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<epsilon>{exp [X] u<xi>（Xi）; u* S '}的（逆）Fourier变换S'，并将超函数空间视为 * <epsilon>0&gt; S '的局部空间<epsilon>。这里S表示Schwartz空间。进一步建立了伪微分算子的（经典）解析理论和超函数的微局部分析，将S'中的伪微分算子和Fourier积分算子的微积分推广到S'_中的微积分<epsilon>。在偏微分算子（和伪微分算子）的研究中，我们可以将相同的论点应用于S '_<epsilon>，特别是超函数空间，就像在分布范畴中使用的那样。我们使研究成为可能，与统一的治疗，偏微分算子在空间的分布，超分布（和Gevrey），和超函数（和解析函数）。例如，我们处理的问题，推导出先验（能量）估计。特别地，我们从先验估计中得到了解析拟微分算子的解析奇点和解析亚椭圆性的传播结果。我们还证明了超函数范畴中亚椭圆性与局部可解性的关系与分布范畴中亚椭圆性与局部可解性的关系相同。从先验估计的推导出发，研究了偏微分算子的先验估计，得到了超函数空间中局部可解性的几个结果。并且我们得到了各种问题的先验估计。本课题的研究人员对相关问题进行了研究。我们相信所得结果对偏微分算子的研究有重要的意义。

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.