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To a plurisubharmonic function $u$ on ${\mathbf{C}}^n$ with logarithmic growth at infinity, we may associate the Robin function $${\displaystyle {\rho }_u\left(z\right)=\underset{\lambda \to \infty }{lim\; sup}u\left(\lambda z\right)-log\left(\lambda z\right)}$$ defined on ${\mathbf{P}}^{n-1}$, the hyperplane at infinity. We study the classes $L_{+}$, and (respectively) $L_p$ of plurisubharmonic functions which have the form $u=log(1+|z\left|\right)+O\left(1\right)$ and (respectively) for which the function ${\rho }_u$ is not identically $-\infty$. We obtain an integral formula which connects the Monge-Ampère measure on the space ${\mathbf{C}}^n$ with the Robin function on ${\mathbf{P}}^{n-1}$. As an application we obtain a criterion on...

Étant donné un système $\left(S\right)$ d’équations différence-différentielles à coefficients constants en deux variables, où les retards sont commensurables, de la forme : ${\mu }_1*f=0$, ${\mu }_2*f=0$, si le système n’est pas redondant (i.e. $V\{{\widehat{\mu }}_1={\widehat{\mu }}_2=0\}$ est discrète dans ${\mathbf{C}}^2$), toute solution $C^{\infty }$ du système admet une représentation $f\left(x\right)=\Sigma a_{\gamma }\left(x\right)e^{i⟨\gamma ,x⟩}$, où $\gamma \in V$, $a_{\gamma }\in \mathbf{C}[x_1,x_2]$ et $a_{\gamma }\left(x\right)e^{i⟨\gamma ,x⟩}$ est une solution du système $\left(S\right)$. La série est de plus convergente dans $ℰ\left({\mathbf{R}}^2\right)$ après un groupement de termes indépendant de la solution $f$.

This note can be considered as a long summary of the invited lecture given by J. Bonet in the Second Functional Analysis Meeting held in Jarandilla de la Vega (Cáceres) in June 1980 and it is based on our joint article [2], which will appear in Studia Mathematica. (...) The main result of the paper [2] is the characterization of those weight functions for which the analogue of Whitney's extension theorem holds.

We prove that an analytic surface $V$ in a neighborhood of the origin in ${ℂ}^3$ satisfies the local Phragmén-Lindelöf condition ${\mathrm{PL}}_{\mathrm{loc}}$ at the origin if and only if $V$ satisfies the following two conditions: (1) $V$ is nearly hyperbolic; (2) for each real simple curve $\gamma$ in ${ℝ}^3$ and each $d\ge 1$, the (algebraic) limit variety $T_{\gamma ,d}V$ satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure $k$-dimensional analytic variety $V$ to satisify ${\mathrm{PL}}_{\mathrm{loc}}$.

It is shown that every closed rotation and translation invariant subspace $V$ of $C\left({\mathbf{R}}^n\right)$ or $\delta \left({\mathbf{R}}^n\right)$, $n\ge 2$, is of spectral synthesis, i.e. $V$ is spanned by the polynomial-exponential functions it contains. It is a classical problem to find those measures $\mu$ of compact support on ${\mathbf{R}}^2$ with the following property: (P) The only function $f\in C\left({\mathbf{R}}^2\right)$ satisfying ${\int }_{{\mathbf{R}}^2}f\circ \sigma d\mu =0$ for all rigid motions $\sigma$ of ${\mathbf{R}}^2$ is the zero function. As an application of the above result a characterization of such measures is obtained in terms of their Fourier-Laplace transforms....

Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P\left(D\right)$ are characterized that admit a continuous linear right inverse on $ℰ\left(\Omega \right)$ or ${𝒟}^{\text{'}}\left(\Omega \right)$, $\Omega$ an open set in ${\mathbf{R}}^n$. For bounded $\Omega$ with $C^1$-boundary these properties are equivalent to $P\left(D\right)$ being very hyperbolic. For $\Omega ={\mathbf{R}}^n$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.

For algebraic surfaces, several global Phragmén-Lindelöf conditions are characterized in terms of conditions on their limit varieties. This shows that the hyperbolicity conditions that appeared in earlier geometric characterizations are redundant. The result is applied to the problem of existence of a continuous linear right inverse for constant coefficient partial differential operators in three variables in Beurling classes of ultradifferentiable functions.