# A multidimensional Tauberian theorem for Laplace transforms of ultradistributions

## TL;DR

This paper establishes a multidimensional Tauberian theorem for Laplace transforms of ultradistributions, providing a new characterization of bounded sets in ultradistribution spaces with support in convex cones.

## Contribution

It introduces a novel multidimensional Tauberian theorem for Laplace transforms of Gelfand-Shilov ultradistributions and characterizes bounded sets in these spaces.

## Key findings

- Derived a Laplace transform characterization of bounded sets in ultradistribution spaces.
- Established a multidimensional Tauberian theorem for Laplace transforms.
- Extended the theory to ultradistributions supported in convex cones.

## Abstract

We obtain a multidimensional Tauberian theorem for Laplace transforms of Gelfand-Shilov ultradistributions. The result is derived from a Laplace transform characterization of bounded sets in spaces of ultradistributions with supports in a convex acute cone of $\mathbb{R}^{n}$, also established here.