About a decade ago, Bourgain, Brezis and van Schaftingen established some borderline Sobolev embeddings on $\mathbb{R}^n$, which lent themselves to the proof of some Gagliardo-Nirenberg inequalities for differential forms on $\mathbb{R}^n$ (see e.g. work of Lanzani and Stein). In an attempt to understand the geometry underlying such estimates, we extend some of these results from the setting of $\mathbb{R}^n$ to the hyperbolic spaces. This is joint work with Sagun Chanillo and Jean van Schaftingen.