Three-term commutator estimates and the regularity of ∖half∖-harmonic maps into spheres

We prove the regularity of weak 1 2 -harmonic maps from the real line into a sphere. A key step is the formulation of the 1 2 -harmonic map equation in the form of a nonlocal linear Schrdinger type equation with three-term commutators on the right-hand side. We then establish a sharp estimate for these threeterm commutators. 1. Introduction 149 2. Definitions and notation 155 3. Three-term commutator estimates: proof of Theorems 1.5 and 1.6 158 4. L-energy decrease controls 163 5. Morrey estimates and Hlder continuity of 1 2 -harmonic maps into the sphere