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

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.