Decay estimates for Rivi\`ere's equation, with applications to regularity and compactness

We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in ${\mathbb{R}}^2$ introduced by Rivi`ere. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmonic and almost-harmonic maps from surfaces.