  We consider the Schr\"odinger map initial value problem into the sphere in
2+1 dimensions with smooth, decaying, subthreshold initial data. Assuming an a
priori $L^4$ boundedness condition on the solution, we prove that the
Schr\"odinger map system admits a unique global smooth solution provided that
the initial data is sufficiently energy-dispersed. Also shown are
global-in-time bounds on certain Sobolev norms of the solution. Toward these
ends we establish improved local smoothing and bilinear Strichartz estimates,
adapting the Planchon-Vega approach to such estimates to the nonlinear setting
of Schr\"odinger maps.
