Dispersive estimates for the three-dimensional Schrödinge equation with rough potentials

The three-dimensional Schrödinger propogator e^itH, H = -Δ + V, is a bounded map from L¹ to L^∞ with norm controlled by |t|^-3/2 provided the potential satisfies two conditions: An integrability condition limiting the singularities and decay of V, and a zero-energy spectral condition on H. This is shown by expressing the spectral measure of H in terms of its resolvents and proving a family of L^p mapping estimates for the resolvents. Previous results in this direction had required V to satisfy explicit pointwise bounds.