Let Dα be a differential
monomial in n variables. We try to identify in this paper those closed, densely
defined linear operators A (in a fairly general class of locally convex spaces) such that
the equation (Dα− A)S = δ ⊗ I has a solution; here S is an operator-valued
distribution in n variables with support in the cone of nonnegative coordinates. The
results are applied to the study of the equation (Dα−A)U = T,T a distribution with
values in the space where A is defined, and to the formulation and solution of an
“abstract Goursat problem” that reduces to the abstract Cauchy problem
when
