone of a linear space $U$ is piecewise polynomial. Choose a basis in the space $T$ of polynomials and consider the corresponding set of convex cones in the space $U$. We claim that $\phi (x)$ is equal to a sum of basis elements in $T$, with the sum being taken over those elements for which the corresponding cones contain $x$.