Decomposition theorems and kernel theorems for a class of functional spaces

We prove new theorems about properties of generalized functions defined on Gelfand–Shilov spaces $S^\beta$ with $0\le\beta<1$. For each open cone $U\subset \mathbb R^d$ we define a space $S^\beta(U)$ which is related to $S^\beta(\mathbb R^d)$ and consists of entire analytic functions rapidly decreasing inside $U$ and having order of growth $\le 1/(1-\beta)$ outside the cone. Such sheaves of spaces arise naturally in non-local quantum field theory, and this motivates our investigation. We prove that the spaces $S^\beta(U)$ are complete and nuclear and establish a decomposition theorem which implies that every continuous functional defined on $S^\beta(\mathbb R^d)$ has a unique minimal closed carrier cone in $\mathbb R^d$. We also prove kernel theorems for spaces over open and closed cones and elucidate the relation between the carrier cones of multilinear forms and those of the generalized functions determined by these forms.