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Microlocal energy methods and pseudo-differential operators

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A functionk(x, u) of the(n+n)-variables is said to be a positive Hermite kernel if\(k(x,u) = \overline {k(u,x)} \), and the matrix(k(x i, xj))i,j is positive semi-definite for every integerN and everyx 1, ..., xN. In this paper, we prove that this positive structure can be microlocalized in the category of microfunctions. Further we obtain a useful theorem concerning the positivity of pseudo-differential operators. This theory will play important roles in the study of analytic singularities of solutions of boundary value problems.

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  1. Department of Mathematics, Faculty of Science, Tokyo Metropolitan University, Fukasawa, Setagaya-Ku, 158, Tokyo, Japan

    Kiyômi Kataoka

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  1. Kiyômi Kataoka
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Kataoka, K. Microlocal energy methods and pseudo-differential operators. Invent Math 81, 305–340 (1985). https://doi.org/10.1007/BF01389055

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