3-Commutators Estimates and the Regularity of 1/2 Harmonic Maps into Spheres

TL;DR

This paper proves the regularity of weak 1/2-harmonic maps into spheres by formulating the problem as a non-local Schrödinger-type equation with 3-commutator estimates, providing sharp bounds for these commutators.

Contribution

It introduces a novel formulation of the 1/2-harmonic map equation using 3-commutators and establishes sharp estimates for these commutators, advancing regularity theory.

Findings

Proved regularity of weak 1/2-harmonic maps into spheres.

Formulated the map equation as a non-local Schrödinger-type equation.

Established sharp estimates for 3-commutators.

Abstract

We prove the regularity of weak 1/2-harmonic maps from the real line into a sphere. The key point in our result is first a formulation of the 1/2-harmonic map equation in the form of a non-local linear Schr\"odinger type equation with a 3-terms commutators in the right-hand-side . We then establish a sharp estimate for these 3-commutators.