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Sharp constants in weighted trace inequalities on Riemannian manifolds. (English) Zbl 1288.46026

Summary: In this paper, we establish some sharp weighted trace inequalities \(W^{1,2}(\rho^{1-2 \sigma}, M) \hookrightarrow L^{\frac{2n}{n-2 \sigma}}(\partial M)\) 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.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J70 Degenerate elliptic equations
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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