Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators

We consider a broad class of modulation spaces $M(\omega ,B)$, parameterized with weight function ω and a normal quasi-Banach function space $B$ of order $r_0\in (0,1]$. Then we prove that $f\in M(\omega ,B)$, if and only if $V_{\varphi }f$ belongs to the Wiener amalgam space $W^r(\omega ,B)$, and${\Vert f\Vert }_{M(\omega ,B)}\asymp {\Vert V_{\varphi }f\cdot \omega \Vert }_B\asymp {\Vert V_{\varphi }f\Vert }_{W^r(\omega ,B)},r\in [r_0,\infty ].$

We use the results to extend and improve continuity and lifting properties for pseudo-differential and Toeplitz operators with symbols in weighted $M^{\infty ,r_0}$-spaces, $r_0\le 1$, when acting on $M(\omega ,B)$-spaces.