A weighted Sobolev–Poincaré-type trace inequality on Riemannian manifolds

Given [Formula: see text] a smooth compact [Formula: see text]-dimensional Riemannian manifold with boundary [Formula: see text]. Let [Formula: see text] be a defining function of [Formula: see text] and [Formula: see text]. In this paper, we study a weighted Sobolev–Poincaré-type trace inequality corresponding to the embedding of [Formula: see text], where [Formula: see text]. More precisely, under some assumptions on the manifold, we prove that there exists a constant [Formula: see text] such that, for all [Formula: see text], [Formula: see text] where [Formula: see text] is a constant depending only on [Formula: see text] and [Formula: see text]. This inequality is sharp in the sense that [Formula: see text] cannot be replaced by any smaller constant. Unlike the classical Sobolev inequality, [Formula: see text] does not depend on [Formula: see text] and [Formula: see text] only, but depends on the manifold.