Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators
Joachim Toft, Christine Pfeuffer, Nenad Teofanov
TL;DR
This work provides a comprehensive framework for norm control in broad modulation spaces $M(\omega, \mathscr B)$ where $\mathscr B$ is a normal quasi-Banach function space, extending known norm equivalences beyond Gabor-frame-based methods. The authors prove that $f\in M(\omega, \mathscr B)$ is equivalent to $V_\phi f \cdot \omega$ belonging to a translation-invariant QBF space $\mathscr B$ and to a Wiener amalgam space $W^r(\omega, \mathscr B)$, with robust, window-tolerant quasi-norm equivalences. Building on these norm estimates, they establish broad continuity results for pseudo-differential operators with symbols in weighted $M^{\infty,r_0}$ spaces acting on modulation spaces, and develop advanced Toeplitz lifting and operator-isomorphism results within this generalized setting. The results yield a powerful, unified toolkit for time-frequency analysis operators, enabling continuity and lifting in highly general function-spaces contexts, with potential implications for quantum mechanics and signal processing. Overall, the paper significantly broadens the landscape of modulation-space theory and operator calculus without relying on Gabor frame decompositions.
Abstract
Let $\mathscr B$ be a normal quasi-Banach function space with respect to $r_0 \in (0,1]$ and $v_0$, $ω$ be $v$-moderate, and let $r\in [r_0,\infty ]$. Then we prove that $f$ belongs to the modulation space $M(ω,\mathscr B )$, iff $V_φf$ belongs to the Wiener amalgam space $W ^r(ω,\mathscr B )$, and $$ \| f \| _{M(ω, \mathscr B)} \asymp \| V _φf \, ω\| _{\mathscr B} \asymp \| V _φf\| _{W ^r(ω, \mathscr B)}. $$ 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 $M(ω,\mathscr B )$-spaces.