C^\infty ({\mathbf R^{2d}}),
\\[1ex]
T\, &:\, & {\mathcal S}_s'({\mathbf R^{d}})\times {\mathcal S}_s'({\mathbf R^{d}}) &\to
{\mathcal S}_s '({\mathbf R^{2d}}),\end{aligned}$$ and similarly when ${\mathcal S}_s$ and ${\mathcal S}_s'$ are replaced by $\Sigma _s$ and $\Sigma _s'$, respectively, or by ${\mathscr S}$ and ${\mathscr S}'$, respectively (cf. [@CPRT10; @Toft8]). We also note that $V_\phi f$ takes the form $$\tag*{(\ref{defstft})$'$}
V_\phi f(x,\xi ) =(2\pi )^{-d/2}\int _{{\mathbf R^{d}}}f(y)\overline {\phi
(y-x)}e^{-i{\langle y,\xi\rangle}}\, dy$$ when $f\in L^p_{(\omega )}({\mathbf R^{d}})$ for some $\omega \in
{\mathscr P}_E({\mathbf R^{d}})$, $\phi \in \Sigma _1({\mathbf R^{d}})$ and $p\ge 1$. Here $L^p_{(\omega )}({\mathbf R^{d}})$, when $p\in (0,\infty ]$ and $\omega \in {\mathscr P}_E({\mathbf R^{d}})$, is the set of all $f\in L^p_{loc} ({\mathbf R^{d}})$ such that $f\cdot \omega \in L^p({\mathbf R^{d}})$.

Mixed quasi-normed space of Lebesgue types
------------------------------------------

Let $p,q\in (0,\infty ]$, and let $\omega \in {\mathscr P}_E({\mathbf R^{2d}})$. Then $L^{p,q}_{(\omega )}({\mathbf R^{2d}})$ and $L^{p,q}_{*,(\omega )}({\mathbf R^{2d}})$ consist of all measurable functions $F$ on ${\mathbf R^{2d}}$ such that $$\begin{aligned}
{3}
{\Vert g_1\Vert _{L^q({\mathbf R^{d}})}}&<\infty ,&
\quad &\text{where} &\quad
g_1(\xi ) &\equiv {\Vert F({\, \cdot \, },\xi )\omega ({\, \cdot \, },\xi )\Vert _{L^p({\mathbf R^{d}})}}
\intertext{and}
{\Vert g_2\Vert _{L^p({\mathbf R^{d}})}}&<\infty ,&
\quad &\text{where} &\quad
g_2(x) &\equiv {\Vert F(x,{\, \cdot \, })\omega (x,{\, \cdot \, })\Vert 