Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with vertex at the origin, and let $\mathscr O_{dec}(W)$ be the "rapidly decreasing" holomorphic functions which decrease faster than any power $e^{\epsilon |z|}$ for all $z\in W$, and let $\mathscr O'_{dec}(W)$ be its dual.

Since $\mathscr B(\mathbb R^2)$ is flabby, and any $\phi(z)\in \mathscr O_{dec}(W)$ can be expressed $\phi(x+iy)$, then does $\mathscr B(\mathbb R^2)$ map surjectively onto $\mathscr O'_{dec}(W)$?