1998 Fiscal Year Final Research Report Summary
Microlocal analysis for operators with infinite degeneracy
| Project/Area Number |
08454027
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
MORIMOTO Yoshinori Kyoto Univ., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (30115646)
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| Co-Investigator(Kenkyū-buntansha) |
KIGAMI Jun Kyoto Univ., Graduate School of Human and Environmental Studies, Ass.Professor, 大学院・人間・環境学研究科, 助教授 (90202035)
HATA Masayoshi Kyoto Univ., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (40156336)
USHIKI Shigehiro Kyoto Univ., Graduate School of Human and Environmental Studies, Professor, 大学院・人間・環境学研究科, 教授 (10093197)
ASANO Kiyoshi Kyoto Univ., Graduate School of Human and Environmental Studies, Professor, 大学院・人間・環境学研究科, 教授 (90026774)
NISHIWADA Kimimasa Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (60093291)
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| Project Period (FY) |
1996 – 1998
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| Keywords | Hypoellipticity / Egorov type / smoothing effect / Huyghens principle / Gevrey order / Ruelle operator / spectre / saddle method |
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| Research Abstract |
The existence and the structure of solutions to partial differential equations with C^* coefficients, admitting the degeneracy of infinite order, were studied microlocally by using the theory of pseudodifferential operators and Fourier integral operators, and the related analytical methods were considered from the viewpoint of the real analysis and the stochastic theory. The head investigator showed that some class of first order pseudodifferential operators which is called "Egorov type" are hypoelliptic by means of some "logarithmic" regularity up estimates though we get only a priori estimate with loss of two derivatives for such operators. Furthermore, the microlocal analytic smoothing effect for the initial value problem of Schrodinger type equations was researched by the head investigator, who found some smoothing effect in the case where the initial data belongs to Gevrey class of order 2, as a joint work with Robbiano-Zuily, by improving their preceding results. The investigator
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Nishiwada studied the Huyghens principle for solutions to the Cauchy problem of second order hyperbolic equations, and investigated the property of gauge invariant tensor fields, defined from the logarithmic term of Hadamard expansion of fundamental solution, for some Euler-Poisson-Darboux-Stellmacher type equations. By using the energy estimate, the global existence of solutions to the mixed problem for Boltzman equations was researched by the investigator Asano. The investigator Ushiki studied the connection between the distribution theory and the ergodic theory concerning the complex dynamical systems, and suceeded in specifying the eigenfunctions of Ruelle operator by regarding the dynamical zeta function as a Fredholm determinant of Ruelle operator complexified. The investigator Hata extended the saddle method into the complex two dimension case, and, as its application, he obtained the remarkable improvement about the non-quadraticity measure of the logarithm. Concerning the structure of solutions to partial differential equations, the investigator Kigami showed that if the infinite self-similar lattice corresponding to Nested fractals is not symmetric then the eigenfunctions with compact support for its Laplace operator compose the orthonormal basis in L^2, and hence the Laplacian spectres are pure points. Less
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