  We prove global well-posedness for the half-wave map with $S^2$ target for
small $\dot{H}^{\frac{n}{2}} \times \dot{H}^{\frac{n}{2}-1}$ initial data. We
also prove the global well-posedness for the equation with $\mathbb{H}^2$
target for small smooth $\dot{B}^{\frac{n}{2}}_{2,1} \times
\dot{B}^{\frac{n}{2}-1}_{2,1}$ initial data.
