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

conference contribution
posted on 31.03.2015, 15:36 by Daniel Grieser, Eugenie Hunsicker
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.

Funding

This work was completed with the support of Leverhulme Trust Project Assistance Grant F/00 261/Z.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Trends in Mathematics

Volume

Pseudo-differential Operators, Time-Frequency Analysis and Partial Differential Equations

Pages

149 - 186

Citation

GRIESER, D. and HUNSICKER, E., 2014. A parametrix construction for the Laplacian on Q-rank 1 locally symmetric spaces. IN: Ruzhansky, M. and Turunen, V. (eds). Fourier Analysis: Pseudo-differential Operators, Time-Frequency Analysis and Partial Differential Equations. Trends in Mathematics. Basel: Birkhauser, pp. 149 - 186.

Publisher

Birkhauser (© Springer International Publishing)

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2014

Notes

Closed access. This paper was published in a book collection of 20 refereed articles based on selected talks from the international conference “Fourier Analysis and Pseudo-Differential Operators,” June 25–30, 2012, at Aalto University, Finland.

ISBN

9783319025490

Language

en

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