This paper offers a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on ℝn. The authors use this idea to extend the result to real hyperbolic spaces ℍn.
Chanillo, S., Van Schaftingen, J., & Yung, P.-L. (2017). Variations on a proof of a borderline Bourgain-Brezis Sobolev embedding theorem. Chinese Annals of Mathematics. Series B, 38(1), 235-252. https://doi.org/10.1007/s11401-016-1069-y (Original work published 2017)