On the space of Laplace transformable distributions

TL;DR

This paper characterizes the space of Laplace transformable distributions over convex open sets as an ultrabornological (PLS)-space and explicitly identifies its topological predual, advancing the understanding of their functional analytic structure.

Contribution

It establishes the ultrabornological (PLS)-space structure of Laplace transformable distributions and explicitly constructs their topological predual, providing new insights into their topology.

Findings

The space of Laplace transformable distributions is ultrabornological (PLS).

An explicit topological predual of this space is determined.

The results deepen the understanding of the functional analytic properties of these distributions.

Abstract

We show that the space $\mathcal{S}'(\Gamma)$ of Laplace transformable distributions, where $\Gamma \subseteq \mathbb{R}^d$ is a non-empty convex open set, is an ultrabornological (PLS)-space. Moreover, we determine an explicit topological predual of $\mathcal{S}'(\Gamma)$.