Studies on differential equations by microlocal analysis

首先，我们从(经典)分析的角度研究了超函数理论，并从这个角度证明了基本结果。在此过程中，我们考虑了{exp[exp]u(Xi)；u*S‘}的(逆)傅立叶变换S；这里，S指的是施瓦茨空间。进一步，我们建立了拟微分算子的(经典)分析理论和超函数的微局部分析，将S的拟微分算子和傅立叶积分算子的微积分推广到S的微积分中。在对偏微分算子(和伪微分算子)的研究中，我们可以将同样的论点应用于S的定理，特别是超函数空间，就像在分布范畴中所使用的那样。我们用统一的处理方法研究了分布空间、超分布空间(和Gevrey空间)、超函数空间(和解析函数空间)中的偏微分算子。例如，我们处理这些问题，得出一个先验(能量)估计。特别地，我们利用先验估计得到了关于解析拟微分算子的解析奇点传播和解析亚椭圆性的结果。我们还证明了超函数范畴中的亚椭圆性与局部可解性之间的关系与分布范畴中的相同。并利用先验估计得到了超函数空间中局部可解性的几个结果。我们从先验估计导出的角度研究了偏微分算子。并得到了各种问题的先验估计。本项目的研究人员对相关问题进行了研究。我们相信，所得到的结果对偏微分算子的研究有很大的帮助。

First, we investigated the theory of hyperfunctions from a viewpoint of (classical) analysis, and proved fundamental results from this viewpoint. In doing so, we considered the (inverse) Fourier transform S'_<epsilon> of {exp [epsilon <xi>] u (xi) ; u*S'}, and regarded the space of hyperfunctions as the local space of *_<epsilon>0>S'_<epsilon>. Here S denotes the Schwartz space. Furthermore, we established the (classical) analytical theory of pseudodifferential operators and microlocal analysis for hyper-functions, generalizing calculus of pseudodifferential operators and Fourier integral operators in S' to calculus in S'_<epsilon>. In the studies of partial differential operators (and pseudodifferential operators), we could apply the same arguments to S'_<epsilon>, especially the space of hyperfunctions, as used in the category of distributions. And we made it possible to investigate, with unified treatments, partial differential operators in the spaces of distributions, ultradistributions (and Gevrey), and hyperfunctions (and analytic functions). For example, we treated the problems, deriving a priori (energy) estimates. In particular, we obtained results on propagation of analytic singularities and analytic hypoellipticity for analytic pseudodifferential operators from a priori estimates. We also proved that the relation between hypoellipticity and local solvability in the category of hyperfunctions is the same as in the category of distributions. And we obtained several results on local solvability in the space of hyperfunctions from a priori estimates.We studied partial differential operators from a viewpoint of derivation of a priori estimates. And we obtained a priori estimates for various problems. The related problems were studied by the investigators of this project. We believe that the results obtained here are of great use for the studies on partial differential operators.