Bourgain-Brezis Estimates on Symmetric Spaces of Non-compact Type

TL;DR

This paper extends Bourgain-Brezis estimates to symmetric spaces of non-compact type, establishing duality and Calderon-Zygmund estimates for divergence-free vector fields and solutions to Poisson's equation.

Contribution

It generalizes Bourgain-Brezis estimates from Euclidean space to non-compact symmetric spaces, providing new duality and sharp Calderon-Zygmund estimates.

Findings

Established duality estimates for divergence-free vector fields on symmetric spaces.

Derived sharp Calderon-Zygmund estimates for Poisson's equation solutions.

Extended Euclidean Bourgain-Brezis results to a broader geometric setting.

Abstract

Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical Sobolev space on M. As a consequence we get a sharp Calderon-Zygmund estimate for solutions to Poisson's equation on M, where the right side data is manufactured from divergence free vector fields which are in L^1. Such a result was proved earlier by Jean Bourgain and Haim Brezis on Euclidean space.