2005 Fiscal Year Final Research Report Summary
Studies on holomorphic mappings on the unit ball in an infinite dimensional space
| Project/Area Number |
15540193
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Hiroshima Institute of Technology (2005)
Ariake National College of Technology (2003-2004) |
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Principal Investigator |
HONDA Tatsuhiro Hiroshima Institute of Technology, Faculty of Engineering, Assistant Professor, 工学部, 助教授 (20241226)
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| Project Period (FY) |
2003 – 2005
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| Keywords | convex mapping / Loewner chain / parametric representation / spirallike mapping / starlike mapping / transition mapping / univalent holomorphic mapping |
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| Research Abstract |
(1)We consider some equalities of the Hamack type and its applications for holomorphic mappings on some infinite dimensional domain.
(2)Let E be a complex Banach space with an unconditional Schauder bass. Let D be a pseudoconvex domain in E and let V be a closed complex submanifold in E. We assume that the dimension of V is finite or the coon of V for E is finite. We denote by O the sheaf of germs of all holomorphic fuctions on D. Then we show that H^P(D＼V,O)=O for 1 □ p<codim_E V -1. Especially, if the dimension of V is finite and the dimension of E is infinite, then H^P(D＼V,O)=0 for p≧1 and D＼V is not pseudoconvex.
By using this insult, we show that H^P(P(E),O) = 0 for 1 □ p<dim E -1, where P(E) is the complex projective space induced from E.
(3)Let B be the unit ball in C^n with respect to an arbitrary norm and let f(z,t) be a g-Loewner chain such that e^<-t>f(z,t)-z has a zero of order k+1 at z=0. We obtain growth and covering theorems for f(・,0).
Moreover, we consider coefficient bounds and examples of mappings in S_<g,k+1>^0(B).
(4)Let B be the unit ball of a complex Banach space with respect to the noem. We obtain growth and covering theorems for some holomorphic mapping with parametric representaion, and consider various examples.
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