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Balayage of Radon measures of infinite energy on locally compact spaces

Natalia Zorii

Abstract

For suitable kernels on a locally compact space $X$, we develop a theory of inner balayage of quite general Radon measures $ω$ (not necessarily of finite energy) to arbitrary $A\subset X$. In the case where $A$ is Borel, this theory provides, as a by-product, a theory of outer balayage. We prove the existence and the uniqueness of inner (outer) swept measures, analyze their properties, and provide a number of alternative characterizations. In spite of being in agreement with Cartan's theory of Newtonian balayage, the results obtained require essentially new methods and approaches, since in the case in question, useful specific features of Newtonian potentials may fail to hold. The theory thereby established generalizes substantially the existing ones, pertaining either to $ω$ of finite energy, or to some particular $A$ (e.g. quasiclosed). This work covers many interesting kernels in classical and modern potential theory, which looks promising for possible applications.

Balayage of Radon measures of infinite energy on locally compact spaces

Abstract

For suitable kernels on a locally compact space , we develop a theory of inner balayage of quite general Radon measures (not necessarily of finite energy) to arbitrary . In the case where is Borel, this theory provides, as a by-product, a theory of outer balayage. We prove the existence and the uniqueness of inner (outer) swept measures, analyze their properties, and provide a number of alternative characterizations. In spite of being in agreement with Cartan's theory of Newtonian balayage, the results obtained require essentially new methods and approaches, since in the case in question, useful specific features of Newtonian potentials may fail to hold. The theory thereby established generalizes substantially the existing ones, pertaining either to of finite energy, or to some particular (e.g. quasiclosed). This work covers many interesting kernels in classical and modern potential theory, which looks promising for possible applications.

Paper Structure

This paper contains 15 sections, 25 theorems, 87 equations.

Table of Contents

  1. Introduction and general conventions
  2. General conventions
  3. Inner and outer balayage
  4. Inner balayage
  5. On the existence and alternative characterizations of $\omega^A$
  6. Outer balayage
  7. Proofs of Theorems \ref{['th1']} and \ref{['th2']}
  8. Proof of Theorem \ref{['th1']}
  9. Proof of Theorem \ref{['th2']}
  10. Some further properties of balayage
  11. Balayage of signed measures
  12. Proof of Theorem \ref{['l-alt-countt']}
  13. Preliminaries
  14. Proof of Theorem \ref{['l-alt-countt']}
  15. Acknowledgements

Key Result

Lemma \oldthetheorem

The inner balayage $\omega^A$ is unique (if it exists).

Theorems & Definitions (54)

  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • ...and 44 more