The Calderón projector for fibred cusp operators

Abstract

A Calderón projector for an elliptic operator P on a manifold with boundary X is a projection from general boundary data to the set of boundary data of solutions u of Pu=0. Seeley proved in 1966 that for compact X and for P uniformly elliptic up to the boundary there is a Calderón projector which is a pseudodifferential operator on ∂X. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity. This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the ϕ-pseudodifferential calculus introduced by Mazzeo and Melrose.In our presentation we provide a setting that may be useful for doing analogous constructions for other types of singularities.