The sharp affine $L^2$ Sobolev trace inequality and variants

TL;DR

This work derives a sharp affine version of the $L^p$ Sobolev trace inequality on the half-space $\\mathbb{R}^n_+$ by coupling the $L_p$ Busemann-Petty centroid inequality with a Legendre-transform based convex-analytic construction $C_f$. The method introduces $C_f$, connected to a convex body $K_f$ and to the $L_p$ centroid framework via $L_f=\\Gamma_p L_f$, and leverages the $L_p$ Petty-Projection inequality to bound the trace term in terms of the energy $\\\\mathcal{E}_p(f)^{p-1}$ and the derivative term $\\int_{\\mathbb{R}^n_+} |\\partial f/\\partial t|^p$. For $p=2$, the inequality is strictly stronger than Escobar-Beckner and admits a complete equality description, while for general $p$ it provides a sharp affine bound with an explicit constant $d_1$; the affine nature makes it invariant under all affine transformations of $\\mathbb{R}^n_+$. An appendix supplies the explicit closed-form expression for $d_1$, highlighting the tight geometric control exercised by the convex-geometry machinery. Overall, the paper reveals the geometric underpinnings of affine Sobolev trace inequalities by exploiting the $L_p$ Busemann-Petty centroid framework.

Abstract

We establish a sharp affine $L^p$ Sobolev trace inequality by using the $L_p$ Busemann-Petty centroid inequality. For $p = 2$, our affine version is stronger than the famous sharp $L^2$ Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all cases of equality in this case. For this new inequality, no Euclidean geometric structure is needed.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Definition 1
• Proposition 1
• proof
• Lemma 2
• proof