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域研究。[2)解的构造和运算演算，解空间的研究。[3]利用非局部伪微分理论对微分方程组进行代数分析研究。首先，对[1]，我们将局部伪微分算子的一般理论推广到纤维丛的派生范畴中的傅里叶-佐藤变换在基空间中有运动的情形。将它应用于复流形上余切丛上的全纯微分形式，我们得到了非局部伪微分算子的复数，并利用它的上同调，定义了非局部伪微分算子簇。因此，我们得到了两个这样的算子的组合，考虑到上面的函数。其次，我们建立了在某一点上对全纯函数的柄进行运算的伪微分算子的概念，即具有符号使得其整函数的阶在所考虑的点上为0的算子.对于[2]和[3]，我们将对[2]和[3]进行研究.此外，我们还刻画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级微分算子.

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.