A Parametrix Construction for the Laplacian on ℚ-rank 1 Locally Symmetric Spaces

Abstract

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, φ-differential operators; that is, they live in the generalised φ-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 φ-calculus. We construct parametrices for them in this paper using a combination of the b-pseudodifferential operator calculus of R. Melrose and the φ-pseudodifferential operator calculus. The construction simplifies and generalizes the construction of a parametrix for the Dirac operator done by Vaillant in his thesis. In addition, we study the mapping properties of these operators and determine the appropriate Hilbert 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.

Mathematics Subject Classification (2010). Primary 58AJ40; Secondary 35J75.