Lorentz space estimates for the Coulombian renormalized energy

In this paper we obtain optimal estimates for the "currents" associated to point masses in the plane, in terms of the Coulombian renormalized energy of Sandier-Serfaty \cite{ss1,ss3}. To derive the estimates, we use a technique that we introduced in \cite{st}, which couples the "ball construction method" to estimates in the Lorentz space $L^{2,\infty}$.