2002 Fiscal Year Final Research Report Summary
Study of the extension of holomorphic functions from submanifolds of a pseudoconvex domain
| Project/Area Number |
13640180
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year 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
|
| Keywords | Extension of holomorphic functions / Integral formula / Pseudoconvex domain |
|---|
| 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 non-smooth 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 non-smooth 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 non-smooth 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.
|
