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Microlocal analysis for operators with infinite degeneracy

无限简并算子的微局域分析

基本信息

批准号：
08454027
负责人：
MORIMOTO Yoshinori
金额：
$ 4.67万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1998
项目状态：
已结题

项目摘要

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 … More 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

利用拟微分算子理论和Fourier积分算子理论研究了具有无穷阶退化的C^* 系数偏微分方程解的存在性和结构，并从真实的分析和随机理论的角度考虑了相关的分析方法.本文的主要研究者证明了一类称为“Egorov型”的一阶伪微分算子是亚椭圆的，虽然我们只得到了这类算子损失两个导数的先验估计，但我们用一些“对数”正则性上估计证明了这类算子是亚椭圆的.此外，作为与Robbiano-Zuily的合作，研究了Schrodinger型方程初值问题的微局部解析光滑效应，改进了他们的已有结果，发现当初始数据属于二阶Gevrey类时，微局部解析光滑效应也存在.研究者 ...更多信息西和田研究了二阶双曲方程柯西问题解的惠更斯原理，并研究了某些Euler-Poisson-Darboux-Stellmacher型方程由基本解的Hadamard展开式的对数项定义的规范不变张量场的性质。利用能量估计，研究者Asano研究了Boltzman方程混合问题解的整体存在性。研究者Ushiki研究了复杂动力系统的分布理论与遍历理论之间的联系，并成功地将动力zeta函数作为Ruelle算子的复化Fredholm行列式来确定Ruelle算子的本征函数。研究者Hata将鞍点方法推广到复杂的二维情形，并作为其应用，得到了对数的非二次性测度的显著改进。关于偏微分方程解的结构，研究者木上浩一指出，如果对应于嵌套分形的无限自相似格不对称，则其拉普拉斯算子具有紧支集的本征函数构成L^2中的正交基，因此拉普拉斯谱是纯点。少