Study of global solutions to Fuchsian equations and local solutions to linear PDE

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1.We tried second microlocal analysis for some class of Turrittin equations with lower terms of homogeneous type. In this case, similarly to the case of Mizohata equations, it became clear that fundamental solutions can be constructed using the theory of second microlocal Fourier transformations. Here we used the result about the growth order for the global solutions to some ordinary differential equations in the space where Fourier transforms live. In more special cases, the inverse Fourier transform satisfies some Fuchsian equations. We got the information of the Fuchsian equations, singularities, exponents, and some behaviors of their solutions.2.On the other hand, we can consider the notion of microfunctions with holomorphic parameters on a product space of a real domain and a complex domain. They are represented as boundary values of holomorphic functions, and their boundary values define the notion of second microfunctions, and so on. The equations in 1 have microlocal ellipticity in rather small region, but we can construct their second microlocal solutions by the actions of their inverses to microfunctions with holomorphic parameters. The relation between this result and the theory of boundary values of pseudo-differential operators due to Kataoka will be a further subject.3.A fundamental solution to linear PDE can be regarded as a kernel function of an integral transformation. We studied the kernels under an expectation that an integral transformation with kernel should be characterized by some notion similar to continuity, and got the kernel theorems in analytic category, by introducing notion of semi-continuity. Here we used the construction of complex kernels with curvilinear Radon transformations.