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.