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

TL;DR

This paper develops a method to construct parametrices for Laplacian operators on noncompact Q-rank 1 locally symmetric spaces, combining advanced pseudodifferential calculi to analyze their properties and solutions.

Contribution

It introduces a new parametrix construction for Laplacians on Q-rank 1 spaces using a combination of $b$- and $$-calculus, extending previous work and simplifying existing methods.

Findings

Constructed parametrices for Gauss-Bonnet and Hodge Laplacians.

Determined Fredholm properties and appropriate Sobolev spaces.

Proved regularity of kernel elements.

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, $\phi$-differential operators, that is, they live in the generalised $\phi$-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 $\phi$-calculus. We construct parametrices for them in this paper using a combination of the $b$-pseudodifferential operator calculus of R. Melrose and the $\phi$-pseudodifferential operator calculus. The construction simplifies 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.