The Calder\'on Projector for Fibred Cusp Operators

TL;DR

This paper extends the concept of the Calderón projector to fibred cusp operators on non-compact manifolds with special fibred structures at infinity, broadening its applicability to singular and symmetric spaces.

Contribution

It generalizes Seeley's Calderón projector construction to fibred cusp operators using the $ ext{phi}$-pseudodifferential calculus, applicable to non-compact and singular spaces.

Findings

Constructed Calderón projector for fibred cusp operators

Applied the $ ext{phi}$-pseudodifferential calculus to this setting

Potentially useful for other singularity types

Abstract

A Calder\'on 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 Calder\'on projector which is a pseudodifferential operator on $\partial X$. 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 $\phi$-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.