2007 Fiscal Year Final Research Report Summary
Real principal-type pseudodifferential operators and dispersive pseudodifferential equations
| Project/Area Number |
17540140
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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 | Tohoku University |
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Principal Investigator |
CHIHARA Hiroyuki Tohoku University, Tohoku University, Mathematical Institute, Associate Professor (70273068)
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| Co-Investigator(Kenkyū-buntansha) |
FUJIIE Setsuro University of Hyogo, Graduate School of Material Science, Associate Professor (00238536)
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| Project Period (FY) |
2005 – 2007
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| Keywords | dispersive partial differential equation / pseudodifferential operator / geometric analysis / Bargmann transformation / Toeplitz operator / Schodinger map |
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| Research Abstract |
We here summarize the research results of the head investigator. At the beginning of this project, he studied linear and nonlinear dispersive partial differential equations. His results are the characterization of well-posedness of the initial value problem for a class of third order linear dispersive PDEs on the torus, time-local existence and the infinite gain of regularity properties of the initial value problem for some nonlinear dispersive PDEs which cannot be handled by the classical energy method, and the resolvent estimates of elliptic Fourier multipliers and their applications to the analysis of time-global smoothing estimates of solutions to corresponding dispersive pseudodifferential equations.
In 2006 academic year, to challenge new research fields, he devoted himself to studying microlocal analysis via the Bargmann transformation, and the harmonic analysis on symmetric spaces. This was a turning point of his research. In 2007 academic year, he studied the initial value prob
… More
lem for Schrodinger maps of a perturbed Euclidean space to a compact hermitian manifold. If the target manifold is not a Kahler manifold, the classical energy method never works. To overcome this difficulty, he introduced pseudodifferential operators acting on sections of the pull-back bundle, and modify the equation of the map to be solvable by the pseudodifferential operators. This result is in preparation for submitting. This research gave him the elementary techniques of complex analysis and complex geometry, and at the end of this project, he began the analysis of Toeplitz operators on the complex Euclidean space by using the general Bargmann transformation and pseudodifferential calculus. He gave the complete generalization of the criterion of the boundedness of Toeplitz operators, which is known in the case of the usual Segal-Bargmann space obtained by Berger and Coburn. He continues working on the deformation estimates by making full use of pseudodifferential calculus.
Finally, graduate students E. Onodera and K. Kaizuka, and the research student R. Mizuhara, who are supervised by the head investigator, got some interesting results on geometric analysis of one-dimensional third-order dispersive flows into almost hermitian manifolds, harmonic and microlocal analysis of regularity of solutions to a class of dispersive PDEs on symmetric spaces of noncompact-type, and microlocal smoothing estimates of Schrodinger evolution equations in a Gevrey class respectively. Less
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Research Products
(17 results)
