Holomorphic extension from the unit sphere in

J Anal Math, 2011

Let f ∈ C ω (∂B n), where B n is the unit ball of C n. We prove that if a, b ∈ B n , a = b, for every complex line L passing through one of a or b, the restricted function f | L∩∂B n has a holomorphic extention to the cross-section L∩B n , then f is the boundary value of a holomorphic function in B n .

Annales Polonici Mathematici, 2019

We study the notions of extendability and domain of holomorphy in the infinite-dimensional case. In this setting it is also true that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. We also prove that the set of non-extendable functions belonging to some classes X(B) ⊂ H(B), B being the open unit ball in a separable complex Banach space, is a lineable and dense G δ. Moreover, when Ω is H b-holomorphically convex (defined in the text), it is shown that the set of non-extendable holomorphic functions on Ω is a lineable and dense G δ set.

Annales Polonici Mathematici, 2003

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

Pacific Journal of Mathematics, 1990

This paper studies the extensions of harmonic and analytic functions defined on the unit disk to continuous functions defined on a certain compactification of the disk.

Annales Polonici Mathematici, 2003

Let D j ⊂ C k j be a pseudoconvex domain and let A j ⊂ D j be a locally pluripolar set, j = 1, . . . , N . Put

Transactions of the American Mathematical Society

Let $D_j\subset\Bbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset\Bbb C^{n_1}\times...\times\Bbb C^{n_N}=\Bbb C^n. $$ Let $U\subset\Bbb C^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,...,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1}) \times(A_{j+1}\times...\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\Bbb C^{n_j}\: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,...,\Sigma_N$ are pluripolar. Put $$ X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times...\times A_{j-1})\times D_j \times(A_{j+1}\times...\times A_N)\: (z',z'')\notin\Sigma_j\}. $$ Then there exists a relatively closed pluripolar subset $\hat M\subset\hat X$ of the `envelope of holomor...

Chinese Annals of Mathematics, Series B, 2017

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions. A more complicated proof, using Cauchy's argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from C n to C n are obtained using Brouwer degree.

Journal of Geometric Analysis, 2003

Let f : D → D ′ be a proper holomorphic mapping between bounded domains D, D ′ in C 2 . Let M, M ′ be open pieces on ∂D, ∂D ′ respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under f is contained in M ′ . It is shown that f extends holomorphically across M . This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in C 2 .

2013

In the theory of one complex variable it is well known that for any domain G ⊂ C there is an f ∈ O(G) (i.e. f is a holomorphic function on G) that cannot be holomorphically extended beyond G. In many variables the situation becomes different. There are pairs of domains D1 $ D2 ⊂ C, n ≥ 2, such that the restriction mapping O(D2) −→ O(D1) is surjective. Domains which carry a non extendible holomorphic function are called domains of holomorphy. They can be characterized by being pseudoconvex, i.e. − log dist(·, ∂D) is a plurisubharmonic function. Moreover, in contrast to the case of real partial differentiability, a separately holomorphic (i.e. a partially complex differentiable) function f : D −→ C (write f ∈ Os(D)) — D ⊂ C, n ≥ 2, a domain — is already continuous (Theorem of Hartogs (1906)) and, therefore, using the Cauchy integral formula holomorphic on D. Recall that a function f : D −→ C is called separately holomorphic if for any a ∈ D and any j ∈ {1, . . . , n} the function of o...

Studia Mathematica, 2006

Let X be a Riemann domain over C k × C ℓ . If X is domain of holomorphy with respect to a family F ⊂ O(X), then there exists a pluripolar set P ⊂ C k such that every slice Xa of X with a / ∈ P is a domain of holomorphy with respect to the family {f | Xa : f ∈ F}.

Proceedings of the American Mathematical Society, 2010

We prove a new cross theorem for separately holomorphic functions. N j=1 A j × D j × A j. One may easily prove that X is connected. We say that a function f : X −→ C is separately holomorphic on X (we write f ∈ O s (X)) if for any j ∈ {1,. .. , N} and (a j , a j) ∈ A j × A j , the function D j z j −→ f (a j , z j , a j) ∈ C is holomorphic in D j .

Advances in Applied Clifford Algebras, 2010

The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions f : R 2n → Cn of the so-

Proceedings - Mathematical Sciences, 2018

Let D, D be arbitrary domains in C n and C N respectively, 1 < n ≤ N , both possibly unbounded and M ⊆ ∂ D, M ⊆ ∂ D be open pieces of the boundaries. Suppose that ∂ D is smooth real-analytic and minimal in an open neighborhood ofM and ∂ D is smooth real-algebraic and minimal in an open neighborhood ofM. Let f : D → D be a holomorphic mapping such that the cluster set cl f (M) does not intersect D. It is proved that if the cluster set cl f (p) of some point p ∈ M contains some point q ∈ M and the graph of f extends as an analytic set to a neighborhood of (p, q) ∈ C n × C N , then f extends as a holomorphic map to a dense subset of some neighborhood of p. If in addition, M = ∂ D, M = ∂ D and M is compact, then f extends holomorphically across an open dense subset of ∂ D.

Bulletin of the Brazilian Mathematical Society, New Series, 2007

We consider the problem of identifying boundary values of holomorphic functions on bounded domains in C 2 . We use the quaternionic analysis techniques to extending the CR structure to a pure function theoretical nature. The advantage of our procedure lies in the fact that it also runs for domains with fractal boundary.

Indiana University Mathematics Journal, 1995

It is shown, that any proper holomorphic map f : D → D between bounded domains D, D C 2 with smooth real-analytic boundaries extends holomorphically to a neighborhood ofD.