A Weakly Polyhomogeneous Calculus for Pseudodifferential Boundary Problems

In a joint work with R. Seeley, a calculus of weakly parametric pseudodifferential operators on closed manifolds was introduced and used to obtain complete asymptotic expansions of traces of resolvents and heat operators associated with the Atiyah–Patodi–Singer problem. The present paper establishes a generalization to pseudodifferential boundary operators, defining weakly polyhomogeneous singular Green operators, Poisson operators, and trace operators associated with a manifold with boundary, as well as a suitable transmission condition for pseudodifferential operators. Full composition formulas are established for the calculus, which contains the resolvents of APS-type problems. The operators in the calculus have complete asymptotic trace expansions in the parameter (when of trace class), with polynomial and logarithmic terms.