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 ${\mathcal{D}}_{L^1}^{\prime }$ 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 ${\mathcal{O}}_C^{\prime }$ 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.

34 pages

Cited by in corpus (4)

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