Sharp trace inequalities for fractional Laplacians

TL;DR

The paper extends Escobar's sharp trace inequality to traces for the fractional Laplacian on $\mathbb{R}^n$ and fully characterizes when equality occurs. Using Fourier analysis, a sharp Sobolev inequality in the $D_\alpha(\mathbb{R}^n)$ framework, and Lieb's sharp Hardy-Littlewood-Sobolev inequality, it derives the best constant for the trace inequality and describes the extremizers, including an explicit Fourier-side form and spatial minimizers. It also develops the trace extension operator $\tau_m$ to connect $D_\alpha(\mathbb{R}^n)$ with $D_{\alpha- m/2}(\mathbb{R}^{n-m})$, extending the result to all relevant $\alpha$, including integer values.

Abstract

The sharp trace inequality of Jose Escobar is extended to traces for the fractional Laplacian on R^n and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Lieb's sharp form of the Hardy-Littlewood-Sobolev inequality.

Theorems & Definitions (7)

• Theorem 1.1: Sobolev trace inequality
• Theorem 2.1: Sobolev inequality
• Remark 2.2
• proof
• Theorem 2.3: Trace inequality
• proof
• proof : Proof of Theorem \ref{['thm: sharp trace nequality']}