(0,\infty ]^{2d}$, $r\in (0,1]$ be such that $r\le \min ({{\boldsymbol p}})$, $\sigma
\in \operatorname {S}_{2d}$, and let $\omega ,v\in {\mathscr P}_E({\mathbf R^{2d}})$ be such that $\omega$ is $v$-moderate. Also let $\phi ,\psi \in M^r_{(v)}({\mathbf R^{d}})$ be such that are dual frames to each other. Then the following is true:

1.  The operators $S_{\phi ,\psi} ^\Lambda \equiv D_\psi \circ C_\phi$ and $S_{\psi ,\phi} ^\Lambda \equiv D_\phi \circ C_\psi$ are both the identity map on $M^{{\boldsymbol p}}_{\sigma ,(\omega )}({\mathbf R^{d}})$, and if $f\in M^{{\boldsymbol p}}_{(\omega )}({\mathbf R^{d}})$, then $$\begin{aligned}
    f &= \sum _{j,\iota \in \Lambda} (V_\phi f)(j,\iota )
    e^{i{\langle {\, \cdot \, },\iota \rangle}}\psi ({\, \cdot \, }-j)\notag
    \\[1ex]
    &=
    \sum _{j,\iota \in \Lambda } (V_\psi f)(j,\iota )
    e^{i{\langle {\, \cdot \, },\iota \rangle}}\phi ({\, \cdot \, }-j),\label{GabExpForm}\end{aligned}$$ with unconditional norm-convergence in $M^{{\boldsymbol p}}_{\sigma ,(\omega )}$ when $\max ({{\boldsymbol p}}) <\infty$, and with convergence in $M^\infty _{(\omega)}$ with respect to the weak$^*$ topology otherwise;

2.  if $f\in M^\infty _{(1/v)}({\mathbf R^{d}})$, then $$\displaystyle{{\Vert f\Vert _{M^{{\boldsymbol p}}_{\sigma ,(\omega )}}}
    \asymp
    {\Vert V_\phi f\Vert _{\ell ^{{\boldsymbol p}}_{\sigma ,(\omega )} (\Lambda ^2) }}
    \asymp
    {\Vert V_\psi f\Vert _{\ell ^{{\boldsymbol p}}_{\sigma ,(\omega )} (\Lambda ^2)}} }.$$

Let $v$, $\phi$ and $\Lambda$ be as in Proposition \[ThmS\]. Then 