The half-wave maps equation is a nonlocal geometric equation arising in the continuum dynamics of Haldane-Shashtry and Calogero-Moser spin systems. Global wellposedness for small data in the critical space \({\dot{B}}^{n/2}_{2,1}\) is known since [11, 15] for dimensions \(n\geqslant 4\) . There is a major obstruction to extending these results to \(n=3\) due to the loss of the \(L^2_tL^\infty _x\) Strichartz estimate. In this work, we prove that the equation admits global solutions for small smooth data in \({\dot{B}}^{3/2}_{2,1}\) which also possesses some angular regularity and weighted decay of derivatives. We use the improved Strichartz estimates of [24] to develop trilinear estimates in Strichartz spaces weighted by commuting vector fields which avoid the use of the \(L^2_tL^\infty _x\) endpoint.
