Pseudodifferential operator calculus for generalized Q-rank 1 locally   symmetric spaces, I
  This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of fibre bundles. The class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary, in directions given by the tower of fibrations. As well as Q-rank 1 locally symmetric spaces, examples include Ricci-flat metrics on the complement of a divisor in a smooth variety constructed by Tian and Yau. In this first part of the calculus construction, parametrices are found for "fully elliptic differential \bfa-operators", which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity. In the second part we will consider operators that do not satisfy this condition. 
Daniel Grieser and Eugenie Hunsicker