On Gevrey property and asymptotic analysis for divergent solutions of analytic partial differential equations

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The purpose of this project is to study Gevrey property and asymptotic analysis for divergent power series solutions of analytic partial differential equations.The first result is that we established the Toeplitz operator method in analytic partial differential equations which enables us to give a precise notion of Fredholmness in the Goursat problem in various Gevrey spaces which has not been studied in explicite way in early studies. We proved further that an index formula for ordinary differnetial operator on Gevrey space is nothing but the geometrical index fornmula for a Toeplitz sumbol associated with the Gevrey filtration n the ring of ordinary differential operators.The second result is that we gave a necessary and sufficient condition for the Borel summability for divergent power series solution of the Cauchy problem of the heat equation, and we proved that the Borel sum is just expressed by an integral with the heat kernel. The condition for the C data we obtained is the well known condition for the uniqueness of solutions of the Cauchy problem. In proving this we made clear that the problem of Borel summability in partial differential equations is not local property of solutions, whereas the notion of the Borel summability is only local one. We also made clear that this problem provides a new kind of problems in partial differential equations.