On fundamental solutions of a class of weak hyperbolic operators

Abstract

We consider a certain class of polyhedrons \( \Re \subseteq \mathbb{E}^n \), multi-anisotropic Jevre spaces \( G^\Re (\mathbb{E}^n) \), their subspaces \( G^\Re_0 (\mathbb{E}^n) \), consisting of all functions \( f \in G^\Re (\mathbb{E}^n) \) with compact support, and their duals \( (G^\Re_0 (\mathbb{E}^n))^* \). We introduce the notion of a linear differential operator \( P(D) \), \( h_\Re \)-hyperbolic with respect to a vector \( N \in \mathbb{E}^n \), where \( h_\Re \) is a weight function generated by the polyhedron \( \Re \). The existence is shown of a fundamental solution \( E \) of the operator \( P(D) \) belonging to \( (G^\Re_0 (\mathbb{E}^n))^* \) with \( \text{supp} E \subseteq \Omega_N \), where \( \Omega_N := \{ x \in \mathbb{E}^n, (x,N) > 0 \} \). It is also shown that for any right-hand side \( f \in G^\Re (\mathbb{E}^n) \) with the support in a cone contained in \( \Omega_N \) and with the vertex at the origin of \( \mathbb{E}^n \), the equation \( P(D)u = f \) has a solution belonging to \( G^\Re (\mathbb{E}^n) \).