Analysis of non-linear singularities by microlocal methods

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Our researches concerning non-linear analysis in our project had almost no progress because we encountered a unexpected difficulties. However our researches concerning linear analysis had a remarkable progress due to KATAOKA's group ; in particular, the second microlocal analysis and an elementary construction of the microlocal theory of systems of equations.1. In the microlocal analysis, it is indispensable to employ expressions of singular solutions or kernel functions by boundary values of pseudodifferential operators. Concerning these expressions, KATAOKA's group obtained a necessary and sufficient condition on growth orders of lower order terms of formal symbols of pseudo-differential operators for taking boundary values. This result was applied to the analysis of microlocal solutions of some Fuchsian type equations. Further this result is deeply connected to the theory of small second hyperfunctions in the second microlocal analysis.2. KATAOKA's group succeeded in obtaining an integral formula of Mellin's type for holomorphic functions with any growth order at the corner in one-dimensional angled domains ; the main idea is to tilte the integral paths. Such a formula was indispensable for extending FUNAKOSHI's resullt on second microlocal analysis along Lagrangian submanifolds.