Bourgain-Brezis inequalities on symmetric spaces of non-compact type

Let M be a global Riemannian symmetric space of non-compact type. We prove a duality estimate, for pairings of divergence-free L1 vector fields, with vector fields in a critical Sobolev space on M:|∫M〈f,ϕ〉dV|≤C‖f‖L1(dV)‖∇ϕ‖Lm(dV). This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces.