spaces, and refer to [@Fei1; @FG1; @Gc2; @Toft5] for proofs.

\[p1.4A\] Let $\sigma \in \operatorname S_{2d}$ and ${{\boldsymbol p}}\in (0,\infty ]^{2d}$. Then the following is true:

1.  if $\omega \in {\mathscr P}_{E}({\mathbf R^{2d}})$, then $\Sigma _1({\mathbf R^{d}})
    \subseteq M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}}) \subseteq \Sigma _1'({\mathbf R^{d}})$;

2.  if $\omega \in {\mathscr P}_{E}({\mathbf R^{2d}})$ satisfies for every $r >0$, then ${\mathcal S}_1({\mathbf R^{d}})\subseteq M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})
    \subseteq {\mathcal S}_1'({\mathbf R^{d}})$;

3.  if $\omega \in {\mathscr P}({\mathbf R^{2d}})$, then $\mathscr S({\mathbf R^{d}})\subseteq M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})
    \subseteq \mathscr S '({\mathbf R^{d}})$.

\[p1.4B\] Let $\sigma \in \operatorname S_{2d}$, $r\in (0,1]$, ${{\boldsymbol p}},{{\boldsymbol p}}_j\in
(0,\infty ] ^{2d}$ and $\omega ,\omega _j,v\in {\mathscr P}_{E}({\mathbf R^{2d}})$, $j=1,2$, be such that $r\le {{\boldsymbol p}}$, ${{\boldsymbol p}}_1\le {{\boldsymbol p}}_2$, $\omega _2\lesssim \omega _1$, and $\omega$ is $v$-moderate. Then the following is true:

1.  if $\phi \in M^r_{(v)}({\mathbf R^{d}})\setminus 0$, then $f\in M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})$, if and only if is finite. In particular, $M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})$ is independent of the choice of $\phi \in M^r_{(v)}({\mathbf R^{d}})\setminus 0$. Moreover, $M^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})$ is a quasi-Banach space under the quasi-norm in , and different choices of $\phi$ give rise to equivalent 