Algebraic study of non-local differential equations and operational calculus

本研究的目的如下：[1]非局部伪微分算子的微局部研究解的构造与运算演算，解空间的研究用非局部伪微分方程理论研究微分-差分方程的代数解析。首先，对于[1]，我们推广了局部伪微分算子的一般理论，将光纤束上导出范畴中的傅里叶-佐藤变换推广到基空间中有移动的情况。将其应用于复流形上的余切束上的全纯微分形式，得到了非局部伪微分算子的复，并取其上同调，定义了非局部伪微分算子群。因此，我们得到了两个这样的算子的组合，从功能上考虑如上所述。其次，我们建立了作用于全纯函数在一点上的柄的伪微分算子的概念，即具有符号使得其作为整个函数的阶在所考虑的点上趋于0的算子。对于[2]和[3]，我们要研究一下。进一步，我们将作用于给定近似阶和半范数的整个函数空间的连续线性算子刻画为无限阶的微分算子。

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.