Weighted Sobolev inequalities under lower Ricci curvature bounds

Hein, Hans-Joachim

doi:10.1090/S0002-9939-2011-10799-8

Weighted Sobolev inequalities under lower Ricci curvature bounds
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by Hans-Joachim Hein PDF

Proc. Amer. Math. Soc. 139 (2011), 2943-2955 Request permission

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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