sets of all $f\in {\mathcal S}_{1/2}'({\mathbf R^{d}})$ such that $V_\phi f\in
L^{p,q}_{(\omega )}({\mathbf R^{2d}})$ and $V_\phi f\in
L^{p,q}_{*,(\omega )}({\mathbf R^{2d}})$, respectively. We equip these spaces with the quasi-norms $${\Vert f\Vert _{M^{p,q}_{(\omega )}}}\equiv {\Vert V_\phi f\Vert _{L^{p,q}_{(\omega )}}}
\quad \text{and}\quad
{\Vert f\Vert _{W^{p,q}_{(\omega )}}}\equiv {\Vert V_\phi f\Vert _{L^{p,q}_{*,(\omega )}}},$$ respectively. One of the most common types of modulation spaces concerns $M^{p,q}_{(\omega )}({\mathbf R^{d}})$ with $\omega \in {\mathscr P}({\mathbf R^{2d}})$, and are sometimes called standard modulation spaces. They were introduced by Feichtinger in [@Fei1] for certain choices of $\omega$.

More generally, for any $\sigma \in \operatorname S_{2d}$, ${{\boldsymbol p}}\in (0,\infty ]^{2d}$ and $\omega \in {\mathscr P}_E({\mathbf R^{2d}})$, the modulation space $M^{{\boldsymbol p}}_{\sigma ,(\omega )}({\mathbf R^{d}})$ is the set of all $f\in {\mathcal S}_{1/2}'({\mathbf R^{d}})$ such that $V_\phi f\in
L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{2d}})$, and we equip $M^{{{\boldsymbol p}}}_{\sigma (\omega )}({\mathbf R^{d}})$ with the quasi-norm $$\label{modnorm2}
{\Vert f\Vert _{M^{{{\boldsymbol p}}}_{\sigma , (\omega )} }}\equiv
{\Vert V_\phi f\Vert _{L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}}}.$$

For conveniency we set $M^p_{(\omega )}=M^{p,p}_{(\omega)}$, and if $\omega =1$ everywhere, then set $$M^{{\boldsymbol p}}=M^{{\boldsymbol p}}_{\sigma ,(\omega )},\quad
M^{p,q} = M^{p,q}_{(\omega )},\quad
W^{p,q} = W^{p,q}_{(\omega )}
\quad \text{and}\quad
M^{p} = M^{p}_{(\omega )}.$$

In the following propositions we list some properties for modulation 