Integrability by compensation for Dirac equation

Abstract

We consider the Dirac operator acting on the Clifford algebra C ℓ m {C\ell }_{m} . We show that under critical assumptions on the potential and the spinor field the equation is subject to an integrability by compensation phenomenon and has a sub-critical behaviour below some positive energy threshold (i.e. ϵ − \epsilon - regularity theorem). This extends in 4 space dimension as well as in 3 dimension a similar result obtained previously by the two first authors in 2D in F. Da Lio and T. Riviére [Critical chirality in elliptic systems, Ann. Inst. H. Poincare Anal. Non Linéaire, 38 (2021), no. 5, 1373–1405].