ALGEBRAIC ANALYTICAL STUDY OF SHEAVES AND INFINITE ORDRE DIFFERENTIAL EQUATIONS

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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.