Fundamental Solutions of Multilinear Differential Operators with Constant Coefficients

Abstract

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 \(x\).