  We prove new interaction Morawetz type (correlation) estimates in one and two
dimensions. In dimension two the estimate corresponds to the nonlinear diagonal
analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we
provide a proof in two different ways. First, we follow the original approach
of Lin and Strauss but applied to tensor products of solutions. We then
demonstrate the proof using commutator vector operators acting on the
conservation laws of the equation. This method can be generalized to obtain
correlation estimates in all dimensions. In one dimension we use the
Gauss-Weierstrass summability method acting on the conservation laws. We then
apply the 2d estimate to nonlinear Schr\"odinger equations and derive a direct
proof of Nakanishi's $H^1$ scattering result for every $L^{2}$-supercritical
nonlinearity. We also prove scattering below the energy space for a certain
class of $L^{2}$-supercritical equations.
