Microlocal analysis for operators with infinite degeneracy

利用拟微分算子理论和傅立叶积分算子理论，在微观局部上研究了具有C^*系数且允许无穷阶退化的偏微分方程解的存在性和结构，并从实分析和随机理论的角度考虑了相关的分析方法.虽然我们只得到了一个损失两个导数的先验估计，但研究人员证明了一类一阶伪微分算子，即“Egorov型”算子，通过一些“对数”正则性上估计是亚椭圆的。在此基础上，研究了Schrodinger型方程初值问题的微域解析光滑化效应，改进了前人的结果，发现在初值属于二阶Gevrey类的情况下，与Robbiano-Zuly共同工作有一定的光滑化效应。调查员…西和田进一步研究了二阶双曲型方程柯西问题解的Huyghens原理，并研究了一些Euler-Poisson-Darboux-Stellmacher型方程的规范不变张量场的性质。利用能量估计，Asano研究了Boltzman方程混合问题解的整体存在性。Ushiki研究了关于复杂动力系统的分布理论和遍历理论之间的联系，并成功地通过将动态Zeta函数作为复杂化的Ruelle算子的Fredholm行列式来描述Ruelle算子的本征函数。研究人员Hata将鞍形方法推广到复杂的二维情形，作为其应用，他在对数的非二次度量方面得到了显著的改进。关于偏微分方程解的结构，Kigami证明了如果对应于嵌套分形的无穷大自相似格子不对称，则对其拉普拉斯算子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。较少

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