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Balayage, equilibrium measure, and Deny's principle of positivity of mass for $α$-Green potentials

Natalia Zorii

Abstract

In the theory of $g_α$-potentials on a domain $D\subset\mathbb R^n$, $n\geqslant2$, $g_α$ being the $α$-Green kernel associated with the $α$-Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,n)$, $α\leqslant2$, we establish the existence and uniqueness of the $g_α$-balayage $μ^F$ of a positive Radon measure $μ$ onto a relatively closed set $F\subset D$, we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for $μ^F(D)=μ(D)$ to hold, given in terms of the $α$-harmonic measure of suitable Borel subsets of $\overline{\mathbb R^n}$, the one-point compactification of $\mathbb R^n$. As a by-product, we find necessary and/or sufficient conditions for the existence of the $g_α$-equilibrium measure $γ_F$, $γ_F$ being understood in an extended sense where $γ_F(D)$ might be infinite. We also discover quite a surprising version of Deny's principle of positivity of mass for $g_α$-potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed.

Balayage, equilibrium measure, and Deny's principle of positivity of mass for $α$-Green potentials

Abstract

In the theory of -potentials on a domain , , being the -Green kernel associated with the -Riesz kernel of order , , we establish the existence and uniqueness of the -balayage of a positive Radon measure onto a relatively closed set , we analyze its alternative characterizations, and we provide necessary and/or sufficient conditions for to hold, given in terms of the -harmonic measure of suitable Borel subsets of , the one-point compactification of . As a by-product, we find necessary and/or sufficient conditions for the existence of the -equilibrium measure , being understood in an extended sense where might be infinite. We also discover quite a surprising version of Deny's principle of positivity of mass for -potentials, thereby significantly improving a previous result by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018). The results thus obtained are sharp, which is illustrated by means of a number of examples. Some open questions are also posed.

Paper Structure

This paper contains 17 sections, 19 theorems, 79 equations, 2 figures.

Table of Contents

  1. General conventions and preliminaries
  2. On the existence and uniqueness of the $g_\alpha$-balayage
  3. How does $\mu(D)$ vary under the $g_\alpha$-balayage?
  4. The case $\alpha<2$
  5. The case $0<\alpha\leqslant2$
  6. On the existence of the $g_\alpha$-equilibrium measure $\gamma_F$
  7. A refined version of Deny's principle of positivity of mass
  8. Proofs
  9. Preliminaries
  10. Proof of Theorem \ref{['bal-ex']}
  11. Proof of Theorem \ref{['th-char']}
  12. Proof of Theorem \ref{['th-balM1']}
  13. Proof of Theorem \ref{['th-balM2']}
  14. Proof of Theorem \ref{['th-eq']}
  15. Proof of Theorem \ref{['th-D']}
  16. ...and 2 more sections

Key Result

Theorem 2.1

For any $\mu\in\mathfrak M^+(D)$ such that $\mu|_{F^r}$ is $c_{\kappa_\alpha}$-absolutely continuous,If either of $I_{\kappa_\alpha}(\mu)$ or $I_{g_\alpha}(\mu)$ is finite, then the assumption $\mu|_{F^r}\in\breve{\mathfrak M}^+(F^r)$ is superfluous. (Note that any $\mu\in\mathfrak M^+(D)$ of finite this $\mu^F_{g_\alpha}$ is said to be the $g_\alpha$-balayage of $\mu$ onto $F$. Actually,For $\mu:

Figures (2)

  • Figure 1: The set $F_1$ in Example \ref{['ex']} with $\varrho_1(x_1)=1/x_1$.
  • Figure 2: The set $F_2$ in Example \ref{['ex']} with $\varrho_2(x_1)=\exp(-x_1)$.

Theorems & Definitions (47)

  • Remark 1.1
  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • ...and 37 more