Soient ${𝔑}_n$ (resp. ${ℰ}_n$) l’anneau des germes de fonctions de Nash (resp. l’anneau des germes de fonctions $C^{\infty }$) à l’origine de ${\mathbf{R}}^n$: ${ℬ}_n$ (resp. ${ℬ}_n^{\text{'}}$) le module sur ${𝔑}_n$ des germes de fonctions de Bernstein $C^{\infty }$ (resp. le module sur ${𝔑}_n$ des germes de distributions de Bernstein) à l’origine de ${\mathbf{R}}^n$. Les deux résultats principaux de l’article sont les suivants : ${ℬ}_n^{\text{'}}$ est un module injectif sur ${𝔑}_n$ et ${ℰ}_n/{ℬ}_n$ est un module plat sur ${𝔑}_n$.

We review recent developments in the theory of inductive limits and use them to give a new and rather easy proof for Hörmander?s characterization of surjective convolution operators on spaces of Schwartz distributions.

Let ${\epsilon }_{\omega }\left(I\right)$ denote the space of all ω-ultradifferentiable functions of Roumieu type on an open interval I in ℝ. In the special case ω(t) = t we get the real-analytic functions on I. For $\mu \in {\epsilon }_{\omega }{\left(I\right)}^{\text{'}}$ with $supp\left(\mu \right)=0$ one can define the convolution operator $T_{\mu }:{\epsilon }_{\omega }\left(I\right)\to {\epsilon }_{\omega }\left(I\right)$, $T_{\mu }\left(f\right)\left(x\right):=⟨\mu ,f(x-·)⟩$. We give a characterization of the surjectivity of $T_{\mu }$ for quasianalytic classes ${\epsilon }_{\omega }\left(I\right)$, where I = ℝ or I is an open, bounded interval in ℝ. This characterization is given in terms of the distribution of zeros of the Fourier Laplace transform $\widehat{\mu }$ of μ.

We show that if Ω is an open subset of ℝ², then the surjectivity of a partial differential operator P(D) on the space of ultradistributions ${}_{\left(\omega \right)}^{\text{'}}\left(\Omega \right)$ of Beurling type is equivalent to the surjectivity of P(D) on $C^{\infty }\left(\Omega \right)$.

Dato un sistema omogeneo di equazioni di convoluzione in spazi dotati di strutture analiticamente uniformi, si forniscono condizioni per ottenere teoremi di rappresentazione per le sue soluzioni.