Bilinear Strichartz estimates for the Schr{\"o}dinger map problem

In this paper we prove bilinear Strichartz estimates for a solution to the Schr{"o}dinger map problem whose size is small in the critical Strichartz space $\mathrm{\mid }\mathrm{\mid }\mathrm{\nabla }{\mathrm{\mid }}^{\frac{d-2}{2}}{\psi }_x{\mathrm{\mid }}_{L_{t,x}^{\frac{2(d+2)}{d}}}$. These estimates will be useful in an upcoming paper in proving a local well - posedness result. Bilinear estimates make use of an argument similar to the argument found in Planchon and Vega (2009). We use the same gauges as in Bejenaru, Ionescu and Kenig (2007), Bejenaru et al. (2011), and Smith.