quasi-norms.

    If in addition ${{\boldsymbol p}}\ge 1$, then $M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}
    ({\mathbf R^{d}})$ is a Banach space with norm ;

2.  $M^{{{\boldsymbol p}}_1}_{\sigma ,(\omega _1)}({\mathbf R^{d}})\subseteq
    M^{{{\boldsymbol p}}_2}_{\sigma ,(\omega _2)}({\mathbf R^{d}})$.

Next we discuss Gabor expansions, and start by recalling some notions. It follows from the analysis in Chapters 11–14 in [@Gc2] that the operators in the following definition are well-defined and continuous.

\[DefAnSynGabOps\] Let $\Lambda =\Lambda _{[\theta ]}$ be as in , $\omega ,v\in {\mathscr P}_E({\mathbf R^{2d}})$ be such that $\omega$ is $v$-moderate, and let $\phi ,\psi \in M^1_{(v)}({\mathbf R^{d}})$.

1.  The *analysis operator* $C^{\Lambda}_\phi$ is the operator from $M^\infty _{(\omega )}({\mathbf R^{d}})$ to $\ell ^{\infty}_{(\omega)}(\Lambda )$, given by $$C^\Lambda _\phi f \equiv \{ V_\phi f(j,\iota ) \} _{j,\iota \in \Lambda} \text ;$$

2.  The *synthesis operator* $D^{\Lambda}_\psi$ is the operator from $\ell ^\infty _{(\omega )}(\Lambda )$ to $M^\infty _{(\omega)}({\mathbf R^{d}})$, given by $$D^\Lambda _\psi c \equiv \sum _{j,\iota \in \Lambda} c_{j,\iota}
    e^{i{\langle {\, \cdot \, },\iota \rangle}}\phi ({\, \cdot \, }-j)\text ;$$

3.  The *Gabor frame operator* $S^{\Lambda}_{\phi ,\psi}$ is the operator on $M^\infty _{(\omega )}({\mathbf R^{d}})$, given by $D^\Lambda _\psi \circ
    C^\Lambda _\phi$, i.e. $$S^{\Lambda}_{\phi ,\psi}f \equiv \sum _{j,\iota \in \Lambda} V_\phi f(j,\iota )
    e^{i{\langle {\, 