Singular Fourier Integral Operators, Micro-hyperbolicity and second microlocalization

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In the study of propagation of (microlocal) singularities of solutions to linear hyperbolic equations, a variety of phenomenon such as branching of singularities and conical refraction. In particular, conical refraction has been studied from many points of vies since it appeared naturally in natural world. Around 1885, in the study of conical refraction, the method of second microlocalization was employed, which blow-up the phase space along an involutive submanifold. This made it possible to obtain a certain successful result by P. Laubin and N. Tose. But there still remained some problems as far as existence of solutions is concerned. In this research, a transformation theory for operators in the category of second microlocalization is constructed.