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

Abstract

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 $r\in [r_0,\infty ]$. Then we prove that $f$ belongs to the modulation space $M(\omega ,\mathcal{B})$, iff $V_{\varphi }f$ belongs to the Wiener amalgam space $W^r(\omega ,\mathcal{B})$, and $$\Vert f{\Vert }_{M(\omega ,\mathcal{B})}\asymp \Vert V_{\varphi }f\omega {\Vert }_{\mathcal{B}}\asymp \Vert V_{\varphi }f{\Vert }_{W^r(\omega ,\mathcal{B})}.$$ We also use the results to deduce continuity for pseudo-differential operators with symbols in weighted $M^{\infty ,r_0}$-spaces, with $r_0\le 1$, when acting on $M(\omega ,\mathcal{B})$-spaces.

Title for 1st version: "On a broad family of quasi-Banach modulation spaces" This is the third version, which is extended with lifting properties for modulation spaces, and with convolution properties. It now has 63 p. (39 p. in 2nd version)