Topological properties of convolutor spaces via the short-time Fourier transform

Abstract: We discuss the structural and topological properties of a general class of weighted $L^1$ convolutor spaces. Our theory simultaneously applies to weighted $\mathcal{D}'{L^1}$ spaces as well as to convolutor spaces of the Gelfand-Shilov spaces $\mathcal{K}{M_p}$. In particular, we characterize the sequences of weight functions $(M_p){p \in \mathbb{N}}$ for which the space of convolutors of $\mathcal{K}{M_p}$ is ultrabornological, thereby generalizing Grothendieck's classical result for the space $\mathcal{O}'{C}$ of rapidly decreasing distributions. Our methods lead to the first direct proof of the completeness of the space $\mathcal{O}{C}$ of very slowly increasing smooth functions.