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A parametrix construction for the Laplacian on Q-rank 1 locally symmetric spaces
conference contribution
posted on 2015-03-31, 15:36 authored by Daniel Grieser, Eugenie HunsickerThis 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 MathematicsVolume
Pseudo-differential Operators, Time-Frequency Analysis and Partial Differential EquationsPages
149 - 186Citation
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
2014Notes
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
9783319025490Publisher version
Language
- en