The relation between the quantitative properties of the solutions of partial differential equations and the geometrical structures of their characteristics

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By the theory of microlocal analysis, the qualitative property of the solutions of partial differential equations, such as the position of their singularity, can be described completely in terms of their characteristics. The aim of this research project was to investigate to what extent the quantitative property is described as well, and to apply it to the problems of the theory of partial differential equations. Especially, I have obtained the following results:The boundedness of Fourier integral operators: A theory of the boundedness properties of Fourier integral operators has been constructed. Especially, we treated the operators which are used to express the canonical transformations, and established their boundedness on weighted L^2-spaces.A weak extension theorem for inhomogeneous equations: An Lp-extension theory of inhomogeneous partial differential equations on a punctured domain has been constructed. A new approach based on the microlocal analysis is used to obtain results, which cannot be covered by the method of Bochner, who first studied this problem for homogeneous equations.A smoothing effect of Schrodinger equations: A smoothing effect on the initial value problem of generalized Schroedinger equations has been investigated, and an unknown relation has been found between the characteristics of operators and the direction in which the solutions gain their extra regularity. This result will be applied to non-linear problem.