Variations on a proof of a borderline Bourgain-Brezis Sobolev embedding theorem

TL;DR

This paper presents a new proof of a critical Sobolev embedding theorem and extends it from Euclidean space to hyperbolic space, broadening the theorem's applicability.

Contribution

It introduces a novel proof technique for the Bourgain-Brezis Sobolev embedding and generalizes the result to hyperbolic spaces.

Findings

New proof of the borderline Bourgain-Brezis Sobolev embedding

Extension of the embedding theorem to hyperbolic spaces

Potential applications in geometric analysis

Abstract

We offer a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on $\mathbb{R}^n$. We use this idea to extend the result to real hyperbolic spaces $\mathbb{H}^n$.