数理解析研究所講究録 1293 巻 2002 年 84-97

84
Beurling’s Minimum Principle in a Cone
千葉大学

千葉大学

・

・

理学部

育子 ( Ikuko Miyamoto )宮本

Department of Mathematics,
Faculty of Science,
Chiba University
自然科学研究科 柳下 稔 ( Minoru Yanagishita )Graduate School of Science and Technology,

Chiba University
千葉大学

・

自然科学研究科吉田英信 ( Hidenobu Yoshida )Graduate School of Science and Technology,
Chiba University

Abstract
This paper shows that some characterizations of the harmonic majorization of the Martinfunction connected with adomain having smooth boundary without acorner e.g aball and ahalf-space also hold for aspecial domain with corners, i.e. acone.

1. Introduction.
Let andbe the set of all real numbers and all positive real numbers, respectively.
We denote by
the -dimensional Euclidean space. Apoint inis denoted
by $P=(X, y),X=(x_{1}, x_{2}, \ldots, x_{n-1})$ . The Euclidean distance of two points $P$ and inis denoted by $|P-Q|$ . Also $|P-O|$ with the origin ofis simply denoted by $|P|$ .
The boundary and the closure of aset inare denoted byand , respectively.
We introduce asystem of spherical coordinates $(r, 9)$ ,, inwhich are related to cartesian coordinates $(x_{1}, x_{2}, \ldots, x_{n-1}, y)$ by$\mathrm{R}$$\mathrm{R}_{+}$$\mathrm{R}^{n}(n\geq 2)$$\mathrm{R}^{n}$$\mathrm{n}$$Q$$\mathrm{R}^{n}$

$O$

$S$

$\mathrm{R}^{n}$$\mathrm{R}^{n}$$\partial S$$\overline{S}$$\Theta=(\theta_{1}, \theta_{2}, \ldots, \theta_{n-1})$$x_{1}=r(\Pi_{j=1}^{n-1}\sin\theta_{j})(n\geq 2)$,$y=r\cos\theta_{1}$$\mathrm{R}^{n}$,and if $n\geq 3$ , then$x_{n+1-k}=r(\Pi_{j=1}^{k-1}\sin\theta_{j})\cos\theta_{k}$$(2\leq k\leq n-1)$ ,where $0\leq r<+\infty$ ,, and if $n\geq 3$ , then $0\leq\theta_{j}\leq\pi(1\leq j\leq n-2)$ .
The unit sphere and the upper half unit sphere are denoted byand,respectively. For simplicity, apointon
and the set
for aset,, are often identified with and , respectively. For two sets
and, the set$- \frac{1}{2}\pi\leq\theta_{n-1}<\frac{3}{2}\pi$$\mathrm{S}^{n-1}$$\mathrm{S}_{+}^{n-1}$$(1, \Theta)$$\Omega$$\Omega\subset \mathrm{S}^{n-1}$$\Theta$$\mathrm{S}^{n-1}$$\{\Theta;(1, \Theta)\in\Omega\}$$\Omega$$\mathrm{A}\subset \mathrm{R}_{+}$$\Omega\subset \mathrm{S}^{n-1}$$\{(r, \Theta)\in \mathrm{R}^{n};r\in\Lambda, (1, \Theta)\in\Omega\}$85
in$\mathrm{R}^{n}$

is simply denoted by$\Lambda\cross\Omega$. In particular, the half-space$\mathrm{R}_{+}\cross \mathrm{S}_{+}^{n-1}=\{(X, y)\in \mathrm{R}^{n};y>0\}$will be denoted by$\mathrm{T}_{n}$.

, Armitage and Kuran [4] said that a
To extend aresult of Beurling [7] for, $|P_{m}|arrow+\infty(marrow+\infty)$ “characterizesof points
sequence
the positive harmonic majorization of $y”$ , if every positive harmonic function inwhich majorizes the function on the set $\{P_{m};m=1,2, ..\}$ majorizes everywhere in, i.e.
$(P=(X, y)\in \mathrm{T}_{n})$ .
$\mathrm{n}=2$$P_{m}=(X_{m}, y_{m})\in \mathrm{T}_{n}$$\{P_{m}\}$$h$$\mathrm{T}_{n}$$y$$\mathrm{y}$$\mathrm{T}_{n}$$\inf\underline{h(P)}.\underline{h(P_{m})}=\mathrm{n}\mathrm{f}$$P\in \mathrm{T}_{n}$

$m$

$y$

$y_{m}$

They proved

THEOREM A(Beurling [7] for $n=2$ , Armitage and Kuran [4, Theorem 1] for $n\geq 2$ ).
,be a sequence of points
Let$\{P_{m}\}$$\{P_{m}\}$$P_{m}=(r_{m}, \Theta_{m})\in \mathrm{T}_{n}$in$\mathrm{T}_{n}$,$\Theta_{m}=(\theta_{1.m}, \theta_{2,m}, \ldots, \theta_{(n-1),m})$satisfying(1.1)$r_{m+1}\geq ar_{m}$a certain $a>1$ . $T/ien$ the sequence
jorization of if and only if$/or$$(m=1, 2, \ldots)$$\{P_{m}\}$characterizes the positive harmonic ma-

$y$

(1.2)

$\sum_{m=1}^{\infty}(\cos\theta_{1,m})^{n}=+\infty$.

ofWe remark that is the Martin function at the infinite Martin boundary point
Tn, i.e. is (up to apositive multiplicative constant) the only positive harmonic function. The “if’ part of Teorem Ais aminimm principle, sinceinwhich vanishes on
if is a positive harmonic function $h(P)$ of $P=(X, y)\in \mathrm{T}_{n}$ , then$\infty$$y$$y$$\partial \mathrm{T}_{n}$$\mathrm{T}_{n}$$\mathrm{h}$$\lim_{P\in \mathrm{T}_{n}}$,$\inf_{Parrow P}$, $\{h(P)-y\}\geq 0$

and the majorization of the function
for every $P’$ onsatisfying (1.1) and (1.2) replaces$\partial \mathrm{T}_{n}$$\lim\inf$$P\in \mathrm{T}_{n}$, $|P|arrow+\infty$$y$$\{h(P)-y\}\geq 0$

.

by

$h$

on the set of points$P_{m}$

86
Hence this sort of sequence was said to be “equivalent to ”in Beurling [7] and thistype of Theorem Awas called “Beurling minimum principle” in Ancona [3, p.18] andMaz’ya [15]. This Theorem Awas also extended by Maz’ya [15] to positive solutions ofasecond-0rder elliptic differential equation in an -dimensional bounded domain withsmooth boundary of class $C^{1,\alpha}(0<\alpha<1)$ .
$\infty$$\mathrm{n}$Let $D$ be adomain inandbe the Martin boundary of $D$ . The Martinfunction at $Q\in\Delta(D)$ is denoted by $K_{Q}(P)(P\in D)$ . Following Armitage and Kuran [4],we say that asubset of $D$ characterizes the positve $ha$ rmonic majorization of $K_{Q}(P)$ , ifevery positive harmonic function in $D$ which majorizes $K_{Q}(P)$ on majorizes $K_{Q}(P)$everywhere in $D$ , i.e.
$\Delta(D)$$\mathrm{R}^{n}$

$E$

$h$

(1.3)

$E$

$\inf_{P\in D}\frac{h(P)}{K_{Q}(P)}=\inf_{P\in E}\frac{h(P)}{K_{Q}(P)}$.

We set$B(P, r)=\{P \in \mathrm{R}^{n};|P -P|<r\}$$(r>0)$

and$d(P)= \inf_{Q\not\in D}|P-Q|$

for any $P\in D$ . For asubset $E$ of $D$ and anumber $\rho(0<\rho<1)$ we put(1.4)$E_{\rho}= \bigcup_{P\in E}B(P, \rho d(P))$.

Dahlberg proved

THEOREM(Dahlberg [10, Theorem 1]). Let $D$ be a Liapunov-Dini domain inand. If
then the following conditions on $E$ are equivalent;(i) $E$ $cha$ racterizes the positve harmonic majorization of(ii) for every , $0<\rho<1$$Q\in\partial D$$\mathrm{B}$$\mathrm{R}^{n}$$E\subset D_{f}$$K_{Q}(P),\cdot$$\rho$(iii)$\int_{E_{\rho}}\frac{dP}{|P-Q|^{n}}=+\infty$,$\int_{E_{\rho}}\frac{dP}{|P-Q|^{n}}=+\infty$.

for some , $0<\rho<1$$\rho$

Since (1.3) is closely related to the notion of minimally thinness ofin (1.4) (seeSj\"ogren [17], Ancona [3] and Zhang [19] , which is also seen in Theorem 1of this paper,
Aikawa and Essen [2, Corollary 7.4.7] also proved Theorem in adifferent way ffom
Dahlberg’s$E_{\rho}$$)$$\mathrm{B}$87
By using asuitable Kelvin transformation which maps
Theorem C follows from Theorem B.

$\mathrm{T}_{n}$onto aball, the following, then the following conditions
THEOREM (Dahlberg [10, Theorem 3]). If$E$
on are equivalent;(i)characterizes the positive harmonic majorization of $yj$(ii) for every , $0<\rho<1$$E\subset \mathrm{T}\mathrm{n}$$\mathrm{C}$$E$$\rho$$I_{E_{\rho}} \frac{dP}{(1+|P|)^{n}}=+\infty$

,

where
$E_{\rho}= \bigcup_{P=(X,y)\in E}B(P, \rho y)$

(iii)

;

for some , $0<\rho<1$$\rho$$\int_{E_{\rho}}\frac{dP}{(1+|P|)^{n}}=+\infty$.

The methods of proving these Theorems Aand were based on the smoothness ofthe boundary having no wedges e.g. aball. For adomain having more rough boundary
e.g. aLipschiz domain, Ancona [3, Theorem 7.4] and Zhang [19, Theorem 3] gave more
complicated results which generalize Theorem A.
$\mathrm{B}$For aLipshitz domain and an NTA domain $D$ , Zhang [19, Corollary 1] and Aikawa[1, Remark and Theorem 1] gave anecessary and sufficient qualitative condition for asubset $E$ of $D$ characterizing the positive harmonic majorization of $K_{Q}(P)$ by connectingin (1.4), respectively. In his paper Aikawa says that sincewith minimally thinness ofageneral NTA domain may have wedges, Theorem does not hold for an NTA domain.
But when we see our results in this paper, we may ask whether quantitative Theorem
can just be extended to aLipshitz domain and an NTA domain.
$E_{\rho}$$\mathrm{B}$$\mathrm{B}$can be extended to aresult at a
In this paper we shall prove Theorems Aandcorner point of awedge i.e. aresult at of acone. We remark that ahalf-space is oneof cones.
$\mathrm{C}$$\infty$

2. Statements of results.
Let be adomain on
problem$\Omega$$\mathrm{S}^{n-1}(n\geq 2)$

with smooth boundary. Consider the Dirichlet$(\Lambda_{n}+\tau)f=0$

$f=0$

where

$\Lambda_{n}$onon$\Omega$$\partial\Omega$is the spherical part of the Laplace operator,$\Delta_{n}$$\Delta_{n}=\frac{n-1}{r}\frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial r^{2}}+r^{-2}\Lambda_{n}$88
We denote the least positive eigenvalue of this boundary value problem bynormalized positive eigenfunction corresponding toby;$d\sigma_{\Theta}$

is the surface element on$\mathrm{S}^{n-1}$

and the,$\int_{\Omega}f_{\Omega}^{2}(\Theta)d\sigma_{\Theta}=1$

where$\tau_{\Omega}$$f_{\Omega}(\Theta)$$\tau_{\Omega}$. We denote the solutions of the equation$t^{2}+(n-2)t-\tau_{\Omega}=0$by$-\beta_{\Omega}(\alpha_{\Omega}, \beta_{\Omega}>0)$$\alpha_{\Omega},$. If$\Omega=\mathrm{S}_{+}^{n-1}$, then$\alpha_{\Omega}=1$,$\beta_{\Omega}=n-1$$f_{\Omega}(\Theta)=(2ns_{n}^{-1})^{1/2}\cos\theta_{1}$

and

,

where
is the surface area $2\pi^{n/2}\{\Gamma(n/2)\}^{-1}$ of.
To make simplify our consideration in the following, we shall assume that if $n\geq 3$ ,then is adomain $(0<\alpha<1)$ on(e.g. see Gilbarg and Trudinger [12, pp.8889] for the definition of
domain By, we denote the set
inwith
the domain on. We call it acone. Thenis acone obtained by putting. It is known that the Martinofis the set, eachof which is aminimal Martin boundary point. The Martin kernel atwith respect toareference point chosen suitably is.
Asubset $E$ of adomain $D$ inis said to be minimally thin at $Q\in\Delta(D)$ (Brelot[8, p.122], Doob [11, p.208] , if there exists apoint $P\in D$ such that$\mathrm{S}^{n-1}$$s_{n}$$\Omega$$C^{2,\alpha}$$\mathrm{S}^{n-1}$$C^{2,\alpha}$$C_{n}(\Omega)$$\mathrm{R}_{+}\mathrm{x}\Omega$$\mathrm{S}^{n-1}(n\geq 2)$$\Omega$$\mathrm{R}^{n}$$\mathrm{T}_{n}$$\Omega=\mathrm{S}_{+}^{n-1}$$\grave{\mathrm{b}}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}$$C_{n}(\Omega)$$\partial C_{n}(\Omega)\cup\{\infty\}$$\infty$$K_{\infty}(P)=r^{\alpha_{\Omega}}f_{\Omega}(\Theta)(P=(r, \Theta)\in C_{n}(\Omega))$$\mathrm{R}^{n}$$)$$\hat{R}_{K_{Q}(\cdot)}^{E}(P)=K_{Q}(P)$where
p.134]).

$\hat{R}_{K_{Q}(\cdot)}^{E}(P)$,is the regularized reduced function of $K_{Q}(P)$ relative to $E$ (Helms [14,The following Theorem 1which is used to prove Theorem 2is aspecialized versionof Aikawa [1, Theorem 1]. Since his proof is so complicated because of an NTA domain
we shall give asimple proof based on afunction which is aconical version of Dahlberg’s[10, pp.240241].

THEOREM 1. Let$E$

be a subset of$C_{n}(\Omega)$. The following conditions on $E$ are equiva-

lent:(i) $E$ characterizes the positive harmonic majorization of(ii) for any , $0<\rho<1$ ,is not minimally thin at ,(iii) for some , $0<\rho<1$ ,is not minimally thin at$E_{\rho}$$\rho$$\rho$$K_{\infty}(P)$;$\infty$$E_{\rho}$$\infty$.

The following Theorem 2extends Theorem C.

THEOREM 2. Suppose thatequivalent:$E\subset C_{n}(\Omega)$. Then the following conditions on $E$ are89(i) E characterizes the positive ha monic majorization of(ii) for every $\rho(0<\rho<1)f$$I_{E_{\rho}} \frac{dP}{(1+|P|)^{n}}=+\infty$,$\int_{E_{\rho}}\frac{dP}{(1+|P|)^{n}}=+\infty$.

$K_{\infty}(P)i$for some $\rho(0<\rho<1)$ ,(Hi)of points $P_{m}\in D$ is said to be separated, if there exists apositive
Asequence
constant such that$\{P_{m}\}$$c$$(i,j=1,2, \ldots,i=j)$$|P_{i}-P_{j}|\geq cd(P_{i})$(e.g. see Ancona [3,18], Aikawa and Essen [2, p.156]).

Prom Theorem 2we immediately have the following Corollary which extends Theorem

A.
COROLLARY. Let$\{P_{m}\}$,$P_{m}\in C_{n}(\Omega)$$6e$

a separated sequence satisfying$\inf_{m}|P_{m}|>0$

The sequence
only if$\{P_{m}\}$.

characterizes the positive harmonic majorization of$\sum_{m=1}^{\infty}(\frac{d(P_{m})}{|P_{m}|})^{n}=+\infty$$K_{\infty}(P)$if and

.

3. Lemmas and proof of Theorem 1.
Let and be two positive real valued functions defined on aset . Then we shall, $A_{2},0<A_{1}\leq A_{2}$ such that $A_{1}g\leq f\leq A_{2}g$, if there exists two constants
write, the interior of and the diameter of are
everywhere on . For asubset in, the distancedenoted by int and diam , respectively. For. two subsets and in
is of the form
is denoted by dist(Si, ). Acube ofandbetween
$Z$

$f$

$g$

$A_{1}$$f\approx g$

$S$

$Z$

$S$

$S$

$S_{1}$$S_{2}$$S_{2}$$S_{1}$$S$$S$$\mathrm{R}^{n}$$[l_{1}2^{-k}, (l_{1}+1)2^{-k}]\mathrm{x}\cdots \mathrm{x}[l_{n}2^{-k}, (l_{n}+1)2^{-k}]$$S_{2}$$\mathrm{R}^{n}$$\mathcal{M}_{k}$$(k=0, \pm 1, \pm 2, \ldots)$. Afamily of the, are integers. Let be anumber satisfying
where ,
with is the set of cubes having the following properties;
Whitney cubes of$(i) \bigcup_{i}W_{i}=C_{n}(\Omega)$ ,$l_{1}$$0< \rho\leq\frac{1}{2}$$l_{n}$$\rho$$\ldots$$C_{n}(\Omega)$$\rho$

90(ii) int(i $=k)$ ,(ii),
where [a] denotes the integer satisfying $[a]\leq a<[a]+1$ (Stein [18, p.167, Theorem 1]).
$W_{i}\cap \mathrm{i}\mathrm{n}\mathrm{t}$$W_{k}=\emptyset$$[ \frac{8}{3\rho}]\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}W_{i}\leq \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\mathrm{P}, \mathrm{R}^{n}\backslash C_{n}(\Omega))\leq 2([\frac{8}{3\rho}]+1)\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}W_{\mathrm{i}}$The following Lemma 1is fundamental in this paper.

LEMMA 1(I.Miyamoto, M.Yanagishita and H.Yoshida [16, Theorems 2and 3]). Leta Borel subset $E$ of
be minimally thin at . Then we have$C_{n}(\Omega)$$\infty$

(3.1)

If

$\int_{E}\frac{dP}{(1+|P|)^{n}}<+\infty$.

is a union of cubes from a family of the Whitney cubes ofthen (3.1) is also sufficient for $E$ to be minimally thin at .
$E$

$C_{n}(\Omega)$with$\rho(0<\rho\leq\frac{1}{2})_{f}$$\infty$

For aset$E\subset C_{n}(\Omega)$

LEMMA 2. Let
exists a subsequence(i),and anumber$\{W_{i}\}_{i\geq 1}$$\rho(0<\rho\leq\frac{1}{2})$, define$E_{\rho}$

be a family of the Whitney cubes ofsuch thatofand$C_{n}(\Omega)$$E_{\frac{\rho}{4}}$as in (1.4).

with . Then there$\rho$$\{W_{i}\}_{i\geq 1}$$\{W_{i_{j}}\}_{j\geq 1}$$\bigcup_{j}W_{i_{j}}\subset E_{\rho}$(ii)$W_{i_{j}}\cap E_{\frac{\rho}{4}}=\emptyset$$(j=1,2, \ldots)$

Proof of Lemma 2. Let$k$$\{W_{i_{\mathrm{j}}}\}_{j\geq 1}$$E_{\frac{\rho}{4}} \subset\bigcup_{j}W_{i_{\mathrm{j}}}$.

be an integer. Let $c=([ \frac{8}{3\rho}]+1)$ and set$I_{k}=\{P\in C_{n}(\Omega)$

Let

,

;

$c\sqrt{n}2^{-k}<\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(P, \partial C_{n}(\Omega))\leq c\sqrt{n}2^{-k+1}\}$be asequence of all Whitney cubes of$W_{i_{j}}\cap Ee4$$=\emptyset$$\{W_{i}\}_{i\geq 1}$$(j=1, 2, \ldots)$.

such that.

Then it is evident that (ii) holds. We shall also show that thissatisfies (i).
Take any
and letbe acube of. Since, there exists apoint
in $E$ such that$\{W_{i_{j}}\}_{j\geq 1}$$W_{i_{j_{0}}}$$\mathcal{M}_{k_{0}}$$W_{i_{j_{0}}}$$W_{i_{j_{0}}}\cap E_{\frac{\rho}{4}}=\emptyset$$P_{j\mathrm{o}}$

(3.2)

Then

$B(P_{j\mathrm{o}}, \frac{\rho}{4}d(P_{j\mathrm{o}}))\cap W_{i_{j_{0}}}=\emptyset$$P_{j\mathrm{o}}\in I_{k_{0}+1}\cup I_{k_{0}}\cup I_{k_{0}-1}$$W_{i_{j}}\cap B(P, ed(4P))=\emptyset$

If$P_{j\mathrm{o}}\in I_{k_{0}+1}$, then. Because, if $P\in I_{k}$ and.

$W_{i_{j}}$$W_{i_{\mathrm{j}}}\in \mathcal{M}_{k+1}\cup \mathcal{M}_{k}\cup \mathcal{M}_{k-1}$.

is aWhitney cube satisfying, then.

$\rho d(P_{j\mathrm{o}})-\frac{\rho}{4}d(P_{j_{0}})=\frac{3}{4}\rho d(P_{j_{0}})>\frac{3}{4}\rho([\frac{8}{3\rho}]+1)\sqrt{n}2^{-(k_{\mathrm{O}}+1)}>\sqrt{n}2^{-k_{0}}$Since the diameter ofisand hence. Even if$W_{i_{j_{0}}}$$W_{i_{j_{0}}}\subset E_{\rho}$$\sqrt{n}2^{-k_{0}}$, we have ffom (3.2) thator
, we similarly have$P_{j\mathrm{o}}\in I_{k_{0}}$$W_{\dot{1}_{j_{0}}}\subset B(P_{j\mathrm{o}},\rho d(P_{j\mathrm{o}}))$$P_{j\mathrm{o}}\in I_{k_{0}-1}$$W_{\dot{\iota}_{\mathrm{j}_{0}}}\subset E_{\rho}$.

91
Thus all cubes of$\{W_{i_{j}}\}_{j\geq 1}$

are contained in$E_{\rho}$, which is just (i).

. First of all, we shall remark the following
fact. Let be apositive constant. Since $E$ characterizes the positive harmonic majoriza, $E_{1}=\{P\in E;K_{\infty}(P)>c\}$ also characterizes the positive harmoniction of. For otherwise there exists apositive harmonic function $h(P)$majorization of, satisfyingon

Proof of Theorem 1. Proof of$(i)\Rightarrow(ii)$$c$$K_{\infty}(P)$$K_{\infty}(P)$$C_{n}(\Omega)$.

$a= \inf_{P\in C_{n}(\Omega)}\frac{h(P)}{K_{\infty}(P)}<\inf_{P\in E_{1}}\frac{h(P)}{K_{\infty}(P)}=b$

If we put $h_{1}(P)=h(P)+bc(P\in C_{n}(\Omega))$ , then $h_{1}(P)\geq bK_{\infty}(P)$ for all $P\in E$ andhence,$\inf_{P\in C_{n}(\Omega)}\frac{h_{1}(P)}{K_{\infty}(P)}=a<b\leq\inf_{P\in E}\frac{h_{1}(P)}{K_{\infty}(P)}$which contradicts the assumption that $E$ characterizes the positive harmonic majoriza. If we can show that for any $\rho(0<\rho<1)(E_{1})_{\rho}$ is not minimally thin attion of, then for any $\rho(0<\rho<1)E_{\rho}$ is also not minimally thin at . Hence by applying
if necessary, we may assume that $K_{\infty}(P)>c$ for every
the following argument to$P\in E$ , without generality.
Suppose that for some number $\rho(0<\rho<1)E_{\rho}$ is minimally thin at . Then to
obtain acontradiction to (i) we shall make apositive harmonic function $h(P)$ on
satisfying$K_{\infty}(P)$$\infty$$\infty$$E_{1}$$\infty$$C_{n}(\Omega)$$\inf_{P\in C_{n}(\Omega)}\frac{h(P)}{K_{\infty}(P)}<\inf_{P\in E}\frac{h(P)}{K_{\infty}(P)}$.

, then let be aconstant function. When $E$ isis abounded subset ofunbounded, we shall follow Dahlberg [10, p.240] to make it.
which are the central
be asequence of points. Let
We can assume
can not accumulate to any finite boundary
in Lemma 2. Thenpoints of cubes
from (i) of Lemma 2andand hence $|P_{j}|arrow+\infty$ , because
point ofand
$K_{\infty}(P)>c$ for any $P\in E$ . Sinceis minimally thin at
If

$E$

$h$

$C_{n}(\Omega)$$\rho\leq\frac{1}{2}$$P_{j}$$\{P_{j}\}$$\{P_{j}\}$$W_{i_{j}}$$P_{j}\in E_{\rho}$$C_{n}(\Omega)$$\infty$$E_{\rho}$$\int_{W_{j}}\dot{.}\frac{dP}{(1+|P|)^{n}}\approx(\frac{d(P_{j})}{|P_{j}|})^{n}$$(j=1,2, \ldots.)$ ,Lemma 1and (i) of Lemma 2give(3.3)$\sum_{j=1}^{\infty}(\frac{d(P_{j})}{|P_{j}|})^{n}<+\infty$.

$(j=1,2, \ldots)$ . The general caseNow we shall assume thattreated at the end of this proof. Take apoint $Qj=$ {Pj}$d(P_{j}) \leq\frac{1}{2}|P_{j}|$$\mathrm{w}\mathrm{i}\mathrm{U}$be$\Phi_{j})\in\partial C_{n}(\Omega)\backslash \{O\}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathfrak{h}^{\gamma}\mathrm{i}\mathrm{n}\mathrm{g}$$|P_{j}-Q_{j}|=d(P_{j})(j=1,2, \ldots)$.

92
Then we also see$h(P)$ by

and hence $|Q_{j}|arrow+\infty(jarrow+\infty)$ . We define afunction$|Q_{j}| \geq\frac{1}{2}|P_{j}|$$h(P)= \sum_{j=1}^{\infty}\mathrm{P}_{Q_{j}}(P)\frac{\{d(P_{j})\}^{n}}{|P_{j}|^{1-\alpha_{\Omega}}}$,$\mathrm{P}_{Q_{j}}(P)=\frac{\partial G(P,Q_{j})}{\partial n_{Q_{j}}}(P\in C_{n}(\Omega))$,

where $G(P_{1}, P_{2})(P_{1}, P_{2}\in C_{n}(\Omega))$ is the Green function ofanddenotesalong the inward normal into
the differentiation at. Then is, because at any fixedwell-defined and hence is apositive harmonic function on$C_{n}(\Omega)$$Q\in\partial C_{n}(\Omega)$$\frac{\partial}{\partial n_{Q}}$$C_{n}(\Omega)$$h$$C_{n}(\Omega)$$P=(r, \Theta)\in C_{n}(\Omega)$$\mathrm{P}_{Q_{j}}(P)\approx r^{\alpha_{\Omega}}f_{\Omega}(\Theta)t_{j}^{-\beta_{\Omega}-1}\frac{\partial}{\partial n_{\Phi_{j}}}f_{\Omega}(\Phi_{j})$satisfying
for every
Now we shaU show
$Q_{j}$$t_{j}\geq 2r$(see Azarin [6, Lemma 1]).
$\inf_{\in E}\frac{h(P)}{K_{\infty}(P)}>0$.

To see first(3.4)$\frac{h(P_{j})}{K_{\infty}(P_{\mathrm{j}})}\geq A\backslash$for some positive constant$A$$\mathrm{P}_{j}(P, Q)(P\in B_{j}, Q\in\partial B_{j})$$(j=1, 2, \ldots)$, denote the Poisson kernel of the ball $B_{j}=B(P_{j}, d(P_{j}))$ by. Then we see$\mathrm{P}_{Q_{j}}(P)\geq \mathrm{P}_{j}(P, Q_{j})$$(P\in B_{j};j=1,2, \ldots)$and hence$\mathrm{P}_{Q_{j}}(P_{j})\geq \mathrm{P}_{j}(P_{j}, Q_{j})=s_{n}^{-1}\{d(P_{j})\}^{1-n}$$(j=1, 2, \ldots)$

.

Since
$f_{\Omega}(\Theta)\approx d(P’)(P’=(1,\Theta),\Theta\in\Omega)$,

we obtain$h(P_{j}) \geq \mathrm{P}_{Q_{j}}(P_{j})\frac{\{d(P_{j})\}^{n}}{|P_{j}|^{1-\alpha_{\Omega}}}\geq AK_{\infty}(P_{j})$$(j=1,2, \ldots)$

Next take any $P\in E$ . Then by (ii) of Lemma 2there exist apoint$|P-P_{j}|< \frac{1}{2}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(W_{i_{j}})\leq\delta d(P_{j})$where$\delta=\frac{1}{2}[\frac{8}{3\rho}]^{-1}$. Hence we$P_{j}$.
such that,$\mathrm{s}\mathrm{e}$$h(P) \geq\frac{1-\delta}{(1+\delta)^{n-1}}h(P_{j})$and$K_{\infty}(P) \leq\frac{1+\delta}{(1-\delta)^{n-1}}K_{\infty}(P_{j})$93from the Harnack’s inequalities (see Armitage and Gardiner [5, Theorem 1.4.1]). Thus
we have$\frac{h(P)}{K_{\infty}(P)}\geq(\frac{1-\delta}{1+\delta})^{n}\frac{h(P_{j})}{K_{\infty}(P_{j})}\geq(\frac{1-\delta}{1+\delta})^{n}A$from (3.4), which shows$\inf_{\in E}\frac{h(P)}{K_{\infty}(P)}>0$

To show$\inf_{P\in C_{n}(\Omega)}\frac{h(P)}{K_{\infty}(P)}=0$, fix aray

$L$.

which is inside$C_{n}(\Omega)$and starts from O.

We shall show
(3.5)$\lim_{|P|arrow+\infty,P\in L}\frac{h(P)}{K_{\infty}(P)}=0$.

Put$g_{j}(P)= \frac{\mathrm{P}_{Q_{j}}(P)}{K_{\infty}(P)}|P_{j}|^{\beta_{\Omega}+1}$$(P\in C_{n}(\Omega),j=1,2, \ldots)$.

Then we have$\frac{h(P)}{K_{\infty}(P)}=\sum_{j=1}^{\infty}g_{j}(P)(\frac{d(P_{j})}{|P_{j}|})^{n}$

Since
(3.6)

$\mathrm{P}_{Q_{j}}(P)\approx t_{j}^{\alpha_{\Omega}-1}r^{-ffi}f_{\Omega}(\Theta)\frac{\partial}{\partial n_{\Phi_{j}}}f_{\Omega}(\Phi_{j})$$(P=(r, \Theta)\in C_{n}(\Omega),r\geq 2t_{j})$(see Azarin [6, Lemma 1]), we see$\lim$$|P|\prec+\infty$, $P\in L$$g_{j}(P)=0$

for any fixed . Hence if we can show that
$j$

(3.7)

$|g_{j}(P)|\leq M$$(P\in L, j=1,2, \ldots)$for some constant $M$ , then we shall have (3.5) bom (3.3).
Now we shall prove (3.7) by dividing into three cases. If$r\leq\lrcorner t2$, then we have$\mathrm{P}_{Q_{\mathrm{j}}}(P)\approx r^{\alpha_{\Omega}}t_{j}^{-\beta_{\Omega}-1}f_{\Omega}(\Theta)\frac{\partial}{\partial n_{\Phi_{j}}}f_{\Omega}(\Phi_{j})$and hence we have$|g_{j}(P)|\leq M$

If$r\geq 2t_{j}$$(P=(r, \Theta)\in C_{n}(\Omega),j=1,2$,$\ldots$).

, then we have$|g_{j}(P)|\leq M$$(P=(r, \Theta)\in C_{n}(\Omega);j=1,2,$ $\ldots)$94
from (3.6). Lastly, put$u^{n-2}G((uR_{1}, _{1})$When$(R_{2}, \Theta_{2})$, $u=t_{j}$ andapproaches to$(1, \Phi_{j})$$((R_{1}, \Theta_{1}),$ $(R_{2}, \Theta_{2})\in C_{n}(\Omega))$$(R_{2}, \Theta_{2}))$.

along the inward normal, we obtain$\frac{\partial G(P,Q_{j})}{\partial n_{Q_{j}}}=\frac{1}{t_{j}^{n-1}}\frac{\partial G}{\partial n_{Q_{\acute{j}}}}($$\frac{1}{2}t_{j}\leq r\leq 2t_{j}$in$\mathrm{O}-_{1}=\Theta$, $(uR_{2}, \Theta_{2}))=G((R_{1}, \Theta_{1}),$

(3.8)

If

$R_{1}= \frac{r}{t_{j}}$( , ),$\Theta$$\frac{r}{t_{j}}$$(1, \Phi j))$.

, then$t_{j}^{n-1}\mathrm{P}_{Q_{j}}(P)\leq M’$$(P=(r, \Theta)\in L;j=1,2,$$\ldots)$

for some constant $M’$ and hence$(P\in L;j=1, 2, \ldots)$ .

$|g_{j}(P)|\leq M$

Finally, even if there is a such thatbyfor every $j\geq J$ . Define$d(P_{j})> \frac{1}{2}|P_{j}|$$j$, there also exists a$J$such that$h_{2}$$d(P_{j}) \leq\frac{1}{2}|P_{j}|$$h_{2}(P)= \sum_{j=J}^{\infty}\mathrm{P}_{Q_{\mathrm{j}}}(P)\frac{\{d(P_{j})\}^{n}}{|P_{j}|^{1-\alpha_{\Omega}}}(P\in C_{n}(\Omega))$,

which satisfies$h_{2}(P_{j})\geq AK_{\infty}(P_{j})(j\geq J)$

md.

$\inf_{P\in C_{\mathfrak{n}}(\Omega)}\frac{h_{2}(P)}{K_{\infty}(P)}=0$

Put $7= \max_{1\leq j<J}K_{\infty}(P_{j})$ . Then the function $h(P)=h_{2}(P)+\gamma$ is apositive harmonicsuch thatfunction on$C_{n}(\Omega)$$\inf_{P\in C_{n}(\Omega)}\frac{h(P)}{K_{\infty}(P)}=0$

and$(j=1, 2, \ldots)$$h(P_{j}) \geq\min\{A, 1\}K_{\infty}(P_{j})$from which it follows in the same way as above that$\inf_{\in E}\frac{h(P)}{K_{\infty}(P)}>0$.

.
Proof of$E$
Suppose that does not characterize the positive harmonic majorization ofsuch that
Then there exists apositive harmonic function $h(P)$ in$(iii)\Rightarrow(i)$$C_{n}(\Omega)$$a= \inf_{P\in C_{n}(\Omega)}\frac{h(P)}{K_{\infty}(P)}<\inf_{P\in E}\frac{h(P)}{K_{\infty}(P)}=b$.

$K_{\infty}(P)$.

95
If we put $v(P)=h(P)-aK_{\infty}(P)$function on$C_{n}(\Omega)$$0<\rho<1$ .

For anysatisfying$P\in E_{\rho}$$(P\in C_{n}(\Omega))$$\inf_{P\in C_{n}(\Omega)}\frac{\mathrm{t}’(P)}{K_{\infty}(P)}=0$, then $v(P)$ is apositive harmonic. Let be any positive number satisfying$\rho$, there exists apoint $P’\in E$ such that $|P-P’|<\rho d(P’)$ andhence$( \frac{1-\rho}{1+\rho})^{n}\frac{v(P’)}{K_{\infty}(P’)}\leq\frac{v(P)}{K_{\infty}(P)}$by Harnack’s inequality, (e.g. Armitage and Gardiner [5, Theorem 1.4.1]). Hence wehave$\inf_{P\in E_{\rho}}\frac{v(P)}{K_{\infty}(P)}\geq(\frac{1-\rho}{1+\rho})^{n}\inf_{\in E}\frac{v(P)}{K_{\infty}(P)}=(\frac{1-\rho}{1+\rho})^{n}(b-a)>0$.

Therefore we obtain$\inf_{P\in C_{n}(\Omega)}\frac{v(P)}{K_{\infty}(P)}<\inf_{P\in E_{\rho}}\frac{v(P)}{K_{\infty}(P)}$.

is minimally thin at
Since $v(P)$ is also apositive superharmonic function,
Miyamoto, Yanagishita and Yoshida [16. Theorem 1] . This contradicts (iii).
$E_{\rho}$$\infty$

(e.g.

$)$

4. Proofs of Theorem 2and Corollary

Proof of Theorem 2. Proof of$(i)\Rightarrow(\mathrm{i}\mathrm{i})$. Suppose that$\int_{E_{\rho}}\frac{dP}{(1+|P|)^{n}}<+\infty$. Let
for some $\rho(0<\rho<1)$ . We can assume that this satisfies
in Lemma 2. Then from (i) of Lemma 2we also haveasubsequence of$0< \rho\leq\frac{1}{2}$$\rho$$\{W_{i_{j}}\}j\geq 1$

be$\{W_{i}\}_{i\geq 1}$$\int_{\bigcup_{j}W_{j}}.\frac{dP}{(1+|P|)^{n}}<+\infty$.

with , we see ffom
is aunion of cubes from the Whitney cubes ofis minimally thin at , and hence from (ii) of
the second part of Lemma 1thatis minimally thin at . Since $E$ characterizes the positive harmonic
Lemma 2thatis not minimally thin at ,, it follows from Theorem 1thatmajorization ofwhich contradicts the conclusion obtained above.

Since$C_{n}(\Omega)$$\bigcup_{j}W_{i_{j}}$$\infty$$\bigcup_{j}W_{i_{\mathcal{J}}}$$\infty$$E_{\frac{\rho}{4}}$$\infty$$K_{\infty}(P)$Proof ofmajorization ofminimally thin at$\rho$$E_{\frac{\rho}{4}}$. Suppose that $E$ does not characterize the positive harmonic. Then we see ffom Theorem 1that for any $\rho(0<\rho<1)E_{\rho}$ is. Lemma 1gives that for any $\rho(0<\rho<1)$$(iii)\Rightarrow(i)$$K_{\infty}(P)$$\infty$$\int_{E_{\rho}}\frac{dP}{(1+|P|)^{n}}<+\infty$.

96This contradicts (Hi).

Proof of Corollary. It is easy to see that if$B(P_{i}, \rho d(P_{i}))\cap$$\{P_{m}\}$$\{Pm\}\rho d(P_{j}))=\emptyset$

is aseparated sequence, then$(i, j=1,2, \ldots; i=j)$

for asufficiently small $\rho(0<\rho<1)$ and hence$\int_{E_{\rho}}\frac{dP}{(1+|P|)^{n}}\approx\sum_{m=1}^{\infty}(\frac{d(P_{m})}{|P_{m}|})^{n}$This corolary immediately follows ffom (iii) of Theorem 2.

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