ON FUNDAMENTAL SOLUTIONS OF A CLASS OF WEAK HYPERBOLIK OPERATORS

Abstract

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