| Co-Investigator(Kenkyū-buntansha) |
平井 武 京都大学, 理学部, 教授 (70025310)
楠 幸夫 京都大学, 理学部, 教授 (90025221)
池部 晃生 京都大学, 理学部, 教授 (00025280)
大鍛治 隆司 京都大学, 理学部, 助手 (20160426)
松本 和一郎 京都大学, 理学部, 助手 (40093314)
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| Budget Amount *help |
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1989: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1988: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Research Abstract |
In this article we define microlocal orders of singularities for distributions and state theorems concerning their important properties. By microlocal orders of singularities we mean not only positions but also directions of singularities. As an application we can clarify how the order of singularity changes by operating Fourier integral operator. For small time t the solution to hyperbolic pseudo- differential equation of order one is described by P_<phi>u , where u is the initial data , phi(x,eta) the phase function and p(x,eta) the amplitude function. Then it holds OS(P_<phi>u;x, xi ) = OS( u; y, eta ) Here OS(u;y, eta ) stands for the order of singularity for u at y in the direction (eta)/(ta1). The mapping (y, eta ) -> (x, xi) is given by the canonical relation generated by phi(x,eta) i.e. y= phi_<eta>(x,eta) and xi= phi_x(x,eta)
Finally we explain OS(u,x_o,xi_o). For f * epsilon' we note OS(f) = inf{r*R, f * H^<-r>} and define OS(f; x_o,xi_o) = inf OS(a(D,X)f) a * S^0, (x_o,xi_o) * (supp a) where (x_o,xi_o) * (supp a) means that a does not vanish in the xi_o - direction. Then it holds the following theorem of Darboux type: For given positive number epsilon there exists delta>O such that OS(f;x_o,xi_o) <less than or equal> OS(s(D,X)f) <less than or equal> OS(f;x_o,xi_o) + epsilon holds for any a * S^0 whose support is contained in delta conic neighborhood of (x_o,xi_o). This inequality plays an important role to prove the first equality.
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