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In this paper we shall construct microlocal parametrices for hyperbolic mixed problems at non-glancing points in the case where their Lopatinski's determinants have real zeros, and we shall investigate the reflection of singularities. It was proved for hyperbolic mixed problems with constant coefficients in a quarter-space that singularities corresponding to boundary waves generally appear when Lopatinski's determinant has real zeros (see [2], [H]3 [12]). We shall show that singularities corresponding to boundary waves appear in variable coefficients cases. Microlocal parametrices for hyperbolic mixed problems were constructed in some cases by using the theory of Fourier integral operators. Chazarain [1] constructed microlocal parametrices for the Dirichlet problem for wave equations at non-glancing points. Microlocal parametrices for the Dirichlet problem for second order operators were constructed at diflfractive points by Melrose [6] and Taylor [10]. But it seems very difficult to construct microlocal parametrices at glancing points which are not diffractive. On the other hand there is a problem of constructing microlocal parametrices when Lopatinski's determinant has real zeros. This problem has no difficulty and we can investigate the reflection of singularities corresponding to boundary waves by constructing microlocal parametrices. We can construct a microlocal parametrix as the

Microlocal parametrices for hyperbolic mixed problems in the case where boundary waves appear