Integral estimates for the Laplace-Beltrami and Green's operators applied to differential forms on manifolds
type: Item
instance of: scholarly article
title: Integral estimates for the Laplace-Beltrami and Green's operators applied to differential forms on manifolds
zbMATH Open document ID: 1044.58002
DOI: 10.4171/ZAA/1181
author: D. Massart
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2004-06-15T00:00:00Z
review text: The author studies the \(L^ p\) theory of the Laplace-Beltrami operator \(\Delta=d\,d^*+d^* d\) and the Green operator \(G\) acting upon differential forms on manifolds. Some norm inequalities for \(\Delta\), \(G\), and their composition \(\Delta\circ G\) that are applied to differential forms on a compact orientable \(C^\infty\)-smooth Riemannian manifold \(M\) without boundary are established. Also, \(A_ r(M)\)-weighted boundedness for the composition \(\Delta\circ G\) is obtained. As applications, the author proves \(A_ r(M)\)-weighted Sobolev-Poincare embedding theorems for the Green operator and norm comparison theorems for solutions of the \(A\)-harmonic equation on manifolds. These results can be used in developing the \(L^ p\) theory of differential forms and the Hodge decomposition.
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 58A14
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 2075238
zbMATH Keywords: differential forms
zbMATH Keywords: Sobolev embedding
zbMATH Keywords: Green operator
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Hodge decomposition
reviewed by: Andrew Bucki
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2075238