\epsilon-regularity for systems involving non-local, antisymmetric operators

TL;DR

This paper establishes an epsilon-regularity theorem for non-local, antisymmetric operator systems, unifying and extending regularity results for various critical and super-critical equations including fractional harmonic maps and conformally invariant variational problems.

Contribution

It provides a new epsilon-regularity theorem applicable to a broad class of non-local systems, offering unified proofs for existing regularity and integrability results.

Findings

Proves epsilon-regularity for non-local antisymmetric systems.

Unifies proofs of regularity for local and non-local equations.

Extends regularity results to fractional harmonic maps and conformally invariant functionals.

Abstract

We prove an epsilon-regularity theorem for critical and super-critical systems with a non-local antisymmetric operator on the right-hand side. These systems contain as special cases, Euler-Lagrange equations of conformally invariant variational functionals as Rivi\`ere treated them, and also Euler-Lagrange equations of fractional harmonic maps introduced by Da Lio-Rivi\`ere. In particular, the arguments presented here give new and uniform proofs of the regularity results by Rivi\`ere, Rivi\`ere-Struwe, Da-Lio-Rivi\`ere, and also the integrability results by Sharp-Topping and Sharp, not discriminating between the classical local, and the non-local situations.