Let $K$ be a compact set in an open set $D$ on a Stein manifold $\Omega$ of dimension $n$. We denote by $H^{\infty }\left(D\right)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm and by $A_K^D$ a compact subset of the space $C\left(K\right)$ consisted of all restrictions of functions from the unit ball ${𝔹}_{H^{\infty }\left(D\right)}$. In 1950ies Kolmogorov posed a problem: does$${\mathscr{H}}_{\epsilon }\left(A_K^D\right)\sim \tau {\left(ln\frac{1}{\epsilon }\right)}^{n+1},\epsilon \to 0,$$where ${\mathscr{H}}_{\epsilon }\left(A_K^D\right)$ is the $\epsilon$-entropy of the compact $A_K^D$. We give here a survey of results concerned with this problem and a related problem on the strict asymptotics of Kolmogorov diameters...

The purpose of this article is twofold. The first is to show a criterion for the normality of holomorphic mappings into Abelian varieties; an extension theorem for such mappings is also given. The second is to study the convergence of meromorphic mappings into complex projective varieties. We introduce the concept of d-convergence and give a criterion of d-normality of families of meromorphic mappings.

Strong pathologies with respect to growth properties can occur for the extension of holomorphic functions from submanifolds $D^{\text{'}}$ of pseudoconvex domains $D$ to all of $D$ even in quite simple situations; The spaces $A^p(D^{\text{'}}):=𝒪(D^{\text{'}})\cap L^p(D^{\text{'}})$ are, in general, not at all preserved. Also the image of the Hilbert space $A^2\left(D\right)$ under the restriction to $D^{\text{'}}$ can have a very strange structure.

Nous montrons qu’une fonction holomorphe sur un sous-ensemble analytique transverse $V$ d’un domaine $D$ borné strictement pseudoconvexe de ${\mathbf{C}}^n$ admet une extension dans $H^p\left(D\right)(1\le p\<+\infty )$ si et seulement si elle vérifie une condition de type $L^p$ à poids sur $V$ ; la démonstration est en partie basée sur la résolution de l’équation $\partial$ avec estimations de type “mesures de Carleson”.

Let D be a bounded strictly pseudoconvex domain with smooth boundary and f = (f1, ..., fp) (fi ∈ Hol(D)) a complete intersection with normal crossing. In this paper we study an extension problem in L∞-norm for holomorphic functions defined on f-1(0) ∩ D and a decomposition formula g = ∑i=1p figi for holomorphic functions g ∈ I(f1, ..., fp)(D) in Lipschitz spaces. We stress that for the two problems the classical theorem cannot be applied because f-1(0) has singularities on the boundary ∂D. This...

We study the problem of extending functions from linear affine subvarieties for the Bergman scale of spaces on convex finite type domains. Our results solve the problem for H¹(D). For other Bergman spaces the result is ϵ-optimal.

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...

We study the extension problem for germs of holomorphic isometries $f:(D;x_0)\to (\Omega ;f\left(x_0\right))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d{s}_D^2$ on $D$ and $d{s}_{\Omega }^2$ on $\Omega$. Our main focus is on boundary extension for pairs of bounded domains $(D,\Omega )$ such that the Bergman kernel $K_D(z,w)$ extends meromorphically in $(z,\overline{w})$ to a neighborhood of $\overline{D}\times D$, and such that the analogous statement holds true for the Bergman kernel $K_{\Omega }(\varsigma ,\xi )$ on $\Omega$. Assuming that $(D;d{s}_D^2)$ and $(\Omega ;d{s}_{\Omega }^2)$ are complete Kähler manifolds, we prove that the germ...

This note is an attempt to describe a part of the historical development of the research on separately holomorphic functions.