# Criteria of Wiener Type for Minimally Thin Sets and Rarefied Sets Associated with the Stationary Schrödinger Operator in a Cone

Authors: Pinhong Long; Zhiqiang Gao; Guantie Deng
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

We give some criteria fora-minimally thin sets and a-rarefied sets associated with the stationary Schrödinger operator at a fixed Martin boundary point or ∞ with respect to a cone. Moreover, we show that a positive superfunction on a cone behaves regularly outside an a-rarefied set. Finally we illustrate the relation between the a-minimally thin set and the a-rarefied set in a cone.


---

## Body

## 1. Introduction

This paper is concerned with some properties for the generalized subharmonic functions associated with the stationary Schrödinger operator. More precisely the minimally thin sets and rarefied sets about these generalized subharmonic functions will be studied. The research on minimal thinness has been exploited a little and attracted many mathematicians. In 1949 Lelong-Ferrand [1] started the study of the thinness at boundary points for the subharmonic functions on the half-space. Then in 1957 Naïm [2] gave some criteria for minimally thin sets at a fixed boundary point with respect to half-space (see [3] for a survey of the results in [1, 2]). In 1980 Essén and Jackson [4] gave the criteria for minimally thin sets at ∞ with respect to half-space, and furthermore they introduced rarefied sets at ∞ with respect to half-space, which is more refined than minimally thin set. Later Miyamoto and Yoshida [5] extended these results of Essén and Jackson from half-space to a cone. In this paper, we will deal with the corresponding questions for the generalized subharmonic functions associated with the stationary Schrödinger operator.To state our results, we will need some notations and preliminary results. As usual, denote byRn(n≥2) the n-dimensional Euclidean space. For an open subset set S⊂Rn, denote its boundary by ∂S and its closure by S¯. Let P=(X,xn), where X=(x1,x2,…,xn-1), and let |P| be the Euclidean norm of P and |P-Q| the Euclidean distance of two points P and Q in Rn. The unit sphere and the upper half unit sphere are denoted by Sn-1 and S+n-1, respectively. For P∈Rn and r>0, let B(P,r) be the open ball of radius r centered at P in Rn, then Sr=∂B(O,r). Furthermore, denote by dSr the (n-1)-dimensional volume elements induced by the Euclidean metric on Sr.ForP=(X,xn)∈Rn, it can be reexpressed in spherical coordinates (r,Θ), Θ=(θ1,θ2,…,θn) via the following transforms:
(1.1)x1=r∏j=1n-1sinθj(n≥2),xn=rcosθ1,
and if n≥3,
(1.2)xn-k+1=rcosθk∏j=1k-1sinθj(2≤k≤n-1),
where 0≤r<∞,0≤θj≤π(1≤j≤n-2;n≥3) and -π/2≤θn-1≤(3π/2)(n≥2).Relative to the system of spherical coordinates, the Laplace operatorΔ may be written as
(1.3)Δ=n-1r∂∂r+∂2∂r2+Δ*r2,
where the explicit form of the Beltrami operator Δ* is given by Azarin (see [6]).LetD be an arbitrary domain in Rn, and 𝒜D denotes the class of nonnegative radial potentials a(P) (i.e., 0≤a(P)=a(r) for P=(r,Θ)∈D) such that a∈Llocb(D) with some b>n/2 if n≥4 and with b=2 if n=2 or n=3.For the identical operatorI, define the stationary Schrödinger operator with a potential a(·) by
(1.4)ℒa=-Δ+a(·)I.
If a∈𝒜D, then ℒa can be extended in the usual way from the space C0∞(D) to an essentially self-adjoint operator on L2(D) (see [7, Chapter 13] for more details). Furthermore ℒa has a Green a-function GDa(·,·). Here GDa(·,·) is positive on D, and its inner normal derivative ∂GDa(·,Q)/∂nQ is nonnegative, where ∂/∂nQ denotes the differentiation at Q along the inward normal into D. We write this derivative by PIDa(·,·), which is called the Poisson a-kernel with respect to D. Denote by GD0(·,·) the Green function of Laplacian. It is well known that
(1.5)GDa(·,·)≤GD0(·,·)
for any potential a(·)≥0. The “inverse” inequality in some sense is much more elaborate. When D is a bounded domain in Rn, Cranston (see [8], the case n=2 is implicitly contained in [9]) have proved that
(1.6)GDa(·,·)≥M(D)GD0(·,·),
where M(D)=M(D,a) is a positive constant and independent of points in D. If a=0, then obviously M(D)≡1.Suppose that a functionu≢-∞ is upper semicontinuous in D. We call u∈[-∞,+∞) a subfunction for the Schrödinger operator ℒa if the generalized mean-value inequality
(1.7)u(P)≤∫S(P,ρ)u(Q)∂GB(P,ρ)a(P,Q)∂nQdσ(Q)
is satisfied at each point P∈D with 0<ρ<infQ∈∂D|P-Q|, where S(P,ρ)=∂B(P,ρ), GB(P,ρ)a(·,·) is the Green a-function of ℒa in B(P,r), and dσ(·) the surface area element on S(P,ρ) (see [10]).Denote bySbH(a,D) the class of subfunctions in D. We call u a superfunction associated with ℒa if -u∈SbH(a,D), and denote by SpH(a,D) the class of superfunctions. If a function u on D is both subfunction and superfunction, then it is called an a-harmonic function associated with the operator ℒa. The class of a-harmonic functions is denoted by H(a,D), and it is obviously SbH(a,D)∩SpH(a,D). Here we follow the terminology from Levin and Kheyfits (see [11–13]).For simplicity, the point(1,Θ) on Sn-1 and the set {Θ;(1,Θ)∈Ω} for a set Ω⊂Sn-1 are often identified with Θ and Ω, respectively. For Ξ⊂R+ and Ω⊂Sn-1, the set {(r,Θ)∈Rn;r∈Ξ,(1,Θ)∈Ω} in Rn is simply denoted by Ξ×Ω. In particular, the half space {(X,xn)∈Rn;xn>0}=R+×S+n-1 will be denoted by Tn. We denote by Cn(Ω) the set R+×Ω in Rn with the domain Ω⊂Sn-1 and call it a cone. For an interval I⊂R+ and Ω⊂Sn-1, write Cn(Ω;I)=I×Ω, Sn(Ω;I)=I×∂Ω, and Cn(Ω;r)=Cn(Ω)∩Sr. By Sn(Ω) we denote Sn(Ω;(0,+∞)), which is ∂Cn(Ω)-{O}. From now on, we always assume D=Cn(Ω) and write GΩa(·,·) instead of GCn(Ω)a(·,·).LetΩ be a domain on Sn-1 with smooth boundary. Suppose that τ is the least positive eigenvalue for -Δ* on Ω and the normalized positive eigenfunction φ(Θ) corresponding to τ satisfies ∫Ωφ2(Θ)dS1=1. Then
(1.8)(Δ*+τ)φ(Θ)=0onΩ,φ(Θ)=0on∂Ω
(see [14, page 41]). In order to ensure the existence of τ and φ(Θ), we pose the assumption on Ω: if n≥3, then Ω is a C2,α-domain (0<α<1) on Sn-1 surrounded by a finite number of mutually disjoint closed hypersurfaces (see e.g., [15, pages 88-89] for the definition of C2,α-domain).LetℬD be the class of the potential a∈𝒜D such that
(1.9)limr→∞r2a(r)=κ0∈[0,∞),r-1|r2a(r)-κ0|∈L(1,∞).
When a∈ℬD, the subfunctions (superfunctions) associated with ℒa are continuous (see, e.g., [16]). In the rest of paper, we will always assume that a∈ℬD.An important role will be played by the solutions of the ordinary differential equation(1.10)-Q′′(r)-n-1rQ′(r)+(τr2+a(r))Q(r)=0(0<r<∞).
When the potential a∈𝒜D, these solutions are well known (see [17] for more references). Equation (1.10) has two specially linearly independent positive solutions V(r) and W(r) such that V is increasing with
(1.11)0≤V(0+)≤V(r)asr→+∞
and W is decreasing with
(1.12)+∞=W(0+)>W(r)↘0asr→+∞.
We remark that both V(r)φ(Θ) and W(r)φ(Θ) are harmonic on Cn(Ω) and vanish continuously on Sn(Ω).Denote(1.13)ικ±=2-n±(n-2)2+4(κ+τ)2.
When a∈ℬD, the normalized solutions V(r) and W(r) of (1.10) satisfying V(1)=W(1)=1 have the asymptotics (see [15]):
(1.14)V(r)≈rικ+,W(r)≈rικ-,asr→∞.
Set
(1.15)χ=ικ+-ικ-=(n-2)2+4(κ+τ),χ′=ω(V(r),W(r))|r=1,
where χ′ is their Wronskian at r=1.Remark 1.1.
Ifa=0 and Ω=S+n-1, then ι0+=1, ι0-=1-n and φ(Θ)=(2nsn-1)1/2cosθ1, where sn=2πn/2{Γ(n/2)}-1 is the surface area of Sn-1.
We recall that(1.16)C1V(r)W(t)φ(Θ)φ(Φ)≤GΩa(P,Q)≤C2V(r)W(t)φ(Θ)φ(Φ),
or
(1.17)C1V(t)W(r)φ(Θ)φ(Φ)≤GΩa(P,Q)≤C2V(t)W(r)φ(Θ)φ(Φ)
for any P=(r,Θ)∈Cn(Ω) and any Q=(t,Φ)∈Cn(Ω) satisfying 0<r/t≤4/5 or 0<t/r≤4/5, where C1 and C2 are two positive constants (see Escassut et al. [11, Chapter 11], and for a=0, see Azarin [6, Lemma 1], Essén, and Lewis [18, Lemma 2]).
The remainder of the paper is organized as follows: in Section2 we will give our main theorems; in Section 3, some necessary lemmas are given; in Section 4, we will prove the main results.


## 2. Statement of the Main Results

In this section, we will state our main results. Before passing to our main results, we need some definitions.Martin introduced the so-called Martin functions associated with the Laplace operator (see Brelot [19] or Martin [20]). Inspired by his spirit, we define the Martin function MΩa associated with the stationary Schrödinger operator as follows:
(2.1)MΩa(P,Q)=GΩa(P,Q)GΩa(P0,Q)(P,Q∈Cn(Ω)×Cn(Ω)∖(P0,P0)),
which will be called the generalized Martin Kernel of Cn(Ω) (relative to P0). If Q=P0, the above quotient is interpreted as 0 (for a=0, refer to Armitage and Gardiner [3]).It is well known that the Martin boundaryΔ of Cn(Ω) is the set ∂Cn(Ω)∪{∞}. When we denote the Martin kernel associated with the stationary Schrödinger operator by MΩa(P,Q)(P∈Cn(Ω),Q∈∂Cn(Ω)∪{∞}) with respect to a reference point chosen suitably, we see
(2.2)MΩa(P,∞)=V(r)φ(Θ),MΩa(P,O)=KW(r)φ(Θ)
for any P∈Cn(Ω), where O is the origin of Rn and K a positive constant.LetE be a subset of Cn(Ω) and let u be a nonnegative superfunction on Cn(Ω). The reduced function of u is defined by
(2.3)RuE(P)=inf{υ(P):υ∈ΦuE},
where ΦuE={υ∈SpH(a,Cn(Ω)):v≥0onCn(Ω),υ≥uonE}. We define the regularized reduced function R^uE of u relative to E as follows:
(2.4)R^uE(P)=limP′→PinfRuE(P′).
It is easy to see that R^uE is a superfunction on Cn(Ω).IfE⊆Cn(Ω) and Q∈Δ, then the Riesz decomposition and the generalized Martin representation allow us to express R^MΩa(·,Q)E uniquely in the form GΩaμ+MΩaν, where GΩaμ and MΩaν are the generalized Green potential and generalized Martin representation, respectively. We say that E is a-minimally thin at Q with respect to Cn(Ω) if ν({Q})=0. At last we remark that Δ0={Q∈Δ:Cn(Ω)isa-minimallythinatQ}, where Δ is the Martin boundary of Cn(Ω).Now we can state our main theorems.Theorem 2.1.
LetE⊆Cn(Ω) and a fixed point Q∈Δ∖Δ0. The following are equivalent:(a)
Eis a-minimally thin at Q;(b)
R^MΩa(·,Q)E≠MΩa(·,Q);(c)
inf{R^MΩa(·,Q)E∩ω:ωisageneralizedMartintopologyneighbourhoodofQ}=0.Ifu is a positive superfunction, then we will write μu for the measure appearing in the generalized Martin representation of the greatest a-harmonic minorant of u.Theorem 2.2.
LetE⊆Cn(Ω) and a fixed point Q∈Δ∖Δ0. Suppose that Q is a generalized Martin topology limit of E. The following are equivalent:(a)
E is a-minimally thin at Q;(b)
there exists a positive superfunctionu such that
(2.5)liminfP→Q,P∈Eu(P)MΩa(P,Q)>μu({Q}),(c)
there is ana-potential u on Cn(Ω) such that
(2.6)u(P)MΩa(P,Q)→∞(P→Q;P∈E).A setE in Rn is said to be a-thin at a point Q if there is a fine neighborhood U of Q which does not intersect E∖{Q}. Otherwise E is said to be not a-thin at Q. A set E in Rn is called a-polar if there is a superfunction u on some open set ω such that E⊆{P∈ω:u(P)=∞}.LetE be a bounded subset of Cn(Ω). Then R^MΩa(·,∞)E(P) is bounded on Cn(Ω), and hence the greatest a-harmonic minorant of R^MΩa(·,∞)E(P) is zero. By the Riesz decomposition theorem there exists a unique positive measure λEa associated with the stationary Schrödinger operator ℒa on Cn(Ω) such that
(2.7)R^MΩa(·,∞)E(P)=GΩaλEa(P)
for any P∈Cn(Ω), and λEa is concentrated on BE, where
(2.8)BE={P∈Cn(Ω):Eisnota-thinatP}.
For a=0, see Brelot [19] and Doob [21]. According to the Fatou's lemma, we easily know the condition (b) in Theorems 2.3 and 2.4.Theorem 2.3.
LetE⊆Cn(Ω) and a fixed point Q∈Δ∖Δ0. Suppose that Q is a generalized Martin topology limit point of E. The following are equivalent:(a)
E is a-minimally thin at Q;(b)
there is ana-potential GΩaμ such that
(2.9)liminfP→Q,P∈EGΩaμ(P)GΩa(P0,P)>∫‍MΩa(P,Q)dμ(P),(c)
there is ana-potential GΩaμ' such that ∫‍MΩa(P,Q)dμ'(P)<∞ and
(2.10)GΩaμ′(P)GΩa(P0,P)→∞(P→Q;P∈E).Theorem 2.4.
LetE⊆Cn(Ω), Q0∈Cn(Ω) and a fixed point Q∈Δ∖Δ0. Suppose that Q is a generalized Martin topology limit point of E. Then E is a-minimally thin at Q if and only if there exists a positive superfunction u such that
(2.11)liminfP→Q,P∈Eu(P)GΩa(Q0,P)>liminfP→Qu(P)GΩa(Q0,P).The generalized Green energyγΩa(E) of λEa is defined by
(2.12)γΩa(E)=∫Cn(Ω)‍(GΩaλEa)dλEa.
Let E be a subset of Cn(Ω) and Ek=E∩Ik(Ω), where Ik(Ω)={P=(r,Ω)∈Cn(Ω):2k≤r≤2k+1}. The previous theorems are concerned with the fixed boundary points. Next we will consider the case at infinity.Theorem 2.5.
A subsetE of Cn(Ω) is a-minimally thin at ∞ with respect to Cn(Ω) if and only if
(2.13)∑k=0∞‍γΩa(Ek)W(2k)V(2k)-1<∞.A subsetE of Cn(Ω) is a-rarefied at ∞ with respect to Cn(Ω), if there exists a positive superfunction υ(P) in Cn(Ω) such that
(2.14)infP∈Cn(Ω)υ(P)MΩa(P,∞)≡0,E⊂Hυ,
where
(2.15)Hυ={P=(r,Θ)∈Cn(Ω):υ(P)≥V(r)}.Theorem 2.6.
A subsetE of Cn(Ω) is a-rarefied at ∞ with respect to Cn(Ω) if and only if
(2.16)∑k=0∞‍W(2k)λΩa(Ek)<∞.Remark 2.7.
Whena=0, Theorems 2.5 and 2.6 reduce to the results by Miyamoto and Yoshida [5]. When a=0 and Ω=S+n-1, these are exactly due to Aikawa and Essén [22].
Set(2.17)c(υ,a)=infP∈Cn(Ω)υ(P)MΩa(P,∞)
for a positive superfunction υ(P) on Cn(Ω). We immediately know that c(υ,a)<∞. Actually let u(P) be a subfunction on Cn(Ω) satisfying
(2.18)limsupP→Q,P∈Cn(Ω)u(P)≤0
for any Q∈∂Cn(Ω)∖{O} and
(2.19)supP=(r,Θ)∈Cn(Ω)u(P)V(r)φ(Θ)=ℓ(a)<∞.
Then we see ℓ(a)>-∞ (for a=0, see Yoshida [23]). If we apply this to u=-υ, we may obtain c(υ,a)<∞.Theorem 2.8.
Letυ(P) be a positive superfunction on Cn(Ω). Then there exists an a-rarefied set E at ∞ with respect to Cn(Ω) such that υ(P)V(r)-1 uniformly converges to c(υ,a)φ(Θ) on Cn(Ω)∖E as r→∞, where P=(r,Θ)∈Cn(Ω).From the definition ofa-rarefied set, for any given a-rarefied set E at ∞ with respect to Cn(Ω) there exists a positive superfunction υ(P) on Cn(Ω) such that υ(P)V(r)-1≥1 on E and c(υ,a)=0. Hence υ(P)V(r)-1 does not converge to c(υ,a)φ(Θ)=0 on E as r→∞.Letu(P) be a subfunction on Cn(Ω) satisfying (2.18) and (2.19). Then
(2.20)υ(P)=ℓ(a)V(r)φ(Θ)-u(P),(P=(r,Θ)∈Cn(Ω))
is a positive superfunction on Cn(Ω) such that c(υ,a)=0. If we apply Theorem 2.8 to this υ(P), then we obtain the following corollary.Corollary 2.9.
Letu(P) be a subfunction on Cn(Ω) satisfying (2.18) and (2.19) for P∈Cn(Ω). Then there exists an a-rarefied set E at ∞ with respect to Cn(Ω) such that υ(P)V(r)-1 uniformly converges to ℓ(a)φ(Θ) on Cn(Ω)∖E as r→∞, where P=(r,Θ)∈Cn(Ω).A coneCn(Ω') is called a subcone of Cn(Ω) if Ω'¯⊂Ω, where Ω'¯ is the closure of Ω'⊂Sn-1.Theorem 2.10.
LetE be a subset of Cn(Ω). If E is an a-rarefied set at ∞ with respect to Cn(Ω), then E is a-minimally thin at ∞ with respect to Cn(Ω). If E is contained in a subcone of Cn(Ω) and E is a-minimally thin at ∞ with respect to Cn(Ω), then E is an a-rarefied set at ∞ with respect to Cn(Ω).


## 3. Some Lemmas

In our arguments we need the following results.Lemma 3.1.
LetE1,E2,…,Em⊆Cn(Ω) and Q∈Δ.(i)
IfE1⊆E2 and E2 is a-minimally thin at Q, then E1 is a-minimally thin at Q.(ii)
IfE1,E2,…,Em are a-minimally thin at Q, then ⋃k=1m‍Ek is a-minimally thin at Q.(iii)
IfE1 is a-minimally thin at Q, then there is an open subset E of Cn(Ω) such that E1⊆E and E is a-minimally thin at Q.Proof.
SinceR^MΩa(·,Q)E1≤R^MΩa(·,Q)E2, we see (i) holds. To prove (ii) we note that R^MΩa(·,Q)Ek is an a-potential for each k and
(3.1)∑k=1m‍R^MΩa(·,Q)Ek≥MΩa(·,Q)quasieverywhereon⋃k=1m‍Ek,
so R^MΩa(·,Q)⋃k‍Ek is an a-potential. Finally, to prove (iii), let u=R^MΩa(·,Q)E1. Then u is an a-potential and u≥MΩa(·,Q) on E1∖F for some a-polar set F. Let υ be a nonzero a-potential such that υ=∞ on F, and let
(3.2)Z={P∈Cn(Ω):u(P)+υ(P)≥MΩa(P,Q)}.
Then Z is open, E1⊆Z and RMΩa(·,Q)Z≤u+υ, so RMΩa(·,Q)Z is an a-potential and Z is a-minimally thin at Q.Lemma 3.2 (see [24]).
Consider(3.3)∂GΩa(P,Q)∂nQ≈t-1V(t)W(r)φ(Θ)∂φ(Φ)∂nΦ,(3.4)∂GΩa(P,Q)∂nQ≈V(r)t-1W(t)φ(Θ)∂φ(Φ)∂nΦ
for any P=(r,Θ)∈Cn(Ω) and any Q=(t,Φ)∈Sn(Ω) satisfying 0<t/r≤4/5(resp.,0<r/t≤4/5). In addition,
(3.5)∂GΩ0(P,Q)∂nQ≲φ(Θ)tn-1∂φ(Φ)∂nΦ+rφ(Θ)|P-Q|n∂φ(Φ)∂nΦ
for any P=(r,Θ)∈Cn(Ω) and any Q=(t,Φ)∈Sn(Ω;((4/5)r,(5/4)r)).Lemma 3.3 (see [24]).
Letμ be a positive measure on Cn(Ω) such that there is a sequence of points Pi=(ri,Θi)∈Cn(Ω), ri→∞(i→∞) satisfying
(3.6)GΩaμ(Pi)=∫Cn(Ω)‍GΩa(Pi,Q)dμ(t,Φ)<∞(i=1,2,3,…;Q=(t,Φ)∈Cn(Ω)).
Then for a positive number ℓ,
(3.7)∫Cn(Ω;(ℓ,∞))‍W(t)φ(Φ)dμ(t,Φ)<∞,limR→∞W(R)V(R)∫Cn(Ω;(0,R))‍V(t)φ(Φ)dμ(t,Φ)=0.Lemma 3.4 (see [24]).
Letν be a positive measure on Sn(Ω) such that there is a sequence of points Pi=(ri,Θi)∈Cn(Ω), ri→∞(i→∞) satisfying
(3.8)∫Sn(Ω)‍∂GΩa(Pi,Q)∂nQdν(Q)<∞(i=1,2,3,…;Q=(t,Φ)∈Cn(Ω)).
Then for a positive number ℓ,
(3.9)∫Sn(Ω;(ℓ,∞))‍W(t)t-1∂φ(Φ)∂nΦdν(t,Φ)<∞,limR→∞W(R)V(R)∫Sn(Ω;(0,R))‍V(t)t-1∂φ(Φ)∂nΦdν(t,Φ)=0.Lemma 3.5.
Letμ be a positive measure on Cn(Ω) for which GΩaμ(P) is defined. Then for any positive number A the set
(3.10){P=(r,Θ)∈Cn(Ω):GΩaμ(P)≥AV(r)φ(Θ)}
is a-minimally thin at ∞ with respect to Cn(Ω).Lemma 3.6.
Letυ(P) be a positive superfunction on Cn(Ω) and put
(3.11)c(υ,a)=infP∈Cn(Ω)υ(P)MΩa(P,∞),cO(υ,a)=infP∈Cn(Ω)υ(P)MΩa(P,O).
Then there are a unique positive measure μ on Cn(Ω) and a unique positive measure ν on Sn(Ω) such that
(3.12)υ(P)=c(υ,a)MΩa(P,∞)+cO(υ,a)MΩa(P,O)+∫Cn(Ω)‍GΩa(P,Q)dμ(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdν(Q),
where ∂/∂nQ denotes the differentiation at Q along the inward normal into Cn(Ω).Proof.
By the Riesz decomposition theorem, we have a unique measureμ on Cn(Ω) such that
(3.13)υ(P)=∫Cn(Ω)‍GΩa(P,Q)dμ(Q)+h(P)(P∈Cn(Ω)),
where h is the greatest a-harmonic minorant of υ on Cn(Ω). Furthermore, by the generalized Martin representation theorem (Lemma 3.8) we have another positive measure ν' on ∂Cn(Ω)∪{∞} satisfying
(3.14)h(P)=∫∂Cn(Ω)∪{∞}‍MΩa(P,Q)dν'(Q)=MΩa(P,∞)ν'({∞})+MΩa(P,O)ν'({O})+∫Sn(Ω)‍MΩa(P,Q)dν'(Q)(P∈Cn(Ω)).
We know from (3.11) that ν'({∞})=c(υ,a) and ν'({O})=cO(υ,a).
Since(3.15)MΩa(P,Q)=limP1→Q,P1∈Cn(Ω)GΩa(P,P1)GΩa(P0,P1)=(∂GΩa(P,Q))/∂nQ(∂GΩa(P0,Q))/∂nQ,
where P0 is a fixed reference point of the generalized Martin kernel, we also obtain
(3.16)h(P)=c(υ,a)MΩa(P,∞)+cO(υ,a)MΩa(P,O)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdν(Q)(P∈Cn(Ω))
by taking
(3.17)dν(Q)={∂GΩa(P0,Q)∂nQ}-1dν′(Q)(Q∈Sn(Ω)).
Hence by (3.13) and (3.16) we get the required.Lemma 3.7.
LetE be a bounded subset of Cn(Ω), and let u(P) be a positive superfunction on Cn(Ω) such that u(P) is represented as
(3.18)u(P)=∫Cn(Ω)‍GΩa(P,Q)dμu(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdνu(Q)
with two positive measures μu(Q) and νu(Q) on Cn(Ω) and Sn(Ω), respectively, and satisfies u(P)≥1 for any P∈E. Then
(3.19)λΩa(E)≤∫Cn(Ω)‍V(t)φ(Φ)dμu(t,Φ)+∫Sn(Ω)‍V(t)t-1φ(Φ)dνu(t,Φ).
When u(P)=R^1E(P)(P∈Cn(Ω)), the equality holds in (3.19).Proof.
SinceλEa is concentrated on BE and u(P)≥1 for any P∈BE, we see that
(3.20)λΩa(E)=∫Cn(Ω)‍dλEa(P)≤∫Cn(Ω)‍u(P)dλEa(P)=∫Cn(Ω)‍R^MΩa(·,∞)Edμu(Q)+∫Sn(Ω)‍(∫Cn(Ω)‍∂GΩa(P,Q)∂nQdλEa(P))dνu(Q).
In addition, we have
(3.21)R^MΩa(·,∞)E(Q)≤MΩa(Q,∞)=V(t)φ(Φ)(Q=(t,Φ)∈Cn(Ω)).
Since
(3.22)∫Cn(Ω)‍∂GΩa(P,Q)∂nQdλEa(P)≤liminfρ→01ρ∫Cn(Ω)‍GΩa(P,Pρ)dλEa(P)
for any Q∈Sn(Ω), where Pρ=(rρ,Θρ)=Q+ρnQ∈Cn(Ω) and nQ is the inward normal unit vector at Q, and
(3.23)∫Cn(Ω)‍GΩa(P,Pρ)dλEa(P)=R^MΩa(·,∞)E(Pρ)≤MΩa(Pρ,∞)=V(rρ)φ(Θρ),
we have
(3.24)∫Cn(Ω)‍∂GΩa(P,Q)∂nQdλEa(P)≤V(t)t-1∂φ(Φ)∂nΦ
for any Q=(t,Φ)∈Sn(Ω). Thus (3.19) follows from (3.20), (3.21), and (3.24). Because R^1E(P) is bounded on Cn(Ω), u(P) has the expression (3.18) by Lemma 3.6 when u(P)=R^1E(P). Then the equalities in (3.20) hold because R^1E(P)=1 for any P∈BE (Doob [21, page 169]). Hence we claim if
(3.25)μu({P∈Cn(Ω):R^MΩa(·,∞)E(P)<MΩa(P,∞)})=0,(3.26)νu({Q=(t,Φ)∈Sn(Ω):∫Cn(Ω)‍∂GΩa(P,Q)∂nQdλEa(P)<V(t)t-1∂φ(Φ)∂nΦ})=0,
then the equality in (3.19) holds.
To see (3.25) we remark that
(3.27){P∈Cn(Ω):R^MΩa(·,∞)E(P)<MΩa(P,∞)}⊂(Cn(Ω))∖BE,μu(Cn(Ω)∖BE)=0.
To prove (3.26) we set
(3.28)BE′={Q∈Sn(Ω):Eisnota-minimallythinatQ},e={P∈E:R^MΩa(·,∞)E(P)<MΩa(P,∞)}.
Then e is an a-polar set, and hence
(3.29)R^MΩa(·,Q)E=R^MΩa(·,Q)E∖e
for any Q∈Sn(Ω). Consequently, for any Q∈BE′, E∖e is not also a-minimally thin at Q, and so
(3.30)∫Cn(Ω)‍MΩa(P,Q)dη(P)=liminfP′→Q,P′∈E∖e∫Cn(Ω)‍MΩa(P,P′)dη(P)
for any positive measure η on Cn(Ω), where
(3.31)MΩa(P,P′)=GΩa(P,P′)GΩa(P0,P′)(P,P′∈Cn(Ω)).
Take η=λEa in (3.30). Since
(3.32)limP→Q,P∈Cn(Ω)MΩa(P,∞)GΩa(P0,P)=V(t)t-1∂φ(Φ)∂nΦ{∂GΩa(P0,Q)∂nQ}-1,(Q=(t,Φ)∈Sn(Ω)),
we obtain from (3.15)
(3.33)∫Cn(Ω)‍∂GΩa(P,Q)∂nΦdλEa(P)=V(t)t-1∂φ(Φ)∂nΦliminfP′→Q,P′∈E∖e∫Cn(Ω)‍GΩa(P,P′)MΩa(P′,∞)dλEa(P)
for any Q∈(t,Φ)∈BE′. Since
(3.34)∫Cn(Ω)‍GΩa(P,P′)MΩa(P′,∞)dλEa(P)=1MΩa(P′,∞)R^MΩa(·,∞)E(P′)=1
for any P'∈E∖e, we have
(3.35)∫Cn(Ω)‍∂GΩa(P,Q)∂nQdλEa(P)=V(t)t-1∂φ(Φ)∂nΦ
for any Q=(t,Φ)∈BE′, which shows
(3.36){Q=(t,Φ)∈Sn(Ω):∫Cn(Ω)‍∂GΩa(P,Q)∂nQdλEa(P)<V(t)t-1∂φ(Φ)∂nΦ}⊂Sn(Ω)∖BE′.
Let h be the greatest a-harmonic minorant of u(P)=R^1E(P), and let νu′ be the generalized Martin representing measure of h. We claim if
(3.37)R^hE(P)=h
on Cn(Ω), then νu′(Sn(Ω)∖BE′)=0. Since
(3.38)dνu′(Q)=∂GΩa(P0,Q)∂nQdνu(Q)(Q∈Sn(Ω))
from (3.15), we also have νu(Sn(Ω)∖BE′)=0, which gives (3.26) from (3.36).
To prove (3.37), we set u*=R^1E(P)-h. Then
(3.39)u*+h=R^1E=R^u*+hE≤R^u*E+R^hE,
and hence
(3.40)R^hE-h≥u*-R^u*E≥0,
from which (3.37) follows.Lemma 3.8 (the generalized Martin representation).
Ifu is a positive a-harmonic function on Cn(Ω), then there exists a measure μu on Δ, uniquely determined by u, such that μu(Δ0)=0 and
(3.41)u(P)=∫Δ‍MΩa(P,Q)dμu(Q)(P∈Cn(Ω)),
where Δ0 is the same as the previous statement.Remark 3.9.
Following the same method of Armitage and Gardiner [3] for Martin representation we may easily prove Lemma 3.8.


## 4. Proofs of the Main Theorems

Proof of Theorem2.1.
First we assume that (b) holds, and letu=R^MΩa(·,Q)E. Since MΩa(·,Q) is minimal, the Riesz decomposition of u is of the form υ+ℓMΩa(·,Q), where υ is an a-potential associated with the stationary Schrödinger operator on Cn(Ω) and 0<ℓ<1. Since u=MΩa(·,Q) quasieverywhere on E and R^υE+ℓu=υ+ℓMΩa(·,Q)=u quasieverywhere on E,
(4.1)R^MΩa(·,Q)E=R^uE≤R^υE+ℓu≤υ+ℓMΩa(·,Q)=R^MΩa(·,Q)E.
Hence ℓ(MΩa(·,Q)-u)≡0, so ℓ=0 by the hypothesis and (a) holds.
Next we assume (a) holds, and letωm be a decreasing sequence of compact neighborhoods of Q in the Martin topology such that ⋂m‍ωm={Q}. Then R^MΩa(·,Q)E⋂‍ωm is a-harmonic on Cn(Ω)∖ωm, and the decreasing sequence {R^MΩa(·,Q)E⋂‍ωm} has a limit h which is a-harmonic on Cn(Ω). Since h is majorized by R^MΩa(·,Q)E, it follows that h≡0 and (c) holds.
Finally we assume (c) holds, then there is a Martin topology neighborhoodω of Q such that R^MΩa(·,Q)E⋂‍ω≠MΩa(·,Q). Since (b) implies (a), the set E⋂‍ω is a-minimally thin at Q and so R^MΩa(·,Q)E⋂‍ω is an a-potential. Then R^MΩa(·,Q)E is an a-potential and we yield (b).Proof of Theorem2.2.
Obviously we see that (c) implies (b). If (b) holds, then there existℓ>μu({Q}) and a Martin topology neighborhood ω of Q such that u≥ℓMΩa(·,Q) on E⋂‍ω. If R^MΩa(·,Q)E⋂‍ω=MΩa(·,Q), then u≥R^uE⋂‍ω≥ℓMΩa(·,Q), and this yields contradictory conclusion that μu=ℓδQ+μu-ℓMΩa(·,Q)>μu({Q})δQ, where δQ is the unit measure with support {Q}. Hence R^MΩa(·,Q)E⋂‍ω≠MΩa(·,Q). Thus E⋂‍ω is a-minimally thin at Q, and so (a) holds.
Finally we assume (a) holds. By Lemma3.1 there is an open subset U of Cn(Ω) such that E⊆U and U is a-minimally thin at Q. By Theorem 2.1 there is a sequence {ωm} of Martin topology open neighborhoods of Q such that R^MΩa(·,Q)E⋂‍ωm(P0)<2-m. The function u1=∑n‍R^MΩa(·,Q)U⋂‍ωm, being a sum of a-potentials, is an a-potential since u1(P0)<∞. Further, since R^MΩa(·,Q)E⋂‍ωm=MΩa(·,Q) on the open set E⋂‍ωm,
(4.2)u1(P)MΩa(P,Q)→∞(P→Q;P∈U),
and so (c) holds.Proof of Theorem2.3.
Clearly (c) implies (b). To prove that (b) implies (a), we suppose that (b) holds and chooseA such that
(4.3)liminfP→Q,P∈EGΩaμ(P)GΩa(P0,P)>A>∫‍MΩa(·,Q)dμ.
Then GΩaμ>AGΩa(P0,·) on E∩ω for some Martin topology neighborhood ω of Q. If ν denotes the swept measure of δP0 onto E∩ω, where δP0 is the unit measure with support {P0}, then it follows that
(4.4)GΩaμ≥AR^GΩa(P0,·)E∩ω=AGΩaν
on Cn(Ω). Let {Kn} be a sequence of compact subsets of Cn(Ω) such that ⋃n‍Kn=Cn(Ω), and let GΩaμn denote the a-potential R^MΩa(·,Q)Kn. Then
(4.5)∫‍R^MΩa(·,Q)Kndν=∫‍GΩaνdμn≤A-1∫‍GΩaμdμn=A-1∫‍R^MΩa(·,Q)Kndμ.
Letting n→∞, we see from our choice of A that
(4.6)R^MΩa(·,Q)E∩ω(P0)=∫‍MΩa(·,Q)dν≤A-1∫‍MΩa(·,Q)dμ<1=MΩa(P0,Q),
then E∩ω is a-minimally thin at Q by Theorem 2.1, and so (a) holds.
Next we suppose that (a) holds. By Lemma3.1 there is an open subset U of Cn(Ω) such that E⊆U and U is a-minimally thin at Q. By Theorem 2.1 there is a sequence {ωn} of Martin topology open neighborhoods of Q such that
(4.7)∑n‍R^MΩa(·,Q)U∩ωn(P0)<∞.
Let μ'=∑nνn, where νn is swept measure of δP0 onto U∩ωn. Then
(4.8)∫‍MΩa(P,Q)dμ′(P)=∑n‍∫‍MΩa(P,Q)dνn(P)=∑n‍R^MΩa(·,Q)U∩ωn(P0)<∞,
and (2.10) holds since
(4.9)GΩaνn=R^GΩa(P0,·)U∩ωn=GΩa(P0,·)
on the open set U∩ωn, so (c) holds.Proof of Theorem2.4.
Since (2.11) is independent of the choice of Q0, we may multiply across by MΩa(Q0,Q). Thus we may assume that Q0=P0 and claim that
(4.10)liminfP→QGΩaμ(P)GΩa(P0,P)=∫‍MΩa(P,Q)dμ(P)
for any a-potential GΩaμ. According to Fatou's lemma, we may yield
(4.11)liminfP→QGΩaμ(P)GΩa(P0,P)≥∫‍MΩa(P,Q)dμ(P).
Since Cn(Ω) is not a-minimally thin at Q, we know that
(4.12)liminfP→QGΩaμ(P)GΩa(P0,P)<∫‍MΩa(P,Q)dμ(P)
from Theorem 2.3. Hence the claim holds.
WhenE is a-minimally thin at Q, we see from (4.10) and the condition (b) of Theorem 2.3 that (2.11) holds for some a-potential u. Conversely, if (2.11) holds, then we can choose A such that
(4.13)liminfP→Q,P∈Eu(P)GΩa(P0,P)>A>liminfP→Qu(P)GΩa(P0,P)
and define GΩaμ by min{u,AGΩa(P0,·)}. Then by (4.10)
(4.14)liminfP→Q,P∈EGΩaμ(P)GΩa(P0,P)=A>liminfP→QGΩaμ(P)GΩa(P0,P)=∫‍MΩa(P,Q)dμ(P),
and it follows from Theorem 2.3 that E is a-minimally thin at Q.Proof of Theorem2.5.
By applying the Riesz decomposition theorem to the superfunctionR^MΩa(·,∞)E on Cn(Ω), we have a positive measure μ on Cn(Ω) satisfying
(4.15)GΩaμ(P)<∞
for any P∈Cn(Ω) and a nonnegative greatest a-harmonic minorant H of R^MΩa(·,∞)E such that
(4.16)R^MΩa(·,∞)E=GΩaμ(P)+H.
We remark that MΩa(·,∞)(P∈Cn(Ω)) is a minimal function at ∞. If E is a-minimally thin at ∞ with respect to Cn(Ω), then R^MΩa(·,∞)E is an a-potential, and hence H≡0 on Cn(Ω). Since
(4.17)R^MΩa(·,∞)E(P)=MΩa(P,∞)
for any P∈BE, we see from (4.16) that
(4.18)GΩaμ(P)=MΩa(P,∞)
for any P∈BE. Take a sufficiently large R from Lemma 3.3 such that
(4.19)C2W(R)V(R)∫Cn(Ω;(0,R])‍V(t)φ(Φ)dμ(t,Φ)<14.
Then from (1.16) or (1.17),
(4.20)∫Cn(Ω;(0,R])‍GΩa(P,Q)dμ(Q)<14MΩa(P,∞)
for any P=(r,Θ)∈Cn(Ω) and r≥(5/4)r, and hence from (4.18)
(4.21)∫Cn(Ω;[R,∞))‍GΩa(P,Q)dμ(Q)≥34MΩa(P,∞)
for any P=(r,Θ)∈BE and r≥(5/4)r. Divide GΩaμ into three parts as follows:
(4.22)GΩaμ(P)=A1(k)(P)+A2(k)(P)+A3(k)(P)(P=(r,Θ)∈Cn(Ω)),
where
(4.23)A1(k)(P)=∫Cn(Ω;(2k-1,2k+2))‍GΩa(P,Q)dμ(Q),A2(k)(P)=∫Cn(Ω;(0,2k-1])‍GΩa(P,Q)dμ(Q),A3(k)(P)=∫Cn(Ω;[2k+2,∞))‍GΩa(P,Q)dμ(Q).
Now we claim that there exists an integer N such that
(4.24)BE∩Ik(Ω)¯⊂{P=(r,Θ)∈Cn(Ω):A1(k)(P)≥14V(r)φ(Θ)}(k≥N).
When we choose a sufficiently large integer N1 by Lemma 3.3 such that
(4.25)W(2k)V(2k)∫Cn(Ω;(0,2k])‍V(t)φ(Φ)dμ(t,Φ)<14C2(k≥N1),∫Cn(Ω;[2k+2,∞))‍W(t)φ(Φ)dμ(t,Φ)<14C2(k≥N1)
for any P=(r,Θ)∈Ik(Ω)¯∩Cn(Ω), we have from (1.16) or (1.17) that
(4.26)A2(k)(P)≤14V(r)φ(Θ)(k≥N1),A3(k)(P)≤14V(r)φ(Θ)(k≥N1).
Put (4.27)N=max{N1,[logRlog2]+2}.
For any P=(r,Θ)∈BE∩Ik(Ω)¯(k≥N), we have from (4.21), (4.22), and (4.26) that
(4.28)A1(k)(P)≥∫Cn(Ω;[R,∞))‍GΩa(P,Q)dμ(Q)-A2(k)(P)-A3(k)(P)≥14V(r)φ(Θ),
which shows (4.24).
Since the measureλEka is concentrated on BEk and BEk⊂BE∩Ik(Ω)¯, finally we obtain by (4.24) that
(4.29)γΩa(Ek)=∫Cn(Ω)‍(GΩaλEka)dλEka(P)≤∫BEk‍V(r)φ(Θ)dλEka(r,Θ)≤4∫BEk‍A1(k)(P)dλEka(P)≤4∫Cn(Ω;(2k-1,2k+2))‍{∫Cn(Ω)‍GΩa(P,Q)dλEka(P)}dμ(Q)≤4∫Cn(Ω;(2k-1,2k+2))V(t)φ(Φ)dμ(t,Φ)(k≥N),
and hence
(4.30)∑k=N∞‍γΩa(Ek)W(2k)V(2k)-1≲∑k=N∞‍∫Cn(Ω;(2k-1,2k+2))‍W(t)φ(Φ)dμ(t,Φ)=∫Cn(Ω;(2N-1,∞))‍W(t)φ(Φ)dμ(t,Φ)<∞
from Lemma 3.3, (1.11) and Lemma C.1 in ([11] or [13]), which gives (2.13).
Next we will prove the sufficiency. Since(4.31)R^MΩa(·,∞)Ek(Q)=MΩa(Q,∞)
for any Q∈BEk as in (4.17), we have
(4.32)γΩa(Ek)=∫BEk‍MΩa(Q,∞)dλEka(Q)≥V(2k)∫BEk‍φ(Φ)dλEka(t,Φ)(Q=(t,Φ)∈Cn(Ω)),
and hence from (1.16) or (1.17), (1.11), and (1.12)
(4.33)R^MΩa(·,∞)Ek(P)≤C2V(r)φ(Θ)∫BEk‍W(t)φ(Φ)dλEka(t,Φ)≤C2V(r)φ(Θ)V-1(2k)W(2k)γΩa(Ek)
for any P=(r,Θ)∈Cn(Ω) and any integer k satisfying 2k≥(5/4)r. Define a measure μ on Cn(Ω) by
(4.34)dμ(Q)={∑k=0∞‍dλEka(Q)(Q∈Cn(Ω;[1,∞))),0(Q∈Cn(Ω;(0,1))).
Then from (2.13) and (4.33)
(4.35)GΩaμ(P)=∫Cn(Ω)‍GΩa(P,Q)dμ(Q)=∑k=0∞‍R^MΩa(·,∞)Ek(P)
is a finite-valued superfunction on Cn(Ω) and
(4.36)GΩaμ(P)≥∫Cn(Ω)‍GΩa(P,Q)dλEka(Q)=R^MΩa(·,∞)Ek(P)=V(r)φ(Θ)
for any P=(r,Θ)∈BEk, and from (1.16) or (1.17)
(4.37)GΩaμ(P)≥C′V(r)φ(Θ)
for any P=(r,Θ)∈Cn(Ω;(0,1]), where
(4.38)C'=C1∫Cn(Ω;[5/4,∞))‍W(t)φ(Φ)dμ(t,Φ).
If we set
(4.39)E′=⋃k=0∞‍BEk,E1=E∩Cn(Ω;(0,1]),C=min(C',1),
then
(4.40)E'⊂{P=(r,Θ)∈Cn(Ω);GΩaμ(P)≥CV(r)φ(Θ)}.
Hence by Lemma 3.5, E' is a-minimally thin at ∞ with respect to Cn(Ω); namely, there is a point P'∈Cn(Ω) such that
(4.41)R^MΩa(·,∞)E′(P′)≠MΩa(P′,∞).
Since E' is equal to E except an a-polar set, we know that
(4.42)R^MΩa(·,∞)E′(P)=R^MΩa(·,∞)E(P)
for any P∈Cn(Ω), and hence
(4.43)R^MΩa(·,∞)E(P′)≠MΩa(P′,∞).
So E is a-minimally thin at ∞ with respect to Cn(Ω).Proof of Theorem2.6.
Let a subsetE of Cn(Ω) be an a-rarefied set at ∞ with respect to Cn(Ω). Then there exists a positive superfunction υ(P) on Cn(Ω) such that c(υ,a)≡0 and
(4.44)E⊂Hυ.
By Lemma 3.6 we can find two positive measures μ on Cn(Ω) and ν on Sn(Ω) such that
(4.45)υ(P)=cO(υ,a)MΩa(P,O)+∫Cn(Ω)‍GΩa(P,Q)dμ(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdν(Q)(P∈Cn(Ω)).
Set
(4.46)υ(P)=cO(υ,a)MΩa(P,O)+B1(k)(P)+B2(k)(P)+B3(k)(P),
where
(4.47)B1(k)(P)=∫Cn(Ω;(0,2k-1])‍GΩa(P,Q)dμ(Q)+∫Sn(Ω;(0,2k-1])‍∂GΩa(P,Q)∂nQdν(Q),B2(k)(P)=∫Cn(Ω;(2k-1,2k+2))‍GΩa(P,Q)dμ(Q)+∫Sn(Ω;(2k-1,2k+2))‍∂GΩa(P,Q)∂nQdν(Q),B3(k)(P)=∫Cn(Ω;[2k+2,∞))‍GΩa(P,Q)dμ(Q)+∫Sn(Ω;[2k+2,∞))‍∂GΩa(P,Q)∂nQdν(Q)(P∈Cn(Ω);k=1,2,3,…).
First we will prove there exists an integer N such that
(4.48)Hυ∩Ik(Ω)⊂{P=(r,Θ)∈Ik(Ω);B2(k)(P)≥12V(r)}
for any integer k≥N. Since υ(P) is finite almost everywhere on Cn(Ω), we may apply Lemmas 3.3 and 3.4 to
(4.49)∫Cn(Ω)‍GΩa(P,Q)dμ(Q),∫Sn(Ω)‍∂GΩa(P,Q)∂nQdν(Q),
respectively; then we can take an integer N such that
(4.50)W(2k-1)V(2k-1)∫Cn(Ω;(0,2k-1])‍V(t)φ(Φ)dμ(t,Φ)≤112JΩC2,(4.51)∫Cn(Ω;[2k+2,∞))‍W(t)φ(Φ)dμ(t,Φ)≤112JΩC2,(4.52)W(2k-1)V(2k-1)∫Sn(Ω;(0,2k-1])‍V(t)t-1∂φ(Φ)∂nΦdν(t,Φ)≤112JΩC2,(4.53)∫Sn(Ω;[2k+2,∞))‍W(t)t-1∂φ(Φ)∂nΦdν(t,Φ)≤112JΩC2
for any integer k≥N, where
(4.54)JΩ=supΘ∈Ωφ(Θ).
Then for any P=(r,Θ)∈Ik(Ω)(k≥N), we have
(4.55)B1(k)(P)≤C2JΩW(r)∫Cn(Ω;(0,2k-1])‍V(t)φ(Φ)dμ(t,Φ)+C2JΩW(r)∫Sn(Ω;(0,2k-1])‍V(t)t-1∂φ(Φ)∂nΦdν(t,Φ)≤V(r)6
from (1.16) or (1.17), (3.3) or (3.4), (4.50), and (4.52), and
(4.56)B3(k)(P)≤C2JΩV(r)∫Cn(Ω;[2k+2,∞))‍W(t)φ(Φ)dμ(t,Φ)+C2JΩV(r)∫Sn(Ω;[2k+2,∞))‍W(t)t-1∂φ(Φ)∂nΦdν(t,Φ)≤V(r)6
from (1.16) or (1.17), (3.3) or (3.4), (4.51), and (4.53). Further we can assume that
(4.57)6κcO(υ,a)JΩ≤V(r)W(r)-1
for any P=(r,Θ)∈Ik(Ω)(k≥N). Hence if P=(r,Θ)∈Ik(Ω)∩Hυ(k≥N), we obtain
(4.58)B2(k)(P)≥υ(P)-V(r)6-B1(k)(P)-B3(k)(P)≥V(r)2
from (4.46) which gives (4.48).
We see from (4.44) and (4.48) that
(4.59)B2(k)(P)≥12V(2k)(k≥N)
for any P∈Ek. Define a function uk(P) on Cn(Ω) by
(4.60)uk(P)=2V(2k)-1B2(k)(P).
Then
(4.61)uk(P)≥1(P∈Ek,k≥N),uk(P)=∫Cn(Ω)‍GΩa(P,Q)dμk(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdνk(Q)
with two measures
(4.62)dμk(Q)={2V(2k)-1dμ(Q)(Q∈Cn(Ω;(2k-1,2k+2))),0(Q∈Cn(Ω;(0,2k-1])∪Cn(Ω;[2k+2,∞))),dνk(Q)={2V(2k)-1dν(Q)(Q∈Sn(Ω;(2k-1,2k+2))),0(Q∈Sn(Ω;(0,2k-1])∪Sn(Ω;[2k+2,∞))).
Hence by applying Lemma 3.7 to uk(P), we obtain
(4.63)λΩa(Ek)≤2V(2k)-1∫Cn(Ω;(2k-1,2k+2))‍V(t)φ(Φ)dμ(t,Φ)+2V(2k)-1∫Sn(Ω;(2k-1,2k+2))‍V(t)t-1∂φ(Φ)∂nΦdν(t,Φ)(k≥N).
Finally we have by (1.11), (1.12), and (1.14)
(4.64)∑k=N∞W(2k)λΩa(Ek)≲∫Cn(Ω;(2N-1,∞))‍W(t)φ(Φ)dμ(t,Φ)+∫Sn(Ω;(2N-1,∞))‍W(t)t-1∂φ(Φ)∂nΦdν(t,Φ).
If we take a sufficiently large N, then the integrals of the right side are finite from Lemmas 3.3 and 3.4.
Suppose that a subsetE of Cn(Ω) satisfies
(4.65)∑k=0∞‍W(2k)λΩa(Ek)<∞.
Then we apply the second part of Lemma 3.7 to Ek and get
(4.66)∑k=1∞‍W(2k){∫Cn(Ω)‍V(t)φ(Φ)dμk*(t,Φ)+∫Sn(Ω)‍V(t)t-1∂φ(Φ)∂nΦdνk*(t,Φ)}<∞,
where μk* and νk* are two positive measures on Cn(Ω) and Sn(Ω), respectively, such that
(4.67)R^1Ek(P)=∫Cn(Ω)‍GΩa(P,Q)dμk*(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdνk*(Q).
Consider a function υ0(P) on Cn(Ω) defined by
(4.68)υ0(P)=∑k=-1∞‍V(2k+1)R^1Ek(P)(P∈Cn(Ω)),
where
(4.69)E-1=E∩{P=(r,Θ)∈Cn(Ω);0<r<1}.
Then υ0(P) is a superfunction or identically ∞ on Cn(Ω). We take any positive integer k0 and represent υ0(P) by
(4.70)υ0(P)=υ1(P)+υ2(P),
where
(4.71)υ1(P)=∑k=-1k0+1‍V(2k+1)R^1Ek(P),υ2(P)=∑k=k0+2∞‍V(2k+1)R^1Ek(P).
Since μk* and νk* are concentrated on BEk⊂Ek¯∩Cn(Ω) and BEk′⊂Ek¯∩Sn(Ω), respectively, we have from (1.16) or (1.17), (3.3) or (3.4), (1.11), and (1.12) that
(4.72)∫Cn(Ω)‍GΩa(P′,Q)dμk*(Q)≤C2V(r′)φ(Θ′)∫Cn(Ω)‍W(t)φ(Φ)dμk*(t,Φ)≤C2W(2k)V(2k)-1V(r′)φ(Θ′)×∫Cn(Ω)‍V(t)φ(Φ)dμk*(t,Φ),∫Sn(Ω)‍∂GΩa(P′,Q)∂nQdνk*(Q)≤C2W(2k)V(2k)-1V(r′)φ(Θ′)∫Sn(Ω)‍V(t)t-1∂φ(Φ)∂nΦdνk*(t,Φ)
for a point P'=(r',Θ')∈Cn(Ω), where r'≤2k0+1 and k≤k0+2. Hence we know by (1.11), (1.12), and (1.14) that
(4.73)υ2(P′)≲V(r′)φ(Θ′)∑k=k0+2∞‍W(2k)∫Cn(Ω)‍V(t)φ(Φ)dμk*(t,Φ)+V(r′)φ(Θ′)∑k=k0+2∞‍W(2k)∫Sn(Ω)‍V(t)t-1∂φ(Φ)∂nΦdνk*(t,Φ).
This and (4.66) show that υ2(P') is finite, and hence υ0(P) is a positive superfunction on Cn(Ω). To see
(4.74)c(υ0,a)=infP∈Cn(Ω)υ0(P)MΩa(P,∞)=0,
we consider the representations of υ0(P), υ1(P), and υ2(P) by Lemma 3.6 as follows:
(4.75)υ0(P)=c(υ0,a)MΩa(P,∞)+cO(υ0,a)MΩa(P,O)+∫Cn(Ω)‍GΩa(P,Q)dμ(0)(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdν(0)(Q),υ1(P)=c(υ1,a)MΩa(P,∞)+cO(υ1,a)MΩa(P,O)+∫Cn(Ω)‍GΩa(P,Q)dμ(1)(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdν(1)(Q),υ2(P)=c(υ2,a)MΩa(P,∞)+cO(υ2,a)MΩa(P,O)+∫Cn(Ω)‍GΩa(P,Q)dμ(2)(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdν(2)(Q).
It is evident from (4.67) that c(υ1,a)=0 for any k0. Since c(υ0,a)=c(υ2,a) and
(4.76)c(υ2,a)=infP∈Cn(Ω)υ2(P)MΩa(P,∞)≤υ2(P′)MΩa(P′,∞)≲∑k=k0+2∞‍W(2k)∫Cn(Ω)‍V(t)φ(Φ)dμk*(t,Φ)+∑k=k0+2∞‍W(2k)∫Sn(Ω)‍V(t)t-1∂φ(Φ)∂nΦdνk*(t,Φ)→0(k0→∞)
from (4.66) and (4.73), we know c(υ0,a)=0 which is (4.74). Since R^1Ek=1 on BEk⊂Ek¯∩Cn(Ω), we know that
(4.77)υ0(P)≥V(2k+1)≥V(r)
for any P=(r,Θ)∈BEk(k=-1,0,1,2,…). We set E'=∪k=-1∞BEk; then
(4.78)E'⊂Hυ0.
Since E' is equal to E except an a-polar set S, we can take another positive superfunction υ3 on Cn(Ω) such that υ3=GΩaη with a positive measure η on Cn(Ω), and υ3 is identically ∞ on S. Define a positive superfunction υ on Cn(Ω) by
(4.79)υ=υ0+υ3.
Since c(υ3,a)=0, it is easy to see from (4.74) that c(υ,a)=0. In addition, we know from (4.78) that E⊂Hυ. Then the subset E of Cn(Ω) is a-rarefied at ∞ with respect to Cn(Ω).Proof of Theorem2.8.
By Lemma3.6 we have
(4.80)υ(P)=c(υ,a)MΩa(P,∞)+cO(υ,a)MΩa(P,O)+∫Cn(Ω)‍GΩa(P,Q)dμ(Q)+∫Sn(Ω)‍∂GΩa(P,Q)∂nQdν(Q)
for a unique positive measure μ on Cn(Ω) and a unique positive measure ν on Sn(Ω), respectively; then
(4.81)υ1(P)=υ(P)-c(υ,a)MΩa(P,∞)-cO(υ,a)MΩa(P,O)(P=(r,Θ)∈Cn(Ω))
also is a positive superfunction on Cn(Ω) such that
(4.82)infP=(r,Θ)∈Cn(Ω)υ1(P)MΩa(P,∞)=0.
Next we will prove there exists an a-rarefied set E at ∞ with respect to Cn(Ω) such that
(4.83)υ1(P)V(r)-1(P=(r,Θ)∈Cn(Ω))
uniformly converges to 0 on Cn(Ω)∖E as r→∞. Let {εi} be a sequence of positive numbers εi satisfying εi→0 as i→∞, and put
(4.84)Ei={P=(r,Θ)∈Cn(Ω);υ1(P)≥εiV(r)}(k=1,2,3,…).
Then Ei(k=1,2,3,…) are a-rarefied sets at ∞ with respect to Cn(Ω), and hence by Theorem 2.6(4.85)∑k=0∞‍W(2k)λΩa((Ei)k)<∞(i=1,2,3,…).
We take a sequence {qi} such that
(4.86)∑k=qi∞‍W(2k)λΩa((Ei)k)<12i(i=1,2,3,…),
and set
(4.87)E=∪i=1∞∪k=qi∞(Ei)k.
Because λΩa is a countably subadditive set function as in Aikawa [25], Essén, and Jackson [4],
(4.88)λΩa(Em)≤∑i=1∞‍∑k=qi∞‍λΩa(Ei∩Ik∩Im)(m=1,2,3,…).
Since
(4.89)∑m=1∞‍λΩa(Em)W(2m)≤∑i=1∞‍∑k=qi∞‍∑m=1∞‍λΩa(Ei∩Ik∩Im)W(2m)=∑i=1∞‍∑k=qi∞λΩa((Ei)k)W(2k)≤∑i=1∞‍12i=1,
by Theorem 2.6 we know that E is an a-rarefied set at ∞ with respect to Cn(Ω). It is easy to see that
(4.90)υ(P)V(r)-1(P=(r,Θ)∈Cn(Ω))
uniformly converges to 0 on Cn(Ω)∖E as r→∞.Proof of Theorem2.10.
SinceλEka is concentrated on BEk⊂Ek¯∩Cn(Ω), we see that
(4.91)γΩa(Ek)=∫Cn(Ω)‍R^MΩa(·,∞)Ek(P)dλEka(P)≤∫Cn(Ω)‍MΩa(P,∞)dλEka(P)≤JΩV(2k+1)λΩa(Ek),
and hence
(4.92)∑k=0∞γΩa(Ek)W(2k)V(2k)-1≲∑k=0∞‍W(2k)λΩa(Ek)
which gives the conclusion of the first part with Theorems 2.5 and 2.6. To prove the second part, we put JΩ′=minΘ∈Ω'¯φ(Θ). Since
(4.93)MΩa(·,∞)=V(r)φ(Θ)≥JΩ′V(r)≥J′ΩV(2k)(P=(r,Θ)∈Ek),R^MΩa(·,∞)Ek(P)=MΩa(·,∞)
for any P=(r,Θ)∈BEk, we have
(4.94)γΩa(Ek)=∫Cn(Ω)‍R^MΩa(·,∞)Ek(P)dλEka(P)≥JΩ′V(2k)λΩa(Ek).
Since
(4.95)JΩ′∑k=0∞‍λΩa(Ek)W(2k)≤∑k=0∞‍V(2k)-1W(2k)γΩa(Ek)<∞
from Theorem 2.5, it follows from Theorem 2.6 that E is a-rarefied at ∞ with respect to Cn(Ω).



---

*Source: 453891-2012-09-09.xml*