ALGEBRAIC ANALYTICAL STUDY OF SHEAVES AND INFINITE ORDRE DIFFERENTIAL EQUATIONS

[1]将研究组长关于复域上微微分方程Cauchy问题的定理推广到伪微分方程系统。[2]用无穷阶微分算子刻划全纯函数层的自同构，而不要求连续性。[3]为了将复域上单卷积方程解的存在性和解析延拓性的结果推广到复域上，首先利用上同调方法定义了一类包含任意线性微分-差分算子的非局部伪微分算子.并进一步给出了这两个算子的合成以及对全纯函数的运算。证明了算子与符号之间的一一对应关系，最后定义了非局部伪微分算子的特征集，证明了非局部伪微分算子的可逆性定理.

The aims of this research were as follows :[1] To generalize the theorem for Cauchy problem of micro-differential equations in the complex domain, obtained by the head investigator, to the sistem of pseudo-differential equations.[2] To characterize the automorphism of the sheaf of holomorphic functions by the infinite ordre differential operators, without assuming the continuity.[3] To generalize the results concerning the existence and the analytic continuation for the single convolution equation in the complex domain to the system.At first, by using the cohomological method, we have defined a natural class of non-local pseudo-differential operators containing any linear differential-difference operators. And furthermore, we gave the composition of two such operators and also the operation to holomorphic functions. We proved the one to one correspondance between the operators and their symbols and finally, defining the characteristic set for the non-local pseudo-differential operator, we proved the invertibility theorem for the non-local pseudo-differential operator.