Classical Balayage of Charges and Measures

TL;DR

This paper explores the properties of balayage of charges and measures within subharmonic functions, emphasizing how these relate to the geometry of domains in Euclidean space.

Contribution

It provides new insights into the relationship between balayage processes and the geometric structure of domains for subharmonic functions.

Findings

Characterization of balayage properties for charges and measures.

Connection between balayage and domain geometry.

Implications for subharmonic function theory.

Abstract

We investigate some properties of balayage of charges and measures for subclasses of subharmonic functions and their relationship to the geometry of domain or open set in finite-dimensional Euclidean space where this balayage is considered.

Equations

$$g(v)\le f(v)\text{for all v∈V}.$$

$$g(v):={\int }_{D}v\mathrm{d}\vartheta \le {\int }_{D}v\mathrm{d}\mu =:f(v)\text{for all v∈V},$$

$${\mathbb{R}}_{-\infty }:=\{-\infty \}\cup \mathbb{R},{\mathbb{R}}_{+\infty }:=\mathbb{R}\cup \{+\infty \},$$

$${\mathbb{R}}^{+}:=\{x\in \mathbb{R}:x\ge 0\},{\mathbb{R}}_{+\infty }^{+}:={\mathbb{R}}^{+}\cup \{+\infty \},$$

$$S^{+}:=\{x\ge 0:x\in S\},S_{\ast }:=S\setminus \{0\}$$

$$x\cdot (\pm \infty ):=\pm \infty =:(-x)\cdot (\mp \infty )$$

$$\frac{x}{\pm \infty }:=0\text{for x∈R},$$

$$B(x,r):=\{x^{\prime }\in {\mathbb{R}}^d:|x^{\prime }-x|<r\},$$

$$B(\infty ,r):=\{x\in {\mathbb{R}}_{\infty }^d:|x|>1/r\},$$

$$B(r):=B(0,r),\mathbb{B}:=B(0,1),$$

$$B_{\circ }(x,r):=B(x,r)\setminus \{x\},$$

$${\mathbb{R}}_{-\infty }^X\stackrel{\text{eqref}df:Rr}{:=}({\mathbb{R}}_{-\infty }{)}^X,{\mathbb{R}}_{+\infty }^X\stackrel{\text{eqref}df:Rr}{:=}({\mathbb{R}}_{+\infty }{)}^X,{\mathbb{R}}_{\pm \infty }^X\stackrel{\text{eqref}df:Rr}{:=}({\mathbb{R}}_{\pm \infty }{)}^X.$$

$$\begin{array}{rl}{\mathrm{Dom}}_{-\infty }:=f^{-1}({\mathbb{R}}_{-\infty })\subset X,& {\mathrm{Dom}}_{+\infty }f:=f^{-1}({\mathbb{R}}_{+\infty })\subset X,\\ \mathrm{Dom}f:=f^{-1}({\mathbb{R}}_{\pm \infty })& ={\mathrm{Dom}}_{-\infty }f\bigcup {\mathrm{Dom}}_{+\infty }f\subset X,\\ \mathrm{dom}f:=f^{-1}(\mathbb{R})=& {\mathrm{Dom}}_{-\infty }f\bigcap {\mathrm{Dom}}_{+\infty }f\subset X,\end{array}$$

$${\mathrm{sbh}}_{\ast }(O):=\mathrm{sbh}(O)\setminus \{-\infty \},{\mathrm{sbh}}^{+}(O):=(\mathrm{sbh}(O){)}^{+}.$$

$${\star }_o:x⟼x^{{\star }_o}$$

$$u^{{\star }_o}(x^{{\star }_o})$$

$$\displaystyle\Bigl{(}u\in\operatorname{sbh}(O)\Bigr{)}\Longleftrightarrow\Bigl{(}u^{\star_{o}}\in\operatorname{sbh}(O^{\star_{o}})\Bigr{)}.$$

$$\operatorname{sbh}^{+}(S)\overset{\eqref{sbh}}{:=}\bigl{\{}u\in\operatorname{sbh}(S)\colon u\geq 0\text{ on }S\bigr{\}}.$$

$${\mathrm{Meas}}^{+}(S)$$

$${\mathrm{Meas}}^{1+}(S)$$

$$L\,{\rm d}\mu:=\bigl{\{}\nu\in\operatorname{Meas}(S)\colon\text{\it there exists $g\in L$ such that\/ $\,{\rm d}\nu=g\,{\rm d}\mu$}\bigr{\}}$$

$$\mu(x,r):=\mu\bigl{(}B(x,r)\bigr{)}\text{ if $B(x,r)\overset{\eqref{B}}{\subset}S$}.$$

$$s_{d-1}:=\frac{2{\pi }^{d/2}}{\Gamma (d/2)}$$

$$\varDelta_{u}:=c_{d}{\bigtriangleup}u\in\operatorname{Meas}^{+}(O),\quad c_{d}\overset{\eqref{sd-1}}{:=}\frac{1}{s_{d-1}(1+(d-3)^{+})}=\frac{\Gamma(d/2)}{2\pi^{d/2}\max\{1,d-2\bigr{\}}}.$$

$${\varkappa }_p(S)$$

$$b_p$$

$$\operatorname{\rm hull-in}_{O}K:=K\bigcup\Bigl{(}\bigcup\{C\in\operatorname{Conn}_{O}(O\setminus K)\colon C\Subset O\}\Bigr{)}.$$

$$\varnothing \ne O={\mathrm{int}}_{{\mathbb{R}}_{\infty }^d}O=\underset{j\in N_O}{⨆}{D}_j\ne {\mathbb{R}}_{\infty }^d,j\in N_O\subset \mathbb{N},D_j={\mathrm{conn}}_{{\mathbb{R}}_{\infty }^d}(O,x_j),$$

$$\varnothing \ne D\ne {\mathbb{R}}_{\infty }^d\text{is an open connected subset, i.e., a domain}.$$

$${\mathrm{hull}-\mathrm{in}}_{O}K=O_{\infty }\setminus {\mathrm{conn}}_{O_{\infty }\setminus K}(\infty );$$

$$\#{\mathrm{Conn}}_{{\mathbb{R}}_{\infty }^d}({\mathbb{R}}^d\setminus {\mathrm{hull}-\mathrm{in}}_{O}K)<\infty .$$

$$\int h\mathrm{d}\vartheta \stackrel{\text{eqref}b0mu}{\le }\int h\mathrm{d}\mu \text{for all/ h∈H.}$$

$$\boxed{H\subset \mathrm{sbh}(S)\subset {\mathbb{R}}_{-\infty }^S}.$$

$$a{\delta }_x+b{\omega }_D(x,\cdot )\in J_x(O),a,b\in {\mathbb{R}}^{+},a+b=1.$$

$$\sum _{k\in \mathbb{N}}{b}_k=1,b_k\in {\mathbb{R}}^{+},D_k,D:=\bigcup _k{D}_k⋐O\text{ are Greenian, then ∑kbkωDk(x,⋅)∈Jx(O).}$$

$$\alpha_{r}^{\infty}\overset{\eqref{MLl}}{\in}\left(C_{0}^{\infty}(r\mathbb{B})\right)^{+}\,{\rm d}\lambda_{d}\in\operatorname{Meas}^{1+}\bigl{(}B(r)\bigr{)},\quad\alpha_{r}^{\infty}(S)=\alpha_{1}^{\infty}(S/r),\quad S\in\operatorname{Borel}(\mathbb{R}^{d}),$$

Full text

MSC 2010: 31B5, 3A05, 3C05, 31C15, 28A25

Keywords: balayage, sweeping out, potential, measure, charge, subharmonic function, harmonic function, polar set, harmonic continuation

1 Introduction

We have are considered in the survey [14] various general concepts of balayage. In this article we deal with a particular case of such balayage with respect to special classes of test subharmonic functions. The general concept of balayage can be defined as follows. Let $F$ be a set and $(R,\leq)$ be a (pre-)ordered set with (pre-)order relation $\leq$. A function $f\colon F\to R$ can be called a balayage of a function $g\in F\to R$ for a subset $\mathcal{V}\subset F$, and we write $g\preceq_{\mathcal{V}}f$, if the function $f$ majorizes the function $g$ on $\mathcal{V}$:

[TABLE]

In this article, we use the balayage when $F$ is a class of integrals defined by charges or positive measures on a subdomain $D$ of finite-dimensional Euclidean space, and classes $\mathcal{V}$ are special subclasses of subharmonic functions on $D$. In this case, relation (1.1) turns into inequalities of the form

[TABLE]

where $\vartheta$ and $\mu$ is a pair of charges or measures.

The main and some special properties of charges and measures are outlined in Sec. 3, Theorems 2–7, Examples 1–5.

2 Definitions, notations and conventions

The reader can skip this Section 2 and return to it only if necessary.

2.1 Sets, order, topology

As usual, $\mathbb{N}:=\{1,2,\dots\}$, $\mathbb{R}$ and $\mathbb{C}$ are the sets of all *natural, real * and complex numbers, respectively; $\mathbb{N}_{0}:=\{0\}\cup\mathbb{N}$ is French natural series, and $\mathbb{Z}:=\mathbb{N}_{0}\cup\mathbb{N}_{0}$.

For $d\in\mathbb{N}$ we denote by $\mathbb{R}^{d}$ the $d$-dimensional real Euclidean space with the standard Euclidean norm $|x|:=\sqrt{x_{1}^{2}+\dots+x_{d}^{2}}$ for $x=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$ and the distance function $\operatorname{dist}(\cdot,\cdot)$. For the real line $\mathbb{R}=\mathbb{R}^{1}$ with Euclidean norm-module $|\cdot|$,

[TABLE]

unless otherwise specified. An open connected (sub-)set of $\mathbb{R}_{\pm\infty}$ is a (sub-)interval of $\mathbb{R}_{\pm\infty}$. The Alexandroff one-point compactification of $\mathbb{R}^{d}$ is denoted by $\mathbb{R}^{d}_{\infty}:=\mathbb{R}^{d}\cup\{\infty\}$.

The same symbol [math] is used, depending on the context, to denote the number zero, the origin, zero vector, zero function, zero measure, etc. The positiveness is everywhere understood as $\geq 0$ according to the context. Given $x\in\mathbb{R}^{d}$ and222A reference mark over a symbol of (in)equality, inclusion, or more general binary relation, etc. means that this relation is somehow related to this reference. $r\overset{\eqref{df:R+}}{\in}\mathbb{R}_{+\infty}^{+}$, we set

[TABLE]

Thus, the basis of open (respectively closed) neighborhood of the point $x\in\mathbb{R}_{\infty}^{d}$ is open (respectively closed) balls $B(x,r)$ (respectively $\overline{B}(x,r)$) centered at $x$ with radius $r>0$.

Given a subset $S$ of $\mathbb{R}^{d}_{\infty}$, the closure $\operatorname{clos}S$, the* interior* $\operatorname{int}S$ and the boundary $\partial S$ will always be taken relative $\mathbb{R}^{d}_{\infty}$. For $S^{\prime}\subset S\subset\mathbb{R}^{d}_{\infty}$ we write $S^{\prime}\Subset S$ if $\operatorname{clos}S^{\prime}\subset\operatorname{int}S$. An open connected (sub-)set of $\mathbb{R}^{d}_{\infty}$ is a (sub-)domain of $\mathbb{R}^{d}_{\infty}$.

2.2 Functions.

Let $X,Y$ are sets. We denote by $Y^{X}$ the set of all functions $f\colon X\to Y$. The value $f(x)\in Y$ of an arbitrary function $f\in X^{Y}$ is not necessarily defined for all $x\in X$. The restriction of a function f to $S\subset X$ is denoted by f\bigm{|}_{S}. We set

[TABLE]

A function $f\in\mathbb{R}_{\pm\infty}^{X}$ is said to be extended numerical. For extended numerical functions $f$, we set

[TABLE]

For $f,g\in\mathbb{R}_{\pm\infty}^{X}$ we write $f=g$ if $\operatorname{Dom}f=\operatorname{Dom}g=:D$ and $f(x)=g(x)$ for all $x\in D$, and we write $f\leq g$ if $f(x)\leq g(x)$ for all $x\in D$. For $f\in\mathbb{R}_{\pm\infty}^{X}$, $g\in\mathbb{R}_{\pm\infty}^{Y}$ and a set $S$, we write ‘‘$f=g$ on $S$ ’’ or ‘‘$f\leq g$ on $S$ ’’ if f\bigm{|}_{S\cap D}=g\bigm{|}_{S\cap D} or f\bigm{|}_{S\cap D}\leq g\bigm{|}_{S\cap D} respectively.

For $f\in F\subset\mathbb{R}_{\pm\infty}^{X}$, we set $f^{+}\colon x\mapsto\max\{0,f(x)\}$, $x\in\operatorname{Dom}f$, $F^{+}:=\{f\geq 0\colon f\in F\}$. So, $f$ is positive on $X$ if $f=f^{+}$, and we write ‘‘$f\geq 0$ on $X$’’.

For topological space $X$, $C(X)\subset\mathbb{R}^{X}$ is the vector space over $\mathbb{R}$ of all continuous functions. We denote the function identically equal to resp. $-\infty$ or $+\infty$ on a set by the symbols $\boldsymbol{-\infty}$ or $\boldsymbol{+\infty}$.

For an open set $O\subset\mathbb{R}^{d}_{\infty}$, we denote by $\operatorname{har}(O)$ and $\operatorname{sbh}(O)$ the classes of all harmonic (locally affine for m = 1) and subharmonic (locally convex for $m=1$) functions on $O$, respectively. The class $\operatorname{sbh}(O)$ contains the minus-infinity function $-\infty$;

[TABLE]

If $o\notin O\ni\infty$, then we can to use the inversion in the sphere $\partial B(o,1)$ centered at $o\in\mathbb{R}^{d}$:

[TABLE]

For a subset $S\subset\mathbb{R}_{\infty}^{d}$, the classes $\operatorname{har}(S)$, $\operatorname{sbh}(S)$, and $C^{k}(S)$ with $k\in\mathbb{N}\cup\{\infty\}$ consist of the restrictions to $S$ of harmonic, subharmonic, and k times continuously differentiable functions in some (in general, its own for each function) open set $O\subset\mathbb{R}_{\infty}^{d}$ containing $S$. A class $\operatorname{sbh}_{*}(S)$ is defined like previous class (2.5),

[TABLE]

By ${\rm const}_{a_{1},a_{2},\dots}\in\mathbb{R}$ we denote constants, and constant functions, in general, depend on $a_{1},a_{2},\dots$ and, unless otherwise specified, only on them, where the dependence on dimension $d$ of $\mathbb{R}_{\infty}^{d}$ will be not specified and not discussed; ${\rm const}^{+}_{\dots}\geq 0$.

2.3 Measures and charges.

Let $\operatorname{Borel}(S)$ be the class of all Borel subsets in $S\in\operatorname{Borel}(\mathbb{R}_{\infty}^{d})$. We denote by $\operatorname{Meas}(S)$ the class of all Borel signed measures, or, charges on $S\in{\operatorname{Borel}}(\mathbb{R}_{\infty}^{d})$; $\operatorname{Meas}_{\operatorname{c}}(S)$ is the class of charges $\mu\in\operatorname{Meas}(S)$ with a compact support $\operatorname{supp}\mu\Subset S$;

[TABLE]

For a charge $\mu\in\operatorname{Meas}(S)$, we let $\mu^{+}$, $\mu^{-}:=(-\mu)^{+}$ and $|\mu|:=\mu^{+}+\mu^{-}$ respectively denote its upper, lower, and total variations. So, $\delta_{x}\in\operatorname{Meas}_{\operatorname{c}}^{1+}(S)$ is the Dirac measure at a point $x\in S$, i.e., $\operatorname{supp}\delta_{x}=\{x\}$, $\delta_{x}(\{x\})=1$. We denote by \mu\bigm{|}_{S^{\prime}} the restriction of $\mu$ to $S^{\prime}\in{\operatorname{Borel}}(\mathbb{R}_{\infty}^{d})$.

If the Kelvin transform (2.6) translates the subharmonic function $u$ into another function $u^{\star}_{o}$ (2.6u), then its Riesz measure $\upsilon$ is transformed common use image under its own mapping-inversion of type $1$ or $2$. These rules are described in detail in L. Schwartz’s monograph [22, Vol. I,Ch.IV, § 6] and we do not dwell on them here, although here interesting questions arise, for example, for the Bernstein – Paley – Wiener –Mary Cartwright classes of entire functions [11], [16], [1], [15] etc.

Given $S\in{\operatorname{Borel}}(\mathbb{R}_{\infty}^{d})$ and $\mu\in\operatorname{Meas}(S)$, the class $L^{1}_{\operatorname{loc}}(S,\mu)$ consists of all extended numerical locally integrable functions with respect to the measure $\mu$ on $S$; $L^{1}_{\operatorname{loc}}(S):=L^{1}_{\operatorname{loc}}(S,\lambda_{d})$. For $L\subset L^{1}_{\operatorname{loc}}(S,\mu)$, we define a subclass

[TABLE]

of the class of all absolutely continuous charges with respect to $\mu$. For $\mu\in\operatorname{Meas}(S)$, we set

[TABLE]

Let ${\bigtriangleup}$ be the the Laplace operator acting in the sense of the theory of distributions, $\Gamma$ be the gamma function,

[TABLE]

be the surface area of the $(d-1)$-dimensional unit sphere $\partial\mathbb{B}$ embedded in $\mathbb{R}^{d}$. For function $u\in\operatorname{sbh}_{*}(O)$, the Riesz measure of $u$ is a Borel (or Radon [20, A.3]) * positive measure *

[TABLE]

In particular, $\Delta_{u}(S)<+\infty$ for each subset $S\Subset O$. By definition, $\Delta_{\boldsymbol{-\infty}}(S):=+\infty$ for all $S\subset O$.

We use different variants of outer Hausdorff $p$-measure $\varkappa_{p}$ with $p\in\mathbb{N}_{0}$:

[TABLE]

Thus, for $p=0$, for any $S\subset\mathbb{R}^{d}$, its Hausdorff [math]-measure $\varkappa_{0}(S)$ is to the cardinality $\#S$ of $S$, for $p=d$ we see that $\varkappa_{d}\overset{\eqref{df:spH}}{=:}\lambda_{d}$ is the Lebesgue measure to Borel proper subsets $S\subset\mathbb{R}_{\infty}^{d}$, where, if $\infty\in S$, we preliminary use the inversion(2.6u), and \sigma_{d-1}:=\varkappa_{d-1}\bigm{|}_{\partial\mathbb{B}} is the $(d-1)$-dimensional surface measure of area on the unit sphere $\partial\mathbb{B}$ in the usual sense.

2.4 Topological concepts. Inward-filled hull of set in open set

Let $O$ be a topological space, $S\subset O$, $x\in O$. We denote by $\operatorname{Conn}_{O}S$ and $\operatorname{conn}_{O}(S,x)\in\operatorname{Conn}_{O}S$ a set of all connected components of $S$ and its connected component containing $x$. We write $\operatorname{clos}_{O}S$, $\operatorname{int}_{O}S$, and $\partial_{O}S$ for the closure, the* interior,* and the boundary of $S$ in $O$. The set $S$ is $O$-precompact if $\operatorname{clos}_{O}S$ is a compact subset of $O$, and we write $S\Subset O$.

Definition 1**.**

An arbitrary $O$-precompact connected component of $O\setminus S$ is called a hole in $S$ with respect to $O$. The union of a subset $K\subset O$ with all holes in it will be called an inward-filled hull of this set $K$ with respect to $O$ and is denoted further as

[TABLE]

Denote by $O_{\infty}$ the Alexandroff one-point compactification of $O$ with underlying set $O\sqcup\{\infty\}$, where $\sqcup$ is the disjoint union of $O$ with a single point $\infty\notin O$. If this space $O$ is a topological subspace of some ambient topological space $T\supset O$, then this point $\infty$ can be identified with the boundary $\partial O\subset T$ , considered as a single point $\{\partial O\}$.

Throughout this article, we use these topological concepts only in cases when $O$ is an open non-empty proper Greenian open set [12, Ch.5, 2] of $\mathbb{R}_{\infty}^{d}=:T$, i. e.,

[TABLE]

For an open set $O$ from (2.15O), we often use statements that are proved in our references only for domains $D$ from (2.15D). This is acceptable since all such cases concern only to individual domains-components $D_{j}$. So, if $S\Subset O$, then $S$ meets only finite many components $D_{j}$. In addition, we give proofs of our statements only for cases $O,D\subset\mathbb{R}^{d}$. If we have $o\notin D_{j}=D\ni\infty$, then we can to use the inversion relative to the sphere $\partial B(o,1)$ centered at $o\in\mathbb{R}^{d}$ as in (2.6).

Theorem 1** ([7, 6.3], [8]).**

Let $K$ be a compact set in an open set $O\subset\mathbb{R}^{d}$. Then

1. (i)

$\operatorname{\rm hull-in}_{O}K$* is a compact subset in $O$;* 2. (ii)

the set $O_{\infty}\setminus\operatorname{\rm hull-in}_{O}K$ is connected and locally connected subset in $O_{\infty}$; 3. (iii)

the inward-filled hull of $K$ with respect to $O$ coincides with the complement in $O_{\infty}$ of connected component of $O_{\infty}\setminus K$ containing the point $\infty$, i. e.,

[TABLE] 4. (iv)

if $O^{\prime}\subset\mathbb{R}_{\infty}^{d}$ is an open subset and $O\subset O^{\prime}$ then $\operatorname{\rm hull-in}_{O}K\subset\operatorname{\rm hull-in}_{O^{\prime}}K$; 5. (v)

$\mathbb{R}^{d}\setminus\operatorname{\rm hull-in}_{O}K$* has only finitely many components, i. e.,*

[TABLE]

3 Properties of balayage of charges and measures

In this section 3 we discuss conventional classical balayage that is particular case of (1.1).

Definition 2**.**

Let $\vartheta,\mu\in\operatorname{Meas}(S)$, $S\subset\operatorname{Borel}(\mathbb{R}^{d}_{\infty})$. Let $H\subset\mathbb{R}_{\pm\infty}^{S}$ be a class of Borel-measurable functions on $S$. Let us assume that the integrals $\int h\,{\rm d}{\vartheta}$ and $\int h\,{\rm d}{\mu}$ are well defined with values in $\mathbb{R}_{\pm\infty}$ for each function $h\in H$. We write ${\vartheta}\preceq_{H}\mu$ and say that the charge $\mu$ is a balayage, or, sweeping (out), of the charge ${\vartheta}$ for $H$, or, briefly, $\mu$ is a $H$-balayage of $\vartheta$, if

[TABLE]

In this article, we consider only balayage for

[TABLE]

In this case, the integrals from (3.1) with values in $\mathbb{R}_{-\infty}$ are well defined for all measures ${\vartheta},\mu\in\operatorname{Meas}_{\operatorname{c}}^{+}(S)$, and, with values in $\mathbb{R}$, for all absolutely continuous (with respect to $\lambda_{d}$) charges ${\vartheta},\mu\overset{\eqref{MLl}}{\in}L_{\operatorname{loc}}^{1}(S)\,{\rm d}\lambda_{d}$ etc.

Theorem 2**.**

Let $O\subset\mathbb{R}^{d}$ be an open set, $\mu\in\operatorname{Meas}(O)$ be a $H$-balayage of ${\vartheta}\in\operatorname{Meas}(O)$, $O^{\prime}\subset\mathbb{R}^{d}$ be an open set, and $H^{\prime}\subset\mathbb{R}_{\pm\infty}^{O^{\prime}}$.

If $1\in H$, then ${\vartheta}(O)\leq\mu(O)$. 2. 2.

If $\pm 1\in H$, then ${\vartheta}(O)=\mu(O)$. 3. 3.

If $H^{\prime}\subset H$, then $\mu$ is a $H^{\prime}$-balayage of ${\vartheta}$. 4. 4.

If $O^{\prime}\subset O$ and $supp\vartheta\cup\operatorname{supp}\mu\subset O^{\prime}$, then \mu\bigm{|}_{O^{\prime}} is a balayage of {\vartheta}\bigm{|}_{O^{\prime}} for H\bigm{|}_{O^{\prime}}.

All statements of Theorem 2 are obvious.

Remark 1**.**

Balayage of charges and measures with a non-compact support is also occur frequently and are used in Analysis. So, a bounded domain $D\subset\mathbb{R}^{d}$ is called a quadrature domain (for harmonic functions) if there is a charge $\mu\in\operatorname{Meas}_{\operatorname{c}}(D)$ such that the restriction \lambda_{d}\bigm{|}_{D} is a balayage of $\mu$ for the class $\operatorname{har}(D)\cap L^{1}(D)$. In connection with the quadrature domains, see very informative overview [10, 3] and bibliography in it.

Theorem 3**.**

If $\mu\in\operatorname{Meas}^{+}_{\operatorname{c}}(O)$ is a balayage of $\vartheta\in\operatorname{Meas}_{\operatorname{c}}^{+}(O)$ for \bigl{(}\operatorname{sbh}(O)\cap C^{\infty}(O)\bigr{)}, then $\mu$ is a balayage of $\vartheta$ for $\operatorname{sbh}(O)$.

Theorem 3 follows from [5, Ch. 4, 10, Approximation Theorem].

Example 1** ([6], [3], [4], [21]).**

Let $x\in O$. If a measure $\mu\in\operatorname{Meas}_{\operatorname{c}}^{+}(O)$ is a balayage of the Dirac measure $\delta_{x}$ for $\operatorname{sbh}(O)$, then this measure $\mu$ is called a Jensen measure for $x$. The class of such measures is denoted by $J_{x}(O)$.

Example 2**.**

We denote by $\omega_{D}\colon D\times{\operatorname{Borel}}(\partial D)\to[0,1]$ the harmonic measure for $D$ with non-polar boundary $\partial D\subset\mathbb{R}_{\infty}^{d}$. Measures $\omega_{D}(x,\cdot)\in\operatorname{Meas}^{+}(\partial D)$, $x\in D$, will also be called a harmonic measure for $D$, but with specification, at $x\in D$. If $D\Subset O$, then measures

[TABLE]

Likewise, if

[TABLE]

So, the surface measure $\sigma_{d-1}$ in the unit sphere $\partial\mathbb{B}$ belong to $J_{0}(r\mathbb{D})$ for any $r>1$.

Example 3**.**

Useful examples of Jensen measures from $J_{0}(B(r))$ are probability measures

[TABLE]

$r\in\mathbb{R}_{*}^{+}$, invariant under the action of the orthogonal group on $\mathbb{R}^{d}$.

Example 4** ([6], [13]).**

Let $x\in O$. If $\mu\in\operatorname{Meas}_{\operatorname{c}}^{+}(O)$ is a balayage of $\delta_{x}$ for $\operatorname{har}(O)$, then such measure $\mu$ is called a Arens – Singer measure for $x$. The class of such measures is denoted by $AS_{x}(O)\supset J_{x}(O)$. Arens – Singer measures are often referred to as representing measures.

Theorem 4**.**

For $H\overset{\eqref{hHS}}{\subset}\operatorname{sbh}(O)$ (resp., $H=-H\subset\operatorname{har}(O)$), let $\mu\in\operatorname{Meas}_{\operatorname{c}}(O)$ be a balayage of $\vartheta\in\operatorname{Meas}_{\operatorname{c}}$ for $H$. Let $\iota_{x}\in J_{x}(O)\subset\operatorname{Meas}^{1+}_{\operatorname{c}}(O)$ (resp., $\iota_{x}\in AS_{x}(O)\subset\operatorname{Meas}^{1+}_{\operatorname{c}}(O)$) with

[TABLE]

be a family Jensen (resp., Arens – Singer) measures for points $x\in O$. The measure $\mu$ and probability measures $\iota_{x}$ are bounded in aggregate, and we can to define the integral of $\iota_{x}$ with respect to $\mu$ [17], [2]**

[TABLE]

In particular, if every Jensen (resp., Arens – Singer) measure $\iota_{x}$ is a parallel shift to a point $x\in O$ of the same Jensen (resp., Arens – Singer) measure $\iota_{0}$ for [math] with the diameter $\operatorname{diam}\operatorname{supp}\iota_{0}$ of $\operatorname{supp}\iota_{0}$ fewer than \frac{1}{2}\operatorname{dist}(\operatorname{supp}\mu,\partial O)\bigr{)}, then integral $\beta$ from (3.6) is a classical convolution $\beta=\iota_{0}*\mu$ of two measures $\iota_{0}$ and $\mu$:

[TABLE]

In these cases both measures $\beta$ from (3.6)–(3.7) also a balayage of $\vartheta$ for $H$ with

[TABLE]

Proof.

Under condition (3.5), for subharmonic function $h\in H\subset\operatorname{sbh}(O)$, we have

[TABLE]

by definitions (3.6)–(3.7). For $H\subset\operatorname{har}(O)$ and $\iota_{x}\in AS_{x}$, by analogy with (3.8), we have equalities in (3.8). ∎

Remark 2**.**

If we choose parallel shifts to $x$ of measures (Example 3, (3.4))

[TABLE]

as measures $\iota_{x}$ for Theorem 4 with a function $r\in C^{\infty}(O)$ and with condition (3.5), then our measures $\beta$ from from (3.6)–(3.7) both measures belong to the class $C_{0}^{\infty}(O)\,{\rm d}\lambda_{d}$ and still $\vartheta\preceq_{H}\beta$, i.e., the measure $\beta\in\operatorname{Meas}_{\operatorname{c}}(O)$ is a balayage of the measure $\vartheta$ for $H$.

Theorem 5**.**

Let $\mu\in\operatorname{Meas}_{\operatorname{c}}(O)$ be a balayage of ${\vartheta}\in\operatorname{Meas}_{\operatorname{c}}(O)$ for $\operatorname{har}(O)$. Then

[TABLE]

(see Subsec. 2.4, Definition 1 of inward-filled hull of compact subset $\operatorname{supp}\mu\cup\operatorname{supp}{\vartheta}$ in $O$).**

Proof.

We set

[TABLE]

By Theorem 1 and [7, Theorem 1.7], if h\in\operatorname{har}\bigl{(}{\operatorname{\rm hull-in}_{O}K}\bigr{)}, then there are functions $h_{k}\in\operatorname{har}(O)$, $k\in\mathbb{N}$, such that the sequence $(h_{k})_{k\in\mathbb{N}}$ converges to $h$ in C\bigl{(}{\operatorname{\rm hull-in}_{O}K}\bigr{)}, and

[TABLE]

Using the opposite function -h\in\operatorname{har}\bigl{(}{\operatorname{\rm hull-in}_{O}K}\bigr{)}, we have the inverse inequality. ∎

Theorem 6**.**

Let $\mu\in\operatorname{Meas}_{\operatorname{c}}(O)$ be a balayage of ${\vartheta}\in\operatorname{Meas}_{\operatorname{c}}(O)$ for $\operatorname{sbh}(O)$. Then

[TABLE]

i. e., if $O^{\prime}\supset\operatorname{\rm hull-in}_{O}(\operatorname{supp}\mu\cup\operatorname{supp}{\vartheta})$ is an open set, then $\mu$ is a $\operatorname{sbh}(O^{\prime})$-balayage of $\vartheta$.

Proof.

We use the notation (3.10). By Theorem 1, if u\in\operatorname{sbh}\bigl{(}{\operatorname{\rm hull-in}_{O}K}\bigr{)}, then there is a function U\in\operatorname{sbh}\bigl{(}O) such that $u=U$ on ${\operatorname{\rm hull-in}_{O}K}$ [7, Theorem 6.1], [9, Theorem 1], [8, Theorem 16], and we have

[TABLE]

that gives (3.11). ∎

Theorem 7**.**

If $\mu\in\operatorname{Meas}_{\operatorname{c}}^{+}(O)$ is a $\operatorname{sbh}(O)$-balayage of a measure ${\vartheta}\in\operatorname{Meas}_{\operatorname{c}}^{+}(O)$, and a set $E$ is polar, then $\mu(O\cap E\setminus\operatorname{supp}{\vartheta})=0$.

Proof.

There is $k_{0}\in\mathbb{N}$ such that $B(x,1/k_{0})\Subset O$ for all $x\in\operatorname{supp}{\vartheta}$. For any $k\in k_{0}+\mathbb{N}_{0}$ there exists an finite cover of $\operatorname{supp}{\vartheta}$ by balls $B(x_{j},1/k)\Subset O$ such that the open subsets

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have complements $\mathbb{R}_{\infty}^{d}\setminus O_{k}$ in $\mathbb{R}_{\infty}^{d}$ without isolated points. Then every open set $O_{k}\Subset O$ is regular for the Dirichlet problem. It suffices to prove that the equality $\mu(\mathcal{O}_{k}\cap E)=0$ holds for every number $k\in k_{0}+\mathbb{N}_{0}$. By definition of polar sets, there is a function $u\in\operatorname{sbh}_{*}(O)$ such that $u(E)=\{-\infty\}$. Consider the functions

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We have $U_{k}\in\operatorname{sbh}_{*}(O)$, and $U_{k}$ is bounded below in $\operatorname{supp}{\vartheta}\Subset O_{k}$. Hence

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Thus, we have $\mu(O_{k}\cap E)=0$. ∎

Generally speaking, Theorem 7 is not true for $\operatorname{har}(O)$-balayage. An implicit example built in [18, Example]. We will indicate in Example 5 one more constructive and general way of building in this direction

Example 5** (development of one example of T. Lyons [19, XIB2]).**

Consider

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Easy to see that $\vartheta\preceq_{\operatorname{sbh}(\mathbb{B})}\mu$. Let $E=(e_{j})_{j\in\mathbb{N}}\Subset r\mathbb{B}\setminus r_{0}\overline{\mathbb{B}}$ be a polar countable set without limit point in $r\mathbb{B}\setminus r_{0}\overline{\mathbb{B}}$. Surround each point $e_{j}\in E$ with a ball $B(e_{j},r_{j})$ of such a small radius $r_{j}>0$ that the union of all these balls is contained in $r\mathbb{B}\setminus r_{0}\overline{\mathbb{B}}$. Consider a measure

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By construction, the measure $\mu_{E}$ is $\operatorname{har}(\mathbb{B})$-balayage of measure $\vartheta$, but

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in direct contrast to Theorem 7.