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

TL;DR

This paper establishes norm equivalences between modulation spaces and Wiener amalgam spaces for broad classes of function spaces, and applies these results to prove continuity of certain pseudo-differential operators.

Contribution

It introduces new norm estimates for modulation spaces related to quasi-Banach spaces and demonstrates operator continuity in this generalized setting.

Findings

Norm equivalence between modulation and Wiener amalgam spaces.

Continuity results for pseudo-differential operators with weighted symbols.

Extension of modulation space theory to quasi-Banach and broad function spaces.

Abstract

Let $\mathscr B$ 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 $M(\omega ,\mathscr B )$, iff $V_\phi f$ belongs to the Wiener amalgam space $W ^r(\omega ,\mathscr B )$, and $$ \| f \| _{M(\omega , \mathscr B)} \asymp \| V _\phi f \, \omega \| _{\mathscr B} \asymp \| V _\phi f\| _{W ^r(\omega, \mathscr B)}. $$ 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 $M(\omega ,\mathscr B )$-spaces.