Weighted $(PLB)$-spaces of ultradifferentiable functions and multiplier spaces

TL;DR

This paper investigates weighted $(PLB)$-spaces of ultradifferentiable functions, characterizing their ultrabornological property via weight systems, and determines the multiplier spaces of Gelfand-Shilov spaces and related ultrahyperfunction spaces.

Contribution

It generalizes Grothendieck's ultrabornological result to ultradifferentiable function spaces and characterizes multiplier spaces for Gelfand-Shilov and ultrahyperfunction spaces.

Findings

Weighted $(PLB)$-spaces are ultrabornological under specific conditions.

Multiplier space of Fourier ultrahyperfunctions is ultrabornological.

Multiplier space of Fourier hyperfunctions is not ultrabornological.

Abstract

We study weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck's classical result that the space $\mathcal{O}_M$ of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not.