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 f : M → M' be a CR homeomorphism between two minimal, rigid polynomial varieties of Cn without holomorphic curves. We show that f extends biholomorphically in a neighborhood of M if f extends holomorphically in a neighborghood of a point p0 ∈ M or if f is of class C1. In the other hand, in case M and M' are two algebraic hypersurfaces, we obtain the extension without supplementary conditions.

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...

Given a multivariate polynomial $h\left(X_1,\cdots ,X_n\right)$ with integral coefficients verifying an hypothesis of analytic regularity (and satisfying $h\left(\mathbf{0}\right)=1$), we determine the maximal domain of meromorphy of the Euler product ${\prod }_{p\mathrm{prime}}h\left(p^{-s_1},\cdots ,p^{-s_n}\right)$ and the natural boundary is precisely described when it exists. In this way we extend a well known result for one variable polynomials due to Estermann from 1928. As an application, we calculate the natural boundary of the multivariate Euler products associated to a family of toric varieties.

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...

In this paper we extend the results on analytic continuation of germs of holomorphic mappings from a real analytic hypersurface to a real algebraic hypersurface to the case when the target hypersurface is of higher dimension than the source. More precisely, we prove the following: Let $M$ be a connected smooth real analytic minimal hypersurface in $C^n$, $M^{\prime }$ be a compact strictly pseudoconvex real algebraic hypersurface in $C^N$, $1\<n\le N$. Suppose that $f$ is a germ of a holomorphic map at a point $p$ in $M$ and $f\left(M\right)$ is in...

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

We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions.