Boundary properties of the kernel of an elliptic pseudodifferential operator on a compact manifold with boundary
Let X be a smooth compact n-dimensional manifold with the boundary \(Y=\partial x\), \(X_ 0=int X\), E and F be complex vector bundles over X. Let A be an elliptic pseudo-differential operator of order m, \(P_ A\) be the orthoprojector on the Ker A in \(H^ 0(X,E)\) and \[ (P_ Af)(x)=\int_{X_ 0}P_ A(x,y)f(y)dy, \] so that \(P_ A(x,y)\) is the reproducing kernel in the Bergman-Aronszajn sense. Let \(P_ A(x)=tr P_ A(x,x)\). The author constructs the asymptotic expansion of \(P_ A(x)\) near the boundary Y. This expansion generalizes some results of S. Bergman and L. Hörmander.
scientific article; zbMATH DE number 4153107
scientific article