Let $K\subset Q$ be compact, convex sets in ${ℝ}^n$ with $\stackrel{\circ }{K}\ne \varnothing$ and let $P\left(D\right)$ be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of $P\left(D\right)$ in the space $ℰ\left(K\right)$ of all $C^{\infty }$-functions on $K$ extends to a zero solution in $ℰ\left(Q\right)$ resp. in $ℰ\left({ℝ}^n\right)$. The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of $P$ in ${ℂ}^n$ and in terms of fundamental solutions for $P\left(D\right)$ with lacunas.

The problem of the existence of extension maps from 0 to ℝ in the setting of the classical ultradifferentiable function spaces has been solved by Petzsche [9] by proving a generalization of the Borel and Mityagin theorems for $C^{\infty }$-spaces. We get a Ritt type improvement, i.e. from 0 to sectors of the Riemann surface of the function log for spaces of ultraholomorphic functions, by first establishing a generalization to some nonclassical ultradifferentiable function spaces.

Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{\mathbb{R}}$, the real underlying manifold to $X$. Let $\mathrm{\Omega }$ be an open subset of $S$ with $\partial \mathrm{\Omega }$ analytic, $Y$ a complexification of $S$. We first recall the notion of $\mathrm{\Omega }$-tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\overline{\partial }}_b$ to the extendability of $CR$ functions on $\mathrm{\Omega }$ to $\mathrm{\Omega }$-tuboids of $X$. Next, if $X$ has complex dimension 2, we give results on extension...

It is proved that any Banach valued distribution on a bounded set can be extended to all of ${\mathbb{R}}^d$ if and only if it is a derivative of a uniformly continuous function. A similar result is given for distributions on an unbounded set. An example shows that this does not extend to Frechet valued distributions. This relies on the fact that a Banach valued distribution is locally a derivative of a uniformly continuous function. For sake of completeness, a global representation of a Banach valued distribution...

In [3], J. Chaumat and A.-M. Chollet prove, among other things, a Whitney extension theorem, for jets on a compact subset E of ℝⁿ, in the case of intersections of non-quasi-analytic classes with moderate growth and a Łojasiewicz theorem in the regular situation. These intersections are included in the intersection of Gevrey classes. Here we prove an extension theorem in the case of more general intersections such that every $C^{\infty }$-Whitney jet belongs to one of them. We also prove a linear extension theorem...

Cet article établit quelques propriétés des distributions sur un ouvert $\Omega$ de ${\mathbf{R}}^N$ dont le hessien est une mesure bornée. Après quelques propriétés topologiques – Compacité faible des bornées de $HB\left(\Omega \right)$ lorsque $\Omega$ est borné, densité des fonctions régulières pour une topologie assez finie – on s’intéresse au comportement sur le bord de $\Omega$ lorsque $\partial \Omega$ est assez régulier; pour ce faire, on est amené à étudier celui des fonctions de $W^{2,1}$. On montre enfin dans une 3ème partie des théorèmes d’injection de Sobolev et notamment...

Soit $a$ un réel de $]0,1[$. Nous étudions le système d’équations de convolution suivant$$(*)\forall x\in {\mathbf{R}}^2,f\left(x\right)=\frac{1}{4}\sum _{\genfrac{}{}{0pt}{}{ϵ=\pm 1}{{ϵ}^{\text{'}}=\pm 1}}f(x+(ϵ,{ϵ}^{\text{'}}\left)\right)=\frac{1}{4}\sum _{\genfrac{}{}{0pt}{}{ϵ=\pm 1}{{ϵ}^{\text{'}}=\pm 1}}f(x+a(ϵ,{ϵ}^{\text{'}}\left)\right).$$Nous démontrons que les exponentielles polynômes solutions de $(*)$ sont denses dans l’espace des solutions $C^{\infty }$ du système d’équations; l’idéal de ${ℰ}^{\text{'}}({\mathbf{R}}^2)$ engendré par les transformées de Fourier des deux mesures intervenant ici est “slowly decreasing” au sens de Berenstein-Taylor. Lorsque $a$ n’est pas un nombre de Liouville, nous montrons qu’il existe un ouvert relativement compact telle que toute solution distribution de $(*)$ régulière...