ALGEBRAIC-ANALYTICAL STUDY OF PSEUDO-DIFFERENTIAL AND CONVOLUTION ERUATIONS

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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.