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Studies on differential equations by microlocal analysis

微分方程的微局部分析研究

基本信息

批准号：
08454023
负责人：
WAKABAYASHI Seiichiro
金额：
$ 5.06万
依托单位：
University of Tsukuba
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 至无数据
项目状态：
已结题

项目摘要

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.

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一些基本结果。在此过程中，我们考虑{exp [<xi>] u （xi）的（反）傅里叶变换S'_<epsilon>；u*S'}，并将超函数空间视为*_<epsilon>0 00 >S'_<epsilon>的局部空间。这里S表示Schwartz空间。建立了伪微分算子的解析理论和超函数的微局部分析，将S‘中的伪微分算子和傅里叶积分算子的演算推广到S’_<epsilon>中的演算。在偏微分算子（和伪微分算子）的研究中，我们可以将同样的论点应用到S'_<epsilon>，特别是超函数空间，就像在分布范畴中使用的那样。我们使得用统一的处理方法来研究分布、超分布（和Gevrey）和超函数（和解析函数）空间中的偏微分算子成为可能。例如，我们处理了这些问题，推导了先验（能量）估计。特别地，我们从先验估计中得到了解析伪微分算子的解析奇异性和解析亚椭圆性的传播结果。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关系相同。并从先验估计中得到了超函数空间的局部可解性的几个结果。我们从先验估计的推导角度研究了偏微分算子。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。我们认为所得结果对偏微分算子的研究具有重要的指导意义。