2004 Fiscal Year Final Research Report Summary
Harmonic analysis in a domain with fractal boundary
| Project/Area Number |
14540157
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Basic analysis
|
|---|
| Research Institution | Ochanomizu University |
|---|
Principal Investigator |
WATANABE Hisako Ochanomizu University, Faculty of Science, Professor, 理学部, 教授 (70017193)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TAKEO Fukiko Ochanomizu University, Faculty of Science, Professor, 理学部, 教授 (40109228)
YOSHIDA Hidenobu Chiba Univ., Graduate School of Natural Sciences, Professor, 大学院・自然科学研究科, 教授 (60009280)
|
|---|
| Project Period (FY) |
2002 – 2004
|
| Keywords | fractal lateral boundary / double layer heat potentials / Whitney decomposition / Besov spaces / Besov norms / maximal functions / uniform domains / boundedness of operators |
|---|
| Research Abstract |
We consider the initial-boundary-value problems in a domain Ω=D×[O,T] with fractal lateral boundary S. It often occurs that an operator K on the Besov space B on the lateral boundary is bounded with respect to the Besov norms on S. We can prove the boundedness of an operator from B to B in the following method.
(1)We extend a function f defined on S to R×[O,T] by using an extension operator E.
(2)The Besov norm of f is estimated by the integral of |▽f(x)|×δ(x)^n over Ω, where δ(x) is the distance from x to S and n is a suitable number.
(3)Instead of the boundedness of K we prove the boundedness of an operator from a function space on Ω to a function space on the outside of Ω by using the maximal functions between Ω and the outside of Ω.
We proved the boundedness of an operator K, which is important to solve the Dirichlet problem by using double layer heat potentials.
|
