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

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