Microlocal energy methods and pseudo-differential operators

A functionk(x, u) of the(n+n)-variables is said to be a positive Hermite kernel if , and the matrix(k(x _i, x_j))_i,j is positive semi-definite for every integerN and everyx ₁, ..., x_N. 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.