Weighted Sobolev inequalities under lower Ricci curvature bounds
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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.References
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Additional Information
- Hans-Joachim Hein
- Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
- MR Author ID: 938594
- ORCID: 0000-0002-3719-9549
- Email: h.hein@imperial.ac.uk
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2943-2955
- MSC (2010): Primary 53C21
- DOI: https://doi.org/10.1090/S0002-9939-2011-10799-8
- MathSciNet review: 2801635