On fundamental solutions of a class of weak hyperbolic operators - Margaryan, Vachagan; Ghazaryan, Haik (Eurasian Mathematical Journal, 2018)
We consider a certain class of polyhedrons \( E^n \), multi-anisotropic Jevre spaces \( G^(E^n) \), their subspaces \( G^_0 (E^n) \), consisting of all functions \( f G^(E^n) \) with compact support, and their duals \( (G^_0 (E^n))^* \). We introduce the notion of a linear differential operator \( P(D) \), \( h_\)-hyperbolic with respect to a vector \( N E^n \), where \( h_\) is a weight function generated by the polyhedron \( \). The existence is shown of a fundamental solution \( E \) of the operator \( P(D) \) belonging to \( (G^_0 (E^n))^* \) with \( supp E Ω_N \), where \( Ω_N := { x E^n, (x,N) > 0 } \). It is also shown that for any right-hand side \( f G^(E^n) \) with the support in a cone contained in \( Ω_N \) and with the vertex at the origin of \( E^n \), the equation \( P(D)u = f \) has a solution belonging to \( G^(E^n) \).