On the weak \(L ^{p }\)-Hodge decomposition and Beurling--Ahlfors transforms on complete Riemannian manifolds
type: Item
instance of: scholarly article
title: On the weak \(L ^{p }\)-Hodge decomposition and Beurling--Ahlfors transforms on complete Riemannian manifolds
zbMATH Open document ID: 1237.58023
publication date: 2011-09-27T00:00:00Z
review text: The author proves a weak \(L^p\)-Hodge decomposition on a (stochastically) complete Riemannian manifold \(M\) with non-negative Weitzenböck curvature on \(k\)-forms, based on the \(L^p\) boundedness of the Beurling-Ahlfors transform. In particular, the boundedness of the Riesz transforms is not needed. As a consequence, a logarithmic Sobolev inequality is obtained in dimension \(n\geq 3\) for vanishing Ricci curvature. The proofs are based on a martingale representation formula for \(dd^* ( a + \square )^{-1}\), where \(a>0\) and \(\square\) denotes the Hodge Laplacian.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 5950527
zbMATH Keywords: Beurling-Ahlfors transforms
zbMATH Keywords: Hodge decomposition
zbMATH Keywords: martingale representation formula
zbMATH Keywords: Weitzenböck curvature
author: Xiang Dong Li
MaRDI profile type: Publication
full work available at URL: s00440-010-0270-2
OpenAlex ID: W4249745838
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DOI: 10.1007/S00440-010-0270-2
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 5950527