  We provide a comprehensive analysis of sharp bilinear estimates of
Ozawa-Tsutsumi type for solutions u of the free Schr\"odinger equation, which
give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we
provide a generalization of their estimates in such a way that provides a
unification with some sharp bilinear estimates proved by Carneiro and
Planchon-Vega, via entirely different methods, by seeing them all as special
cases of a one parameter family of sharp estimates. We show that the extremal
functions are solutions of the Maxwell-Boltzmann functional equation and
provide a new proof that this equation admits only Gaussian solutions. We also
make a connection to certain sharp estimates on $u^2$ involving certain
dispersive Sobolev norms.
