A weakly polyhomogeneous calculus for pseudodifferential boundary problems

In a joint work with R. Seeley, a calculus of weakly parametric pseudodiffe rential operators on closed manifolds was introduced and used to obtain com plete asymptotic expansions of traces of resolvents and heat operators asso ciated with the Atiyah- Patodi-Singer problem. The present paper establishe s a generalization to pseudo differential boundary operators, defining weak ly polyhomogeneous singular Green operators, Poisson operators, and trace o perators associated with a manifold with boundary, as well as a suitable tr ansmission condition for pseudodifferential operators. Full composition for mulas are established for the calculus, which contains the resolvents of AP S-type problems. The operators in the calculus have complete asymptotic tra ce expansions in the parameter (when of trace class), with polynomial and l ogarithmic terms. (C) 2001 Academic Press.