[@Gc2], with certain pre-results given already in [@GH1; @Sj1], and is in some sense obtained by Feichtinger already in [@Fei1].

There are also several extensions and modifications of these results. For example, in [@GH2; @Toft2] it was proved that $$\label{ModCont1C}
\begin{gathered}
{\operatorname{Op}}(a)\, :\, M^{p_1,q_1}\to M^{p_2,q_2}
\quad \text{when} \quad
a\in M^{p,q}, \ q\le \min (p,p')
\\[1ex]
\quad \text{and} \quad
\frac 1{p_1} - \frac 1{p_2} = \frac 1{q_1} - \frac 1{q_2} = 1-\frac 1{p} - \frac 1{q},
\ q\le p_2,q_2\le p,
\end{gathered}$$ which covers both and . See also [@Toft3; @Toft5; @Toft8; @Toft11] for extensions of the latter result to weighted spaces, and [@MoPf; @Wa] for related results with other types of modulation spaces as symbol classes. Furthermore, in [@DeRu1; @DeRu2; @DeRu3; @FiRu], related analysis in background of compact or local-compact Lie groups can be found.

In the literature, it is usually assumed that $p$ and $q$ here above belong to $[1,\infty ]$ instead of the larger interval $(0,\infty ]$. An important reason for excluding the cases $p<1$ or $q<1$ is that the involved spaces fail to be local convex, leading in general to several additional difficulties compared to the situation when $p,q\in [1,\infty ]$. On the other hand, in view of 