Paley-Wiener-Schwartz type theorems for function spaces on an unbounded closed convex set and their applications

In the talk there will be considered some problems of operator theory and Fourier analysis in spaces of rapidly decreasing functions on unbounded convex sets of a multidimensional real space of the form
$$ U (b, C) = \{\xi \in {\mathbb R}^n: -\langle \xi, y \rangle \ \le b(y), \, \forall y \in C \}, $$
where $C$ is an open convex acute cone in ${\mathbb R}^n$ with the vertex at the origin, $b$ is a convex continuous positively homogeneous of order 1 function on ${\overline C}$.
One of the problems is as follows. For an unbounded closed convex set $\Omega \subset {\mathbb R}^n$ ($\Omega \ne {\mathbb R}^n$) let $S(\Omega)$ be the Schwartz space on $\Omega$. Let $D \subset {\mathbb R}^n$ be a bounded convex domain, $K$ is a closure of $D$, $G = U (b, C) + K$. Let $\mu$ be a linear continuous functional on $C^{\infty} (K)$. The operator $M_{\mu}: f \in S(G) \to \mu*f$ acts from $S(G)$ into $S(U)$. The problem is to find conditions for surjectivity of the operator $M_{\mu}$.