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Extension of meromorphic distribution

Let $W$ be a topological vector space (e.g. Frechet) with a dense subspace $V$. Let $D_s$ be a distribution on $V$ that is meromorphic in $s\in\mathbb C$ and extends continuously to $W$ with respect ...
Tian An's user avatar
  • 3,799
2 votes
0 answers
85 views

Proving that $u_0$ is a canonical solution

I've asked this question on stack exchange before but no one could help me so I wish I can get some help here. Let's first start with the definition of the canonical solution: Consider $\frac{\...
User-123's user avatar
1 vote
0 answers
271 views

Starlike sets in $\mathbb{C}^n$

Let $S$ be a bounded domain in $\mathbb{C}^n$. $S$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. If $S$ ...
user332912's user avatar
4 votes
0 answers
197 views

Approximation of a holomorphic function vanishing at a submanifold by polynomials

Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). ...
Sergei Akbarov's user avatar
3 votes
1 answer
144 views

Coefficient problem in the class $\Sigma$

Let $\Sigma$ be the class of univalent (injective) holomorphic functions on $\mathbb{C}\backslash \mathbb{D}$ where $\mathbb{D}$ is the closed unit disk. Analogous to the famous Bieberbach conjecture ...
user700974's user avatar
5 votes
1 answer
159 views

$C^j$-topology considered by Greene and Krantz

My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
Pita's user avatar
  • 113
2 votes
0 answers
55 views

A holomorphic map into a Hilbert space with prescribed orthogonality

This is a variation of my previous question. Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is ...
erz's user avatar
  • 5,529
13 votes
1 answer
408 views

Does the $\overline{\partial}$ operator have closed image?

Let $X$ be a complex-analytic manifold, not necessarily compact. Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
Daniel Bruegmann's user avatar
6 votes
2 answers
240 views

Continuity of a differential of a Banach-valued holomorphic map

Originally posted on MSE. Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
erz's user avatar
  • 5,529
2 votes
0 answers
119 views

Covariant derivative of the Monge-Ampere equation on Kähler manifolds

I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
BenjaminRaj's user avatar
1 vote
0 answers
142 views

Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$

Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$: $$ V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}. $$ ...
Sergei Akbarov's user avatar
6 votes
1 answer
259 views

Do analytic functionals form a cosheaf?

Let $X$ be a complex-analytic manifold. Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
Daniel Bruegmann's user avatar
2 votes
1 answer
113 views

Norm of vector-valued holomorphic functions

Let $G$ be a connected simply connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space. Q1. Which functions $F:G\to(0,+\infty)$ are such that there is a holomorphic $f:G\to H\backslash ...
erz's user avatar
  • 5,529
6 votes
1 answer
319 views

Derivatives of norm of vector-valued holomorphic functions

Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think ...
erz's user avatar
  • 5,529
3 votes
1 answer
305 views

The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
Sergei Akbarov's user avatar
4 votes
0 answers
714 views

Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
user6818's user avatar
  • 1,893
0 votes
2 answers
1k views

Question on Hartogs's Extension Theorem

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)? For Hartogs's Extension Theorem see here: http://en.wikipedia.org/wiki/...
bernard's user avatar
  • 53
3 votes
1 answer
1k views

Constructing the imaginary part of a holomorphic function

Hallo, Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function ...
hapchiu's user avatar
  • 339
3 votes
2 answers
904 views

Corona Theorem in several variables

Hallo, I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit ...
hapchiu's user avatar
  • 339
-2 votes
1 answer
1k views

holomorphic extension of a function [closed]

hi, I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{...
bruno's user avatar
  • 29