Lorentz space estimates for the Coulombian renormalized energy

TL;DR

This paper provides optimal Lorentz space estimates for currents related to point masses in the plane, utilizing a novel coupling of the ball construction method with Lorentz space estimates to analyze Coulombian renormalized energy.

Contribution

It introduces a new technique combining ball construction with Lorentz space estimates to improve bounds on currents associated with Coulombian energies.

Findings

Optimal Lorentz space estimates for currents in Coulombian energy context

New coupling technique for ball construction and Lorentz space analysis

Enhanced understanding of point mass interactions in the plane

Abstract

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 $L^{2,\infty}$.