Algebraic study of non-local differential equations and operational calculus

本文的主要研究内容如下：[1]非局部伪微分算子层的微局部研究。[2]构造解和运算微积分，研究解空间。[3]利用非局部伪微分方程理论研究微分-差分方程的代数分析，首先对文[1]推广了局部伪微分算子的一般理论，将纤维丛上导出范畴中的Fourier-Sato变换推广到基空间中有移动的情形。将其应用于复流形上余切丛上的全纯微分形式，得到了非局部伪微分算子复形，并取其上同调，定义了非局部伪微分算子层.因此，我们得到了两个这样的算子的复合，考虑函式如上所述。其次，我们建立了在一点上作用于全纯函数的茎的伪微分算子的概念，即一个算子的符号使得它作为一个整函数的阶随着所考虑的点而趋于0。对于[2]，[3]，我们将进行研究。进一步地，本文刻划了作用于具有给定近似阶和半范数的整函数空间的连续线性算子为无穷阶微分算子。

The aims of this research were as follows:[1] Microlocal study of sheaves of non-local pseudo-differential operators.[2] Construction of solutions and operational calculus, study of the solution space.[3] Algeblaic analytical study of differential-difference equations by means of the theory of non-local pseudo-differential equations.First for [1], we generalized the general theory of local pseudo-differential operators generalizing the Fourier-Sato transformation in the derived category on fiber bundle to the case having a move in the base space. Applying it to holomorphic differential forms on the cotangent bundle over a complex manifold, we obtained the complex of non-local pseudo-differential operators and taking its cohomology, we defined the sheaf of non-local pseudo-differential operators. We thus obtained composition of two such operators, considering functorially as above. Next, we established the notion of pseudo-differential operators operating to the stalk of holomorphic functions at a point, that is, an operator with the symbol such that its order as an entire function goes to 0 with the considered point.For [2], and [3], we are going to investigate.Furtheremore, we characterized continuous linear operator acting to the space of entire functions with given proximate ordre and a semi-norm as differential operator of infinite order.