| Budget Amount *help |
¥3,470,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
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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
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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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