Sharp constants in weighted trace inequalities on Riemannian manifolds

TL;DR

This paper proves sharp weighted trace inequalities on compact Riemannian manifolds with boundary, extending fractional Laplacian theory and addressing a conjecture in conformal geometry.

Contribution

It establishes the first sharp weighted trace inequalities involving fractional Sobolev spaces on manifolds with boundary, advancing the understanding of fractional conformal Laplacians.

Findings

Proved sharp weighted trace inequalities on manifolds with boundary.

Connected inequalities to fractional Laplacian problems in conformal geometry.

Addressed a conjecture of Aubin in the context of weighted inequalities.

Abstract

We establish some sharp weighted trace inequalities $W^{1,2}(\rho^{1-2\sigma}, M)\hookrightarrow L^{\frac{2n}{n-2\sigma}}(\pa M)$ 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.