| Project/Area Number |
10640182
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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 | Himeji Institute of Technology |
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Principal Investigator |
HOSHIRO Toshihiko Himeji Institute of Technology, Faculty of Science, Assistant Professor, 理学部, 助教授 (40211544)
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| Co-Investigator(Kenkyū-buntansha) |
USA(FUJIWARA) Takeshi Himeji Institute of Technology, Faculty of Science, Assistant Professor, 理学部, 助教授 (10202293)
UMEDA Tomio Himeji Institute of Technology, Faculty of Science, Professor, 理学部, 教授 (20160319)
IWASAKI Chisato Himeji Institute of Technology, Faculty of Science, Professor, 理学部, 教授 (30028261)
SUGIMOTO Mitsuru Osaka University, Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (60196756)
HIRANO Katsuhiro Himeji Institute of Technology, Faculty of Science, Lecturer, 理学部, 講師 (90316034)
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| Project Period (FY) |
1998 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2000: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1999: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1998: ¥1,500,000 (Direct Cost: ¥1,500,000)
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| Keywords | Partial Different Equations / Harmonic Analysis / Spectral Analysis / Smoothing Effect / Restriction Theorem / Limiting Absorption Principle / 初期値問題 |
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| Research Abstract |
In this project, we are mainly concerned with smoothing properties of solutions to dispersive equations. We used not only methods of real analysis but also methods of the other areas, especially spectral theory. Our subjects are smoothing effect of dispersive initial value problem, the Fourier restriction theorem of harmonic analysis, and the limiting absorption principle of spectral theory. All of these can be regarded as bounded properties of Fourier multipliers with singular symbols in some function spaces. So a method of one area might be useful to obtain new results of the other areas. This is the idea of how to make progress on this project. In short, our results are as follows :
(1) We applied Mourre's method to obtain global smoothing properties of dispersive equations. We discovered that the smoothing effect occurs quite generally, even in the case of variable coefficients.
(2) We proved new estimates of Schrodinger initial value problem, by using classical results of Bessel functions. We also proved some smoothing properties in spherical directions.
(3) We discovered an interesting phenomenon of fundamental solutions to wave equations with non-convex characteristics in the case of space dimension 3.
(4) We proved some results on the action of √<-Δ> in weighted Sobolev spaces.We also proved the eigenfunction expansion theorem of the relativistic Schrodinger operators.
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