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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weighted Sobolev inequalities under lower Ricci curvature bounds


Author: Hans-Joachim Hein
Journal: Proc. Amer. Math. Soc. 139 (2011), 2943-2955
MSC (2010): Primary 53C21
Published electronically: January 4, 2011
MathSciNet review: 2801635
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Abstract: We obtain sharp weighted Poincaré and Sobolev inequalities over complete, noncompact Riemannian manifolds with polynomial volume growth and a quadratically decaying lower bound on Ricci. This improves and extends earlier work of Tian-Yau and Minerbe. We deduce a sharp existence result for bounded solutions of the Poisson equation on such manifolds, highlighting the well-known distinction between spaces of volume growth $ \leq 2$ and $ > 2$ in terms of their Green's functions. We also show that if the manifold is nonparabolic and carries a smooth function which behaves like the radius function of a cone, then these solutions almost decay at the rates expected from a cone.


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Additional Information

Hans-Joachim Hein
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
Email: h.hein@imperial.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10799-8
PII: S 0002-9939(2011)10799-8
Received by editor(s): April 29, 2010
Received by editor(s) in revised form: July 23, 2010
Published electronically: January 4, 2011
Communicated by: Michael Wolf
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.