The Structure of the Resolvent of Elliptic Pseudodifferential Operators

We show that the resolvent kernel of an elliptic b-pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is a polyhomogeneous, or classical, function on a certain manifold with corners. The singularities of the resolvent kernel are shown to localize near the diagonal as the resolvent parameter goes to infinity. Explicit descriptions of the expansions, including logarithmic terms, are given. In particular, the asymptotics of the resolvent restricted to the diagonal follows as a corollary. Applications to the asymptotic behavior of the heat kernel and to the analysis of the poles of complex powers are also given.