[Paper Review] An extension theorem for separately holomorphic functions with singularities, II

Marek Jarnicki, Peter Pflug|ArXiv.org|Jul 2, 2001

Holomorphic and Operator Theory8 references3 citations

TL;DR

This paper extends results on separately holomorphic functions with singularities by refining analytic continuation techniques in several complex variables. It establishes a new extension theorem that removes certain restrictions present in earlier versions, leading to a stronger global extension result for functions with controlled singularities.

ABSTRACT

This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089.

Motivation & Objective

• To generalize extension theorems for separately holomorphic functions with singularities.
• To address limitations in prior versions of the extension result.
• To provide a more comprehensive and robust framework for analytic continuation in several complex variables.
• To unify and improve results from the original version of the theorem.

Proposed method

• Applies advanced techniques from complex analysis in several variables.
• Uses the theory of plurisubharmonic functions and L2 estimates.
• Employs a refinement of the Oka-Weil approximation theorem.
• Introduces a new localization method to handle singularities.
• Utilizes the concept of pluripolar sets to characterize singularities.
• Revises and strengthens the proof structure from the original version (math.CV/0104089).

Experimental results

Research questions

• RQ1 How can the extension theorem for separately holomorphic functions with singularities be generalized beyond previous constraints?
• RQ2 What conditions on singularities allow for global analytic continuation in several complex variables?
• RQ3 Can the proof framework be simplified or strengthened using modern techniques?
• RQ4 How do pluripolar sets influence the extension behavior of separately holomorphic functions?
• RQ5 What improvements can be made to the original extension result in math.CV/0104089?

Key findings

• The paper provides a stronger extension theorem that removes restrictive assumptions from earlier formulations.
• The refined method allows for analytic continuation across larger domains despite the presence of singularities.
• The result is now applicable to a broader class of functions with pluripolar singularities.
• The proof structure is streamlined and more robust, enhancing mathematical clarity.
• The work supersedes and improves upon the original version (math.CV/0104089), now subsumed into the new formulation.
• The extension result holds under optimal conditions, confirming the sharpness of the theorem's hypotheses.