Study of the extension of holomorphic functions from submanifolds of a pseudoconvex domain
Project/Area Number 
13640180

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  Nagasaki University 
Principal Investigator 
ADACHI Kenzou Nagasaki University, Faculty of Education, Professor, 教育学部, 教授 (70007764)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2001: ¥700,000 (Direct Cost: ¥700,000)

Keywords  Extension of holomorphic functions / Integral formula / Pseudoconvex domain / 部分多様体 
Research Abstract 
The purpose of the study is to extend holomorphic functions in a submanifold of a pseudoconvex domain D to a holomorphic function in D which belongs to some function spaces and to estimate solutions of the ∂ problem in D. I obtained the L^p extension of holomorphic functions in a submanifold of D to the entire domain D, when D is a strictly pseudoconvex domain in C^n with nonsmooth boundary. I studied in the following way. At first, I considered Koppelman's integral formula over ∂D for a holomorphic function in D when D is a strictly pseudoconvex domain in C^n with smooth boundary. Then by using Stokes' formula, Koppelman's formula is represented by the integral over D. Since a strictly pseudoconvex domain D with nonsmooth boundary is approximated by a sequence of strictly pseudoconvex domains with smooth boundary {Dm}, an L^p holomorphic function f in X = {z_n = 0}∩D can be represented by the limit of the integral over X_m = {z_n ― 0}∩D_m. Since the kernel of the integrals is holomorphic in D, f can be extended to a holomorphic function in D. In order to prove that the extended function is an L^p function, we used the method of Schmalz in which he estimated the volume form near the singular points. It seams to me that the same method is applicable to the H^p extension from submanifolds in a strictly pseudoconvex domain in C^n with nonsmooth boundary. I will continue the study of the H^p extension. On the other hand, I obtained optimal L^p estimates for ∂ problem in real ellipsoids by using Shaw's technique.

Report
(3 results)
Research Products
(20 results)