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Parametrix Construction and Invertibility of Δg + 1
Abstract
Given a manifold M with an asymptotically cylindrical end, we construct the parametrix for the shifted Laplace operator Δg + 1 on the blown-up b-double space Mb2. We will review the geometry of manifold with a cylindrical end and construct the inverse directly via separation of variables, which motivates the parametrix construction. Next, we review the mapping properties and show how to get elliptic regularity from the mapping properties of the parametrix. Through this explicit example, we hope this paper provides another reference for the theory of b-operators introduced by Melrose \cite{Mel93}.
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BibTeX
@misc{chang2026parametrix,
title = {{Parametrix Construction and Invertibility of $Δ_g + 1$}},
author = {Shih-Yu Chang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10315},
eprint = {2610.10315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10315},
}
RIS
TY - UNPB
TI - Parametrix Construction and Invertibility of Δ_g + 1
AU - Shih-Yu Chang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10315
UR - https://arxiv.org/abs/2610.10315
AB - Given a manifold M with an asymptotically cylindrical end, we construct the parametrix for the shifted Laplace operator Δ_g + 1 on the blown-up b-double space M_b^2. We will review the geometry of manifold with a cylindrical end and construct the inverse directly via separation of variables, which motivates the parametrix construction. Next, we review the mapping properties and show how to get elliptic regularity from the mapping properties of the parametrix. Through this explicit example, we hope this paper provides another reference for the theory of b-operators introduced by Melrose \cite{Mel93}.
ER -