In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold $X$. Then, we define stratified ultradistributions of Beurling and Roumieu type on $X$. In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if $X$ is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to $X$. Second, the tempered-stratified...

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.

We consider the space ${}^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on ${}^{ℳ}$. A description of the space ${(}^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

A kernel theorem for spaces of Laplace ultradistributions supported by an n-dimensional cone of product type is stated and proved.