ALGEBRAIC-ANALYTICAL STUDY OF PSEUDO-DIFFERENTIAL AND CONVOLUTION ERUATIONS

本研究的目的如下：[1]利用薄层的微局部研究，对复域上的卷积方程进行代数解析研究。[2)研究Fabry-Ehrenpreis-Kawai定理，将解的解析延拓理论应用于卷积方程。[3)将复域上微微分方程柯西问题的理论推广到拟微分情形。对于问题[1]和[2]，我们首先构造了多元椭圆条件下的卷积方程的例子。同时，对于复域上齐次卷积方程全纯解的解析延拓问题，我们引入了它的特征集作为通常的常系数线性偏微分方程解的自然推广，并利用特征集给出了任何解在其上解析延拓的区域的表达式。特别地，这几乎完全解决了作为泛函微分方程解的重要例子的无穷阶差分方程解的解析延拓问题。在管域的情况下，证明了在自然条件下，特征集与符号零点在无穷远处的累积方向重合。然而，对于[3]，我们还没有成功地得到一个普遍的理论。但是，我们现在正在用归纳极限的方法来研究这个问题，这个问题还在发展之中。

The aims of this research were as follows :[1] The algebraic-analytical study of convolution equations in the complex domains, using micro-local study of sheaves.[2] Study of the Fabry-Ehrenpreis-Kawai Theorem, applying the theory of analytical continuation of solutions to convolution equations.[3] The extension of the theory of the Cauchy problem for micro-differential equationts in the complex domains to the pseudo-differential case.For the problem [1] and [2], at first, we constructed good examples of convolution equations with elliptic codition in the several variables. And also, for the problem of the analytic continuation of the holomorphic solutions to the homogeneous convolution equation in the complex domains, we introduced its characteristic sets to be the natural extension of the case of usual constant coefficients linear partial differential equations and we could present the expicit form of the domains to which any solution is continued analytically, using the characteristic set. In particular, this resolves almost completely the problem of the analytic continuation to the infinite ordre differential-difference equations which are important examples of the functional-differential equaion. In the case of tube domains, one proved that, in a natural condition, the characteristic set coincides with the accumulating directions at infinity of the zeros of the symbol. However, for [3], we did not yet succeed to get an general theory. But we are now studying the problem using the sheaf theoritical study by means of inductive limits which is in the course of developpement.