Sharp constants in weighted trace inequalities on Riemannian manifolds

Abstract

We establish some sharp weighted trace inequalities $W^{1,2}(ρ^{1-2σ}, M)\hookrightarrow L^{\frac{2n}{n-2σ}}(\pa M)$ on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $\rho$ is a defining function of $M$ and $\sigma \in (0,1)$. This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.

34 pages

References in corpus (4)

• An extension problem related to the fractional Laplacian
• On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions
• A Harnack inequality for fractional Laplace equations with lower order terms
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• Higher order Sobolev trace inequalities on balls revisited
• Compactness of solutions to nonlocal elliptic equations
• Conformal metrics with prescribed fractional scalar curvature on conformal infinities with positive fractional Yamabe constants
• A fractional conformal curvature flow on the unit sphere
• Sharp Sobolev inequalities involving boundary terms revisited