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 construct a solution T0 in the distribution sense of equation fT = 1 near a critical point of f and we give an upper bound for the order of T0 in terms of f's Newton Polyhedron, provided f is non degenerate in some sense. The order of T0 is equal to this upper bound when f is non-negative.

Étant donnés $r$ champs de vecteurs $X_1,...,X_r$, réels, de classe $C^{\infty }$ dans ${\mathbf{R}}^n$, nous étudions l’existence de traces sur une variété de classe $C^{\infty }$, de dimension $(n-1)$, frontière d’un ouvert $\Omega$, des distributions $u\in {𝒟}^{\text{'}}\left(\Omega \right)$ telles que:$$u\in L^2\left(\Omega \right);X_{j}u\in L^2\left(\Omega \right),j=1,...,r.$$

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.

Propagation of regularity is considered for solutions of rectangular systems of infinite order partial differential equations (resp. convolution equations) in spaces of hyperfunctions (resp. C∞ functions and distributions). Known resulys of this kind are recovered as particular cases, when finite order partial differential equations are considered.