Construction of the Green function on a Riemannian manifold using harmonic coordinates
type: Item
instance of: scholarly article
title: Construction of the Green function on a Riemannian manifold using harmonic coordinates
zbMATH Open document ID: 0867.58066
author: Tsutomu Hiroshima
published in: Journal of Mathematics of Kyoto University
publication date: 1997-08-14T00:00:00Z
review text: Let \((M,g)\) be a compact Riemannian manifold of dimension \(n\geq 3\) without boundary. If \(\nabla\), resp. \(\Delta\), denote the Levi-Civita connection of \((M,g)\), resp. the Laplace operator, the author proves the estimate \(|\nabla^2 u|_p\leq C|\Delta u|_p\), where \(C\) depends on the geometric data of \((M,g)\). This \(L^p\)-estimate is derived via a Calderon-Zygmund type inequality. In the main result of the paper it is estimated the constant \(C\) in terms of the diameter, the injectivity radius and the lower bound of the Ricci tensor. For this aim the author constructs a parametrix of the Green function by using harmonic coordinates.
reviewed by: Vicenţiu D. Rădulescu
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 58E07
zbMATH DE Number: 950576
zbMATH Keywords: \(L^ p\)-estimates
zbMATH Keywords: Levi-Civita connection
zbMATH Keywords: Laplace operator
zbMATH Keywords: Ricci tensor
zbMATH Keywords: Green function
Wikidata QID: Q115240426
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518601
OpenAlex ID: W1568872129
DOI: 10.1215/KJM/1250518601
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description: scientific article; zbMATH DE number 950576