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

We deduce factorization properties for Wiener amalgam spaces $W{L}^{p,q}$, an extended family of modulation spaces $M(\omega ,ℬ)$, and for Schatten symbols $s_p^w$ in pseudo-differential calculus under e. g. convolutions, twisted convolutions and symbolic products. Here $M(\omega ,ℬ)$ can be any quasi-Banach Orlicz modulation space. For example we show that $W{L}^{1,r}\ast W{L}^{p,q}=W{L}^{p,q}$ and $W{L}^{1,r}\#s_p^w=s_p^w$ when $r\in (0,1]$, $r\le p,q<\infty$. In particular we improve Rudin’s identity $L^1\ast L^1=L^1$.