On wave systems with antisymmetric potential in dimension d >= 4 and well-posedness for (half-)wave maps

We prove a priori estimates for wave systems of the type \[ ∂ₜₜ u - Δ u = Ω · du + F(u) in Rᵈ × R \] where d ≥ 4 and Ω is a suitable antisymmetric potential. We show that the assumptions on Ω are applicable to wave- and half-wave maps, the latter by means of the Krieger-Sire reduction. We thus obtain well-posedness of those equations for small initial data in Ḣd/2(Rᵈ).