# On point-mass Riesz external fields on the real axis

## TL;DR

This paper extends the understanding of Riesz $s$-equilibrium measures on the real axis under external point-mass fields, introducing new techniques and examples that challenge standard growth assumptions at infinity.

## Contribution

It generalizes equilibrium measure results for $0<s<1$, explores non-concave positive parts of signed measures, and introduces innovative methods like iterated balayage in this context.

## Key findings

- Extended equilibrium measure results for $0<s<1$
- Provided examples with unconventional growth at infinity
- Applied signed measures and balayage techniques

## Abstract

The purpose of this work is twofold. First, we aim to extend for $0<s<1$ the results of one of the authors about equilibrium measures in the real axis in external fields created by point-mass charges for the case of logarithmic potentials ($s=0$). Our second motivation comes from the work of the other two authors on Riesz $s$-equilibrium problems on finitely many intervals on the real line in the presence of external fields. They have shown that when the signed equilibrium measure has concave positive part on every interval, then the $s$-equilibrium support is also a union of finitely many intervals, with one of them at most included in each of the initial intervals. As the positive part of the signed equilibrium for point-mass external fields is not necessarily concave, the investigation of the corresponding $s$-equilibrium support is of comparative interest. Moreover, we provide simple examples of compactly supported equilibrium measures in external fields $Q$ not satisfying the usual requirements about the growth at infinity, that is, $\displaystyle \lim_{|x|\rightarrow \infty}\,Q(x) = \infty\,.$ Our main tools are signed equilibrium measures and iterated balayage algorithm in the context of Riesz $s$-equilibrium problems on the real line. As these techniques are not mainstream work in the field and can be applied in other contexts we highlight their use here.