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

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.