1990 Fiscal Year Final Research Report Summary
Analytic Functions of Several Variables on Bounded Domains
| Project/Area Number |
01540163
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Osaka Medical College |
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Principal Investigator |
NISHIMURA Yasuchiro Osaka Medical College, Department of Mathematics Lecturer, 教養部, 講師 (90156117)
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| Co-Investigator(Kenkyū-buntansha) |
YASUDA Reiko Osaka Medical College, Department of Mathematics Assistant Professor, 教養部, 助教授 (90084855)
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| Project Period (FY) |
1989 – 1990
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| Keywords | Uniform estimate for THETA-equation / THETA-closed extension / Corona problem / Invariant Poison kernel / Poisson Sego kernel / Automorphisms of C^n |
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| Research Abstract |
1. We examined precisely the method of Wolff which gives the uniform estimate of the solutions of THETA-equations in the unit disk. We tried to generalize this method to the THETA-equation 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 THETA-equation THETA=f is reduced to an estimate of some integral over S of the differential forms in H^1_, _1 (S).
(3) A THETA-closed 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 THETA-closed (2, 1) form whose tangential part is M-harmonic. 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 THETA-data 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.
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