A parametrix construction for the Laplacian on Q-rank 1 locally symmetric spaces

This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, -di erential operators, that is, they live in the generalised -calculus studied by the authors in a previous paper, which extends work of Melrose and Mazzeo. However, because they are not totally elliptic elements in this calculus, it is not possible to construct parametrices for these operators within the -calculus. We construct parametrices for them in this paper using a combination of the b-pseudodi erential operator calculus of R. Melrose and the -pseudodi erential operator calculus. The construction simpli es and generalizes the construction done by Vaillant in his thesis for the Dirac operator. In addition, we study the mapping properties of these operators and determine the appropriate Hlibert spaces between which the Gauss-Bonnet and Hodge Laplace operators are Fredholm. Finally, we establish regularity results for elements of the kernels of these operators.