Regularity of n/2-harmonic maps into spheres

TL;DR

This paper establishes Hoelder continuity for n/2-harmonic maps into spheres across all dimensions, extending previous one-dimensional results through advanced compensation techniques and fractional calculus tools.

Contribution

It generalizes Hoelder regularity results for harmonic maps into spheres from one dimension to higher dimensions using novel fractional and localization methods.

Findings

Proves Hoelder continuity for n/2-harmonic maps in arbitrary dimensions.

Develops fractional Hodge decomposition and higher order Poincare inequalities.

Introduces localization effects for nonlocal operators like the fractional Laplacian.

Abstract

We prove Hoelder continuity for n/2-harmonic maps from subsets of Rn into a sphere. This extends a recent one-dimensional result by F. Da Lio and T. Riviere to arbitrary dimensions. The proof relies on compensation effects which we quantify adapting an approach for Wente's 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 Poincare inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied.