Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications

Abstract

In this paper we extend Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds for dimension $n\ne 2$. As one application, we solve a generalized Yamabe problem on locally conforamlly flat manifolds via a new designed energy functional and a new variational approach. Even for the classic Yamabe problem on locally conformally flat manifolds, our approach provides a new and relatively simpler solution.

References in corpus (8)

• An extension problem related to the fractional Laplacian
• On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions
• On a fractional Nirenberg problem, part II: existence of solutions
• Sign of Green's function of Paneitz operators and the Q curvature
• Q curvature on a class of manifolds with dimension at least 5
• Prescribing integral curvature equation
• One Dimensional Conformal Metric Flows
• One dimensional conformal metric flow II

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• Sharp Hardy-Littlewood-Sobolev inequalities on compact CR manifold
• Extremal problem of Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds
• An optimal Hardy-Littlewood-Sobolev inequality on ${\mathbf{R}}^{n-k}\times {\mathbf{R}}^n$ and its consequences
• Subcritical Approach to Sharp Hardy-Littlewood-Sobolev Type Inequalities on the Upper Half Space
• Sharp reversed Hardy-Littlewood-Sobolev inequality with extended kernel