and under (partial) Fourier transformations.

From now on we let $\mathscr F$ be the Fourier transform, given by $$(\mathscr Ff)(\xi )= \widehat f(\xi ) \equiv (2\pi )^{-d/2}\int _{{\mathbf R^{d}}} f(x)e^{-i{\langle x,\xi\rangle} }\, dx$$ when $f\in L^1({\mathbf R^{d}})$. Here ${\langle {\, \cdot \, },{\, \cdot \, }\rangle}$ denotes the usual scalar product on ${\mathbf R^{d}}$. The map $\mathscr F$ extends uniquely to homeomorphisms on $\mathscr S'({\mathbf R^{d}})$, $\mathcal
S_s'({\mathbf R^{d}})$ and $\Sigma _s'({\mathbf R^{d}})$, and restricts to homeomorphisms on $\mathscr S({\mathbf R^{d}})$, $\mathcal S_s({\mathbf R^{d}})$ and $\Sigma _s({\mathbf R^{d}})$, and to a unitary operator on $L^2({\mathbf R^{d}})$.

Next we recall some mapping properties of Gelfand-Shilov spaces under short-time Fourier transforms. Let $\phi \in \mathscr S({\mathbf R^{d}})$ be fixed. For every $f\in
\mathscr S'({\mathbf R^{d}})$, the *short-time Fourier transform* $V_\phi
f$ is the distribution on ${\mathbf R^{2d}}$ defined by the formula $$\label{defstft}
(V_\phi f)(x,\xi ) =\mathscr F(f\, \overline{\phi ({\, \cdot \, }-x)})(\xi ) =
(f,\phi ({\, \cdot \, }-x)e^{i{\langle {\, \cdot \, },\xi\rangle}}).$$ We recall that if $T(f,\phi )\equiv V_\phi f$ when $f,\phi \in {\mathcal S}_{1/2}({\mathbf R^{d}})$, then $T$ is uniquely extendable to sequentially continuous mappings $$\begin{aligned}
{2}
T\, &:\, & {\mathcal S}_s'({\mathbf R^{d}})\times {\mathcal S}_s({\mathbf R^{d}}) &\to
{\mathcal S}_s '({\mathbf R^{2d}})\bigcap 