Let $F$ be a local field (e.g. $\mathbb{R}$, $\mathbb{C}$, or finite extensions of $\mathbb{Q}_p$). Let $X$ (resp. $Y$) be a non-degenerate quadratic (resp. symplectic) space over $F$. Then $\mathrm{SO}(X) \times \mathrm{Mp}(Y)$ form a reductive dual pair in $\mathrm{Mp}(Z)$ for $Z = X \otimes Y$ (with natural symplectic form on it). For a fixed additive character $\psi: F \to \mathbb{C}^\times$ and a polarization $Z = Z^+ \oplus Z^-$, let $\omega_\psi$ be the Weil representation of $\mathrm{SO}(X) \times \mathrm{Mp}(Y)$ acting on the space of Schwartz functions $\mathcal{S}(Z^+)$ on $Z^+$ restricted from $\mathrm{Mp}(Z)$. Now for $f \in \mathcal{S}(\mathrm{SO}(X))$ and $f' \in \mathcal{S}(\mathrm{Mp}(Y))$, define $$ \omega_\psi(f)\phi(X) = \int_{\mathrm{SO}(X)} f(g) \omega_\psi(g, 1) \phi(X) \mathrm{d} g \\ \omega_\psi(f')\phi(X) = \int_{\mathrm{Mp}(X)} f'(h) \omega_\psi(1,h) \phi(X) \mathrm{d} h. $$ I wonder if $\omega_\psi: \mathcal{S}(\mathrm{SO}(X)) \times \mathcal{S}(Z^+) \to \mathcal{S}(Z^+)$ and $\omega_\psi: \mathcal{S}(\mathrm{Mp}(Z)) \times \mathcal{S}(Z^+) \to \mathcal{S}(Z^+)$ are surjective or not. In the proof of Theorem 1 of the paper Howe duality and the trace formula, the author says that
Given $f$, there exists $f', \phi$ such that $\omega_\psi(f')\phi = \phi_f$. This is clear.
where $\phi_f \in \mathcal{S}(Z^+)$ is a certain function associated to $f$. It seems that the general claim (surjectivity of $\omega_\psi$) is also true, but I can't figure our why this should be true. Thanks in advance.