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 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. (2000). Primary 58AJ40; Secondary 35J75. Mathematics Subject Classification

Dimension of monopoles on asymptotically conic 3-manifolds: Figure 1.

“…Here the behavior of the operator at infinity leads to the use of 'hybrid' b-/scattering-type Sobolev spaces adapted to a splitting of the vector bundle there. Similar hybrid Sobolev spaces have also appeared in [14,16]. One novelty of the problem presented here is that some of the indicial roots themselves depend on the parameter γ, so that the index can both increase and decrease as γ decreases (see Figure 1 and the associated discussion in § 4.2).…”

Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half‐space

“…Proposition in fact tells us that and define b ‐pseudodifferential operators, compare Definition or [, pp. 7ff.…”

“…As is noted in [19], this definition does not depend on the choice of dvol , and coordinates and can be generalised to ∈ ℝ by arguments of duality and interpolation.…”

“…The following material can be found in different forms and flavours in various sources, for instance in [15,18,19,25,26,[29][30][31]. We will in particular make use of [31] and [18,19]. Any manifold with corners has, locally, product coordinates ( , ) ∈ ℝ + × ℝ − = ℝ , where the vanishing of the determines the boundary hypersurfaces.…”

Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent

“…allowing the fibrewise Laplacians to have non-trivial kernel. A parametrix construction for a closely related operator, the Hodge Laplacian for the cusp metric x 2 g φ , was given by Grieser and Hunsicker in [GrHu14], and a similar construction for the spin Dirac operator (assuming constant dimension for the kernel of the operators induced on the fibres) was given by Vaillant in [Vai01]. Vaillant also analyzed the low energy behavior of the resolvent for this operator.…”