In this paper, we study a class of first order nonlinear degenerate partial differential equations with singularity at $(t,x)=(0,0)\in {\mathbf{C}}^2$. Using exponential-type Nagumo norm approach, the Gevrey asymptotic analysis is extended to case of holomorphic parameters in a natural way. A sharp condition is then established to deduce the $k$-summability of the formal solutions. Furthermore, analytical solutions in conical domains are found for each type of these nonlinear singular PDEs.

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...