Fredholm theory for degenerate pseudodifferential operators on manifolds with fibered boundaries

We consider the calculus psi (de)*(,)* (X, (de)Omega (1/2)) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm elements in psi (de)*(,)* (X)by means of the invertibility of an appropriate symbol map, and describe a K-theoretical formula for the Fredholm index extending the Atiyah-Singer formula for closed manifolds. The algebra of operators of order (0, 0) is shown to be a psi*-algebra, hence its K-theory coincides with that of its C*-closure, and we give a description of the corresponding cyclic 6-term exact sequence. We define a Wodzicki-type residue trace on an ideal in psi (de)*(,)*(X,(de)Omega (1/2)), and we show that it coincides with Dixmier's trace for operators of order -dim X in this ideal. This extends a result of Connes for the closed case.