A weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds

Abstract

Given (M, g) a smooth compact (n + 1)-dimensional Riemannian manifold with boundary ∂M . Let ρ be a defining function of M and σ ∈ (0, 1). In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of W 1,2 (ρ 1−2σ , M ) ֒→ L p (∂M ), where p = 2n n−2σ . More precisely, under some assumptions on the manifold, we prove that there exists a constant B > 0 such that, for all u ∈ W 1,2 (ρ 1−2σ , M ),. This inequality is sharp in the sense that µ −1 cannot be replaced by any smaller constant. Moreover, unlike the classical Sobolev inequality, µ −1 does not depend on n and σ only, but depends on the manifold.