Factorizations for quasi-Banach time-frequency spaces and Schatten classes

TL;DR

This paper establishes factorization properties for various quasi-Banach time-frequency spaces and Schatten classes, extending classical results and providing new identities for Wiener amalgam and modulation spaces.

Contribution

It extends factorization results to quasi-Banach modules over quasi-Banach algebras, including Wiener amalgam, modulation, and Schatten symbol spaces, under bounded approximate identities.

Findings

Wiener amalgam space factorization: WL^{1,r}*WL^{p,q}=WL^{p,q} for r in (0,1]

Improved Rudin's identity: L^1*L^1=L^1

Extension of factorization to a broad class of pseudo-differential symbol classes

Abstract

We deduce factorization properties for a quasi-Banach module over a quasi-Banach algebra. Especially we extend a result by Hewitt and prove that if any such algebra which possess a bounded left approximate identity, then any element in the module can be factorized. As applications, we deduce factorization properties for Wiener amalgam spaces, for an extended family of modulation spaces and for Schatten symbol classes in pseudo-differential calculus under multiplications like convolutions, twisted convolutions and symbolic products. For example we show for Wiener amalgam spaces that WL^{1,r}*WL^{p,q}=WL^{p,q} when r in (0,1], and p and q are finite and larger than r. In particular we improve Rudin's identity L^1*L^1=L^1.