Microlocal analysis for operators with infinite degeneracy

利用伪微分算子理论和傅立叶积分算子理论，在微局部研究了允许无限阶退化的C^*系数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解析方法。主要研究者证明了一类称为“Egorov型”的一阶伪微分算子是半椭圆型的，尽管我们只得到了这类算子损失两个导数的先验估计。此外，首席研究员还研究了薛定谔型方程初值问题的微局部解析平滑效应，通过与Robbiano-Zuily的合作，在初始数据属于2阶Gevrey类的情况下发现了一定的平滑效应。作者研究了二阶双曲型方程Cauchy问题解的惠更斯原理，并研究了一些Euler-Poisson-Darboux-Stellmacher型方程基本解的Hadamard展开的对数项所定义的规范不变张量场的性质。研究者Asano利用能量估计研究了玻尔兹曼方程混合问题解的整体存在性。研究者Ushiki研究了复动力系统的分布理论与遍历理论之间的联系，并将动态zeta函数看作Ruelle算子复化的Fredholm行列式，成功地确定了Ruelle算子的特征函数。研究者Hata将鞍法推广到复二维情况，并在其应用中，对对数的非二次测度有了显著的改进。关于偏微分方程解的结构，研究者Kigami证明了如果巢形分形所对应的无限自相似格是不对称的，则其拉普拉斯算子的紧支持本征函数构成L^2中的标准正交基，因此拉普拉斯谱是纯点。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