2003 Fiscal Year Final Research Report Summary
Study of Smoothing Effects and Related Problems
| Project/Area Number |
13640187
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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, Professor, 大学院・理学研究科, 教授 (40211544)
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| Co-Investigator(Kenkyū-buntansha) |
UMEDA Tomio Himeji Institute of Technology, Faculty of Science, Professor, 大学院・理学研究科, 教授 (20160319)
IWASAKI Chisato Himeji Institute of Technology, Faculty of Science, Professor, 大学院・理学研究科, 教授 (30028261)
AKAHORI Tatsuo Himeji Institute of Technology, Faculty of Science, Professor, 大学院・理学研究科, 教授 (40117560)
SUGIMOTO Mitsuru Osaka University, Graduate School of Science, Assistant Professor, 大学院・理学研究科, 助教授 (60196756)
HIRANO Katsuo Himeji Institute of Technology, Faculty of Science, Lecturer, 大学院・理学研究科, 講師 (90316034)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Partial Differential Equations / Harmonic Analysis / Spectral Analysis / Smoothing Effect / Restriction Theorem / Limiting Absorption Principle |
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| Research Abstract |
In this project, we are concerned with smoothing properties of solutions to dispersive equations and related topics. Our approach is besed on not only methods of real analysis but also methods of the other areas, especially limiting absorption principle in 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.
Also we investigated our problems as pure mathematical ones, especially we considered what conditions are necessary for the smoothing effects. Thus our project can be regarded as the first step of research, which is analogous to the sophisticated research of Cauchy problems in 60's and 70's (which is initiated by the work of J.Hadamard in 20's, but the development of miclolocal analysis enabled results like Lax-Mizohata theorem).
In short, our results are as follows:
(1)We investigated the smoothing effect of dispersive initial value problems in the case of constant coefficients. We proved the smoothing effect occurs in the case where the principal symbol is not elliptic.
(2)We considered the necessary condition of the smoothing effect. We proved the condition for the gradient of the principal symbol is necessary for the maximal smoothing effect.
(3)We investigated the smoothing effects in Besov type function spaces.
(4)We introduced the approach using the boundednes property of Fourier integral operators in weighted L^2 Such approach will enable us to reduce the operator to prototype ones.
(5)We proved the unconditional convergence of wavelet expansion in L^p space, even the mother wavelet is weakly localized as Shannon's wavelet.
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