A note on the barrelledness of weighted PLB-spaces of ultradifferentiable functions (Q6039903)

In a recent paper [\textit{A. Debrouwere} et al., Proc. Am. Math. Soc. 148, No.~12, 5171--5180 (2020; Zbl 1477.46028)], the authors studied weighted (PLB)-spaces of ultradifferentiable functions of Beurling and Roumieu type in the sense of Braun, Meise and Taylor, via a weight function and a weight system. They characterized barrelledness of these spaces in terms of a condition of the defining weight system, using a growth assumption on the weight function and the weight system in order to prove the necessity. In the present paper, they improve their previous work by getting rid of the aforementioned growth constraint. In particular, they show that the multiplier space of the Gelfand-Shilov space \(\Sigma_r^s(\mathbb{R}^d)\) of Beurling type is ultrabornological, whereas the one of the Gelfand-Shilov space \(\mathcal{S}_s^r(\mathbb{R}^d) \) of Roumieu type is not barrelled.