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

Natalia Zorii

TL;DR

The paper develops a generalized theory of inner and outer pseudo-balayage for signed Radon measures, including measures of infinite energy, onto general sets with kernels satisfying the energy principle and related regularity conditions. It defines the f-weighted Gauss functional I_f and proves the existence and uniqueness of the inner pseudo-balayage hat{ω}^A, along with equivalent variational characterizations and quasi-everywhere potential equalities; it then extends these results to the outer pseudo-balayage and to signed, infinite-energy contexts. The work applies these concepts to the inner Gauss variational problem, establishing solvability criteria in terms of inner capacity c_*(A) and the mass of hat{ω}^A(X), and provides convergence results under monotone set approximations as well as a detailed treatment of preparatory and solvability results. Together, these results extend Fuglede’s framework to signed measures of infinite energy and connect pseudo-balayage with inner Gauss-type equilibrium problems, with implications for weighted minimum energy problems using kernels such as α-Riesz and Green kernels.

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.

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

TL;DR

The paper develops a generalized theory of inner and outer pseudo-balayage for signed Radon measures, including measures of infinite energy, onto general sets with kernels satisfying the energy principle and related regularity conditions. It defines the f-weighted Gauss functional I_f and proves the existence and uniqueness of the inner pseudo-balayage hat{ω}^A, along with equivalent variational characterizations and quasi-everywhere potential equalities; it then extends these results to the outer pseudo-balayage and to signed, infinite-energy contexts. The work applies these concepts to the inner Gauss variational problem, establishing solvability criteria in terms of inner capacity c_*(A) and the mass of hat{ω}^A(X), and provides convergence results under monotone set approximations as well as a detailed treatment of preparatory and solvability results. Together, these results extend Fuglede’s framework to signed measures of infinite energy and connect pseudo-balayage with inner Gauss-type equilibrium problems, with implications for weighted minimum energy problems using kernels such as α-Riesz and Green kernels.

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.
Paper Structure (14 sections, 19 theorems, 122 equations)

This paper contains 14 sections, 19 theorems, 122 equations.

Table of Contents

  1. Introduction and general conventions
  2. General conventions
  3. Inner and outer pseudo-balayage
  4. Proofs of Theorems \ref{['th1']} and \ref{["th1'"]}
  5. Proof of Theorem \ref{['th1']}
  6. Proof of Theorem \ref{["th1'"]}
  7. Convergence of pseudo-balayage for monotone families of sets
  8. Pseudo-balayage in the inner Gauss variational problem
  9. Proof of Theorem \ref{['th-solv1']}
  10. Preparatory results
  11. Proof of Theorem \ref{['th-solv1']}
  12. Acknowledgements
  13. A data availability statement
  14. Funding and Competing interests

Key Result

Theorem 2.1

There is one and the same measure $\widehat{\omega}^A\in\mathcal{E}^+_f(A)$, called the inner pseudo-balayage of $\omega$ onto $A$, that satisfies any one of the following three assertions.

Theorems & Definitions (45)

  • Example 1.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.1
  • proof
  • ...and 35 more