Regularity of n/2-harmonic maps into spheres

We prove Hölder continuity for n/2-harmonic maps from arbitrary subsets of Rn into a sphere. This extends a recent one-dimensional result by F. Da Lio and T. Rivière to arbitrary dimensions. The proof relies on compensation effects which we quantify adapting an approach for Wenteʼs inequality by L. Tartar, instead of Besov space arguments which were used in the one-dimensional case. Moreover, fractional analogues of Hodge decomposition and higher order Poincaré inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied.This work was the authorʼs PhD thesis, written in March 2010.