Analytic Functions of Several Variables on Bounded Domains
Project/Area Number  01540163 
Research Category 
GrantinAid for General Scientific Research (C)

Allocation Type  Singleyear Grants 
Research Field 
解析学

Research Institution  Osaka Medical College 
Principal Investigator 
NISHIMURA Yasuchiro Osaka Medical College, Department of Mathematics Lecturer, 教養部, 講師 (90156117)

CoInvestigator(Kenkyūbuntansha) 
YASUDA Reiko Osaka Medical College, Department of Mathematics Assistant Professor, 教養部, 助教授 (90084855)

Project Period (FY) 
1989 – 1990

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1990 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1989 : ¥300,000 (Direct Cost : ¥300,000)

Keywords  Uniform estimate for THETAequation / THETAclosed extension / Corona problem / Invariant Poison kernel / Poisson Sego kernel / Automorphisms of C^n / 〓ー方程式に対する一様評価 / 〓ー閉拡張 / ポアッソンーセゲー核 / 〓方程式 / 積分公式(Henkin,Skodaの) / 有界正則関数 
Research Abstract 
1. We examined precisely the method of Wolff which gives the uniform estimate of the solutions of THETAequations in the unit disk. We tried to generalize this method to the THETAequation THETAu=f in the 2 dimensional unit ball. (1) Several spaces of functions or differential forms in the unit ball B or on the unit sphere S are introduced. Specially, H^1_, _1 (S) on S and H^1_, _1 (B) in B are important. (2) The problem of the estimation of the solutions u of THETAequation THETA=f is reduced to an estimate of some integral over S of the differential forms in H^1_, _1 (S). (3) A THETAclosed extension operator E : H^1_, _1 (S)>H^1_, _1 (B) is constructed. An operator of the same kind was formerly given by Henkin and Skoda in 1970's. But ours is different from theirs. The method of construction is also different from their method. Our operator has the advantage of giving a THETAclosed (2, 1) form whose tangential part is Mharmonic. We will publish this result after we examine whether our method is also applicable to the general n dimensional case. (4) An analogue of the Green's formula is given. This formula suggests that what we need to estimate is the differential in the normal direction of the wedge product of the THETAdata f and forms PHI in H^1_, _1 (B). 2. We investigated the group AX of holomorphic automorphisms of C^2 which preserve the coordinate axes. (1) Subgroups AX_K (O<K<*) of AX are defined. A condition for an automorphism T in AX to belong to AX_K is given in terms of its Jacobian JT. (2) A subgroup AP of AX is defined and a generator system of AP in the sense of uniform convergence is decided.

Report
(3results)
Research Output
(3results)