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

Abstract

We prove the regularity of weak $\frac{1}{2}$-harmonic maps from the real line into a sphere. A key step is the formulation of the $\frac{1}{2}$-harmonic map equation in the form of a nonlocal linear Schrödinger type equation with three-term commutators on the right-hand side. We then establish a sharp estimate for these three-term commutators.