# Fractional div-curl quantities and applications to nonlocal geometric equations

## TL;DR

This paper develops a fractional calculus framework for div-curl quantities, applying it to nonlocal geometric equations to establish regularity results for fractional harmonic maps and related systems.

## Contribution

It introduces a fractional div-curl theory, generalizes key estimates, and applies these to prove regularity of fractional harmonic maps and solutions to nonlocal geometric PDEs.

## Key findings

- Established a nonlocal version of Wente's lemma.
- Proved regularity for fractional harmonic maps into spheres.
- Provided new proofs for regularity of half-harmonic maps into manifolds.

## Abstract

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente's lemma. We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a theory for fractional harmonic maps analogous to the local theory. Firstly, regarding fractional harmonic maps into spheres, we obtain a conservation law analogous to Shatah's conservation law and give a new regularity proof analogous to H\'elein's for harmonic maps into spheres. Secondly, we prove regularity for solutions to critical systems with nonlocal antisymmetric potentials on the right-hand side. Since the half-harmonic map equation into general target manifolds has this form, as a corollary, we obtain a new proof of the regularity of half-harmonic maps into general target manifolds following closely Rivi\`{e}re's celebrated argument in the local case. Lastly, the fractional div-curl quantities provide also a new, simpler, proof for H\"older continuity of $W^{s,n/s}$-harmonic maps into spheres and we extend this to an argument for $W^{s,n/s}$-harmonic maps into homogeneous targets. This is an analogue of Strzelecki's and Toro-Wang's proof for $n$-harmonic maps into spheres and homogeneous target manifolds, respectively.