What question did this study set out to answer?

The aim is to study a weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds.

January 18, 2026

A Weighted Sobolev-Poincare Type Trace Inequality on Riemannian Manifolds

Key Points

The aim is to study a weighted Sobolev-Poincaré type trace inequality on Riemannian manifolds.
Analyzed smooth compact Riemannian manifolds with boundary conditions.
Proved the existence of a constant relating to the embedding.
Investigated the dependency of constants on the manifold.
A weighted Sobolev-Poincaré trace inequality was established.
The inequality is sharp, meaning smaller constants cannot replace the one found.
The inequality's dependency extends beyond classical Sobolev parameters.

Abstract

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.

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