1999 Fiscal Year Final Research Report Summary
Variational Formula for Quasiconformal Mappings
| Project/Area Number |
10640173
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Yamaguchi University |
|---|
Principal Investigator |
YANAGIHARA Hiroshi Yamaguchi University, Faculty of Engineering, Associate Professor, 工学部, 教授 (30200538)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MASUMOTO Makoto Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (50173761)
KATO Takao Yamaguchi University, Faculty of Science, Professor, 理学部, 教授 (10016157)
KURIYAMA Ken Yamaguchi University, Faculty of Engineering, Professor, 工学部, 教授 (10116717)
GOUMA Tomomi Yamaguchi University, Faculty of Science, Research Associate, 理学部, 助手 (70253135)
KIUCHI Isao Yamaguchi University, Faculty of Science, Associate Professor, 理学部, 助教授 (30271076)
|
|---|
| Project Period (FY) |
1998 – 1999
|
| Keywords | singular integral / quasiconformal mapping / variational formula / Beltrami differential / Bloch functions / distortion estimate / growth estimate / global solutions |
|---|
| Research Abstract |
Let SィイD1pィエD1(C) = LィイD1pィエD1(C, (π(1 + |z|ィイD12ィエD1)ィイD12ィエD1)ィイD1-1ィエD1dxdy) be the space of p-th integrable functions with respect to the spherical metric on the Riemann sphere C. Let T be an variant of the Ahlfors-Beurling operator defined by
<<numerical formula>>.
Theorem 1 Let p 【not a member of】(2, ∞). Then for u 【not a member of】 SィイD1pィエD1(C), Tu(z) exists almost everywhere. Furthermore There exists CィイD2pィエD2 > 0 such that
<<numerical formula>>.
Theorem 2 Let p 【not a member of】 (2, ∞) Then for all μ 【not a member of】 LィイD1∞ィエD1(C) with ||μ||ィイD2∞ィエD2 < min{1/C(p), 1/3} μ-quasiconformal automorphism f of C can be expanded as follows,
<<numerical formula>>,
where the series converges absolutely in SィイD1pィエD1(C).
|
