On Fuglede's problem on pseudo-balayage for signed Radon measures of infinite energy

TL;DR

This paper extends Fuglede's pseudo-balayage theory to include signed Radon measures of infinite energy, providing new tools for potential theory and solving an open problem posed by Fuglede.

Contribution

The paper develops a comprehensive theory of inner and outer pseudo-balayage for signed Radon measures of infinite energy on general sets, extending previous finite-energy results.

Findings

Extended pseudo-balayage theory to measures of infinite energy

Solved Fuglede's open problem from 2016

Applied the theory to weighted minimum energy problems

Abstract

For suitable kernels on a locally compact space, we develop a theory of inner (outer) pseudo-balayage of quite general signed Radon measures (not necessarily of finite energy) onto quite general sets (not necessarily closed). Such investigations were initiated in Fuglede's study (Anal. Math., 2016), which was, however, mainly concerned with the outer pseudo-balayage of positive measures of finite energy. The results thereby obtained solve Fuglede's problem, posed to the author in a private correspondence (2016), whether his theory could be extended to measures of infinite energy. An application of this theory to weighted minimum energy problems is also given.