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We study the regularity of weak solutions for two elliptic systems involvingthe $n$-Laplacian and a critical nonlinearity in the right hand side:$H$-systems and $n$-harmonic maps into compact Riemannian manifolds. Under theassumptions that the solutions belong to $W^{n/2,2}$ in an even dimension $n$,we prove their continuty.  The tools used in the proof involve Hardy spaces and BMO, and theRivi\`{e}re--Uhlenbeck decomposition (with estimates in Morrey spaces). Aprominent role is played by the Coifman--Rochberg--Weiss commutator theorem.

Regularity for solutions of H-systems and n-harmonic maps with n/2 square integrable derivatives