Global structure of some ultradistributions

Abstract

Given $p \in \mathbb{N}$, a non empty open subset $\Omega$ of $\mathbb{R}^k$ and a semi-regular matrix $\mathfrak{M}$, we characterize the elements of the duals of the Beurling classes $\mathcal{D}{(\mathfrak{M})}{\Omega}$ and $\\mathcal{D}_{L_p}{(\mathfrak{M})}{\Omega}$ of ultradifferentiable functions. We provide a global representation of these ultradistributions with and without compact support by means of series involving measures in the first case and elements of $\L_{loc}^{q}{\Omega}$ in the second.