Limiting Bourgain-Brezis estimates for systems: theme and variations

TL;DR

This paper reviews Bourgain and Brezis's 2002 estimates for linear differential systems, discusses recent developments, and highlights open problems in the field of endpoint estimates and critical Sobolev spaces.

Contribution

It provides an overview of key results, recent progress, and open challenges related to Bourgain-Brezis estimates for systems of differential equations.

Findings

Summarizes the original Bourgain-Brezis estimates from 2002.

Discusses advancements made in the last decade.

Identifies open problems in endpoint estimates and Sobolev spaces.

Abstract

J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case $\mathrm{L}^1$ of singular integral estimates and the critical Sobolev space $\mathrm{W}^{1, n} (\mathbf{R}^n)$. This paper presents an overview of the results, further developments over the last ten years and challenging open problems.