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

Let $\mathcal{B}$ be a normal quasi-Banach function space with respect to $r_0\in (0,1]$ and $v_0$, $\omega$ be $v$-moderate, and let $r\in [r_0,\mathrm{\infty }]$. Then we prove that $f$ belongs to the modulation space $M(\omega ,\mathcal{B})$, iff $V_{\phi }f$ belongs to the Wiener amalgam space $W^r(\omega ,\mathcal{B})$, and $$\mathrm{\parallel }f{\mathrm{\parallel }}_{M(\omega ,\mathcal{B})}\asymp \mathrm{\parallel }{V}_{\phi }f\omega {\mathrm{\parallel }}_{\mathcal{B}}\asymp \mathrm{\parallel }{V}_{\phi }f{\mathrm{\parallel }}_{W^r(\omega ,\mathcal{B})}\mathrm{.}$$ We also use the results to deduce continuity for pseudo-differential operators with symbols in weighted $M^{\mathrm{\infty },r_0}$-spaces, with $r_0\le 1$, when acting on $M(\omega ,\mathcal{B})$-spaces.