Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.0884

Extension theorems of Sakai type for separately holomorphic and meromorphic functions

Peter Pflug·Viet-Anh Nguyen
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Peter Pflug, Viet-Anh Nguyen. 2007-04-06. Extension theorems of Sakai type for separately holomorphic and meromorphic functions. https://arxiv.org/abs/0704.0884

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Reciprocal Problem on Weighted Bergman Spaces

We study the reciprocal problem for weighted Bergman spaces: if $f\in A_α^p(\Bn)$ and $\inf_{\Bn}|f|>0$, does it follow that $1/f\in A_α^p(\Bn)$? We determine the range of parameters for which the answer is affirmative. In particular, we prove that functions in $A_α^p(\D)$ have the reciprocal property for all $α\in \R$ and $p\geq 1,$ where $\D$ denotes the unit disk in $\C.$ Moreover, functions in $A_α^p(\Bn)$ have the reciprocal property for all $α\in \R$ when $n=2$ and $p=2.$ In addition, we resolve the reciprocal problem in the three-dimensional Drury--Arveson space $H_3^2$ and obtain an equivalent condition in the four-dimensional space $H_4^2$. We also settle the reciprocal problem for general $A_α^p(\Bn)$ under some additional conditions.

math.CV↗
arXiv2026

Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure

We give affirmative answers to both parts of Question 2.6 posed by Grosse-Erdmann (2025) concerning Wiman--Valiron inequalities in the unit disk. For every unbounded analytic function in the disk, we establish the proposed iterated-logarithm inequalities outside exceptional sets of finite logarithmic measure. The corresponding power estimate is a corollary. Both conclusions follow from a variance bound for Khinchin families and a classical estimate for their largest atom. The multiplicative constants in the main inequalities can be chosen absolute. We also obtain a disk analogue of Rosenbloom's composition estimate, with an explicit boundary prefactor. The key step combines a boundary change of variable with a monotone auxiliary function whose derivative is exactly the variance of a rescaled member of the Khinchin family. A classical example shows that the leading logarithmic exponent $1/2$ cannot be decreased.

math.CV↗
arXiv2026

Minimal m-subharmonic functions with nonmaximal Hessian measures

For every $2\le m\le n$, we construct Hölder continuous functions on the closed unit ball that are minimal in the Cegrell class $\F_m$, although their Hessian measures are not maximal in the $m$-subharmonic ordering. This answers a question posed by the second author. We also obtain a minimality criterion based on partial pluricomplex energy and an explicit family of Monge--Ampère examples on a product domain. For $m=1$, minimality and maximality are equivalent on a ball.

math.CV↗