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Abstract and Applied Analysis
Volume 2012, Article ID 453891, 29 pages
http://dx.doi.org/10.1155/2012/453891
Research Article

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

Laboratory of Mathematics and Complex Systems, School of Mathematical Science, Beijing Normal University, Beijing 100875, China

Received 8 April 2012; Accepted 18 June 2012

Academic Editor: Ondřej Došlý

Copyright © 2012 Pinhong Long et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give some criteria for a-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.

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 by 𝐑𝑛(𝑛2) the 𝑛-dimensional Euclidean space. For an open subset set 𝐒𝐑𝑛, denote its boundary by 𝜕𝐒 and its closure by 𝐒. Let 𝑃=(𝑋,𝑥𝑛), where 𝑋=(𝑥1,𝑥2,,𝑥𝑛1), and let |𝑃| be the Euclidean norm of 𝑃 and |𝑃𝑄| the Euclidean distance of two points 𝑃 and 𝑄 in 𝐑𝑛. The unit sphere and the upper half unit sphere are denoted by 𝐒𝑛1 and 𝐒+𝑛1, respectively. For 𝑃𝐑𝑛 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 (𝑛1)-dimensional volume elements induced by the Euclidean metric on 𝑆𝑟.

For 𝑃=(𝑋,𝑥𝑛)𝐑𝑛, it can be reexpressed in spherical coordinates (𝑟,Θ), Θ=(𝜃1,𝜃2,,𝜃𝑛) via the following transforms: 𝑥1=𝑟𝑛1𝑗=1sin𝜃𝑗(𝑛2),𝑥𝑛=𝑟cos𝜃1,(1.1) and if 𝑛3, 𝑥𝑛𝑘+1=𝑟cos𝜃𝑘𝑘1𝑗=1sin𝜃𝑗(2𝑘𝑛1),(1.2) where 0𝑟<,0𝜃𝑗𝜋(1𝑗𝑛2;𝑛3) and 𝜋/2𝜃𝑛1(3𝜋/2)(𝑛2).

Relative to the system of spherical coordinates, the Laplace operator Δ may be written as Δ=𝑛1𝑟𝜕+𝜕𝜕𝑟2𝜕𝑟2+Δ𝑟2,(1.3) where the explicit form of the Beltrami operator Δ is given by Azarin (see [6]).

Let 𝐷 be an arbitrary domain in 𝐑𝑛, and 𝒜𝐷 denotes the class of nonnegative radial potentials 𝑎(𝑃) (i.e., 0𝑎(𝑃)=𝑎(𝑟) for 𝑃=(𝑟,Θ)𝐷) such that 𝑎𝐿𝑏loc(𝐷) with some 𝑏>𝑛/2 if 𝑛4 and with 𝑏=2 if 𝑛=2 or 𝑛=3.

For the identical operator 𝐼, define the stationary Schrödinger operator with a potential 𝑎() by 𝑎=Δ+𝑎()𝐼.(1.4) If 𝑎𝒜𝐷, then 𝑎 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 𝑎 has a Green 𝑎-function 𝐺𝑎𝐷(,). Here 𝐺𝑎𝐷(,) is positive on 𝐷, and its inner normal derivative 𝜕𝐺𝑎𝐷(,𝑄)/𝜕𝑛𝑄 is nonnegative, where 𝜕/𝜕𝑛𝑄 denotes the differentiation at 𝑄 along the inward normal into 𝐷. We write this derivative by 𝑃𝐼𝑎𝐷(,), which is called the Poisson 𝑎-kernel with respect to 𝐷. Denote by 𝐺0𝐷(,) the Green function of Laplacian. It is well known that 𝐺𝑎𝐷(,)𝐺0𝐷(,)(1.5) 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 𝐺𝑎𝐷(,)𝑀(𝐷)𝐺0𝐷(,),(1.6) 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 function 𝑢 is upper semicontinuous in 𝐷. We call 𝑢[,+) a subfunction for the Schrödinger operator 𝑎 if the generalized mean-value inequality 𝑢(𝑃)𝑆(𝑃,𝜌)𝑢(𝑄)𝜕𝐺𝑎𝐵(𝑃,𝜌)(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜎(𝑄)(1.7) is satisfied at each point 𝑃𝐷 with 0<𝜌<inf𝑄𝜕𝐷|𝑃𝑄|, where 𝑆(𝑃,𝜌)=𝜕𝐵(𝑃,𝜌), 𝐺𝑎𝐵(𝑃,𝜌)(,) is the Green 𝑎-function of 𝑎 in 𝐵(𝑃,𝑟), and 𝑑𝜎() the surface area element on 𝑆(𝑃,𝜌) (see [10]).

Denote by 𝑆𝑏𝐻(𝑎,𝐷) the class of subfunctions in 𝐷. We call 𝑢 a superfunction associated with 𝑎 if 𝑢𝑆𝑏𝐻(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 𝑎. 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 [1113]).

For simplicity, the point (1,Θ) on 𝐒𝑛1 and the set {Θ;(1,Θ)Ω} for a set Ω𝐒𝑛1 are often identified with Θ and Ω, respectively. For Ξ𝐑+ and Ω𝐒𝑛1, the set {(𝑟,Θ)𝐑𝑛;𝑟Ξ,(1,Θ)Ω} in 𝐑𝑛 is simply denoted by Ξ×Ω. In particular, the half space {(𝑋,𝑥𝑛)𝐑𝑛;𝑥𝑛>0}=𝐑+×𝐒+𝑛1 will be denoted by 𝐓𝑛. We denote by 𝐶𝑛(Ω) the set 𝐑+×Ω in 𝐑𝑛 with the domain Ω𝐒𝑛1 and call it a cone. For an interval 𝐼𝐑+ and Ω𝐒𝑛1, 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 𝐺𝑎Ω(,) instead of 𝐺𝑎𝐶𝑛(Ω)(,).

Let Ω be a domain on 𝐒𝑛1 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 Δ𝜑+𝜏(Θ)=0onΩ,𝜑(Θ)=0on𝜕Ω(1.8) (see [14, page 41]). In order to ensure the existence of 𝜏 and 𝜑(Θ), we pose the assumption on Ω: if 𝑛3, then Ω is a 𝐶2,𝛼-domain (0<𝛼<1) on 𝑆𝑛1 surrounded by a finite number of mutually disjoint closed hypersurfaces (see e.g., [15, pages 88-89] for the definition of 𝐶2,𝛼-domain).

Let 𝐷 be the class of the potential 𝑎𝒜𝐷 such that lim𝑟𝑟2𝑎(𝑟)=𝜅0[0,),𝑟1||𝑟2𝑎(𝑟)𝜅0||𝐿(1,).(1.9) When 𝑎𝐷, the subfunctions (superfunctions) associated with 𝑎 are continuous (see, e.g., [16]). In the rest of paper, we will always assume that 𝑎𝐷.

An important role will be played by the solutions of the ordinary differential equation 𝑄(𝑟)𝑛1𝑟𝑄𝜏(𝑟)+𝑟2+𝑎(𝑟)𝑄(𝑟)=0(0<𝑟<).(1.10) When the potential 𝑎𝒜𝐷, 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 0𝑉(0+)𝑉(𝑟)as𝑟+(1.11) and 𝑊 is decreasing with +=𝑊(0+)>𝑊(𝑟)0as𝑟+.(1.12) 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 𝜄±𝜅=2𝑛±(𝑛2)2+4(𝜅+𝜏)2.(1.13) When 𝑎𝐷, the normalized solutions 𝑉(𝑟) and 𝑊(𝑟) of (1.10) satisfying 𝑉(1)=𝑊(1)=1 have the asymptotics (see [15]): 𝑉(𝑟)𝑟𝜄+𝜅,𝑊(𝑟)𝑟𝜄𝜅,as𝑟.(1.14) Set 𝜒=𝜄+𝜅𝜄𝜅=(𝑛2)2+4(𝜅+𝜏),𝜒||=𝜔(𝑉(𝑟),𝑊(𝑟))𝑟=1,(1.15) where 𝜒 is their Wronskian at 𝑟=1.

Remark 1.1. If 𝑎=0 and Ω=𝐒+𝑛1, then 𝜄+0=1, 𝜄0=1𝑛 and 𝜑(Θ)=(2𝑛𝑠𝑛1)1/2cos𝜃1, where 𝑠𝑛=2𝜋𝑛/2{Γ(𝑛/2)}1 is the surface area of 𝐒𝑛1.
We recall that 𝐶1𝑉(𝑟)𝑊(𝑡)𝜑(Θ)𝜑(Φ)𝐺𝑎Ω(𝑃,𝑄)𝐶2𝑉(𝑟)𝑊(𝑡)𝜑(Θ)𝜑(Φ),(1.16) or 𝐶1𝑉(𝑡)𝑊(𝑟)𝜑(Θ)𝜑(Φ)𝐺𝑎Ω(𝑃,𝑄)𝐶2𝑉(𝑡)𝑊(𝑟)𝜑(Θ)𝜑(Φ)(1.17) for any 𝑃=(𝑟,Θ)𝐶𝑛(Ω) and any 𝑄=(𝑡,Φ)𝐶𝑛(Ω) satisfying 0<𝑟/𝑡4/5 or 0<𝑡/𝑟4/5, 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 Section 2 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 𝑀𝑎Ω associated with the stationary Schrödinger operator as follows: 𝑀𝑎Ω𝐺(𝑃,𝑄)=𝑎Ω(𝑃,𝑄)𝐺𝑎Ω𝑃0,𝑄𝑃,𝑄𝐶𝑛(Ω)×𝐶𝑛𝑃(Ω)0,𝑃0,(2.1) 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 boundary Δ of 𝐶𝑛(Ω) is the set 𝜕𝐶𝑛(Ω){}. When we denote the Martin kernel associated with the stationary Schrödinger operator by 𝑀𝑎Ω(𝑃,𝑄)(𝑃𝐶𝑛(Ω),𝑄𝜕𝐶𝑛(Ω){}) with respect to a reference point chosen suitably, we see 𝑀𝑎Ω(𝑃,)=𝑉(𝑟)𝜑(Θ),𝑀𝑎Ω(𝑃,𝑂)=𝐾𝑊(𝑟)𝜑(Θ)(2.2) for any 𝑃𝐶𝑛(Ω), where 𝑂 is the origin of 𝐑𝑛 and 𝐾 a positive constant.

Let 𝐸 be a subset of 𝐶𝑛(Ω) and let 𝑢 be a nonnegative superfunction on 𝐶𝑛(Ω). The reduced function of 𝑢 is defined by 𝑅𝐸𝑢𝜐(𝑃)=inf(𝑃)𝜐Φ𝐸𝑢,(2.3) where Φ𝐸𝑢={𝜐𝑆𝑝𝐻(𝑎,𝐶𝑛(Ω))𝑣0on𝐶𝑛(Ω),𝜐𝑢on𝐸}. We define the regularized reduced function 𝑅𝐸𝑢 of 𝑢 relative to 𝐸 as follows: 𝑅𝐸𝑢(𝑃)=lim𝑃𝑃inf𝑅𝐸𝑢𝑃.(2.4) It is easy to see that 𝑅𝐸𝑢 is a superfunction on 𝐶𝑛(Ω).

If 𝐸𝐶𝑛(Ω) and 𝑄Δ, then the Riesz decomposition and the generalized Martin representation allow us to express 𝑅𝐸𝑀𝑎Ω(,𝑄) uniquely in the form 𝐺𝑎Ω𝜇+𝑀𝑎Ω𝜈, where 𝐺𝑎Ω𝜇 and 𝑀𝑎Ω𝜈 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={𝑄Δ𝐶𝑛(Ω)is𝑎-minimallythinat𝑄}, where Δ is the Martin boundary of 𝐶𝑛(Ω).

Now we can state our main theorems.

Theorem 2.1. Let 𝐸𝐶𝑛(Ω) and a fixed point 𝑄ΔΔ0. The following are equivalent:(a)𝐸is 𝑎-minimally thin at 𝑄;(b)𝑅𝐸𝑀𝑎Ω(,𝑄)𝑀𝑎Ω(,𝑄);(c)𝑅inf{𝑀𝐸𝜔𝑎Ω(,𝑄)𝜔isageneralizedMartintopologyneighbourhoodof𝑄}=0.

If 𝑢 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. Let 𝐸𝐶𝑛(Ω) and a fixed point 𝑄ΔΔ0. 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 superfunction 𝑢 such that liminf𝑃𝑄,𝑃𝐸𝑢(𝑃)𝑀𝑎Ω(𝑃,𝑄)>𝜇𝑢({𝑄}),(2.5)(c)there is an 𝑎-potential 𝑢 on 𝐶𝑛(Ω) such that 𝑢(𝑃)𝑀𝑎Ω(𝑃,𝑄)(𝑃𝑄;𝑃𝐸).(2.6)

A set 𝐸 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{𝑃𝜔𝑢(𝑃)=}.

Let 𝐸 be a bounded subset of 𝐶𝑛(Ω). Then 𝑅𝐸𝑀𝑎Ω(,)(𝑃) is bounded on 𝐶𝑛(Ω), and hence the greatest a-harmonic minorant of 𝑅𝐸𝑀𝑎Ω(,)(𝑃) is zero. By the Riesz decomposition theorem there exists a unique positive measure 𝜆𝑎𝐸 associated with the stationary Schrödinger operator 𝑎 on 𝐶𝑛(Ω) such that 𝑅𝐸𝑀𝑎Ω(,)(𝑃)=𝐺𝑎Ω𝜆𝑎𝐸(𝑃)(2.7) for any 𝑃𝐶𝑛(Ω), and 𝜆𝑎𝐸 is concentrated on 𝐵𝐸, where 𝐵𝐸=𝑃𝐶𝑛(Ω)𝐸isnot𝑎-thinat𝑃.(2.8) 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. Let 𝐸𝐶𝑛(Ω) and a fixed point 𝑄ΔΔ0. 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 an 𝑎-potential 𝐺𝑎Ω𝜇 such that liminf𝑃𝑄,𝑃𝐸𝐺𝑎Ω𝜇(𝑃)𝐺𝑎Ω𝑃0>𝑀,𝑃𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑃),(2.9)(c)there is an 𝑎-potential 𝐺𝑎Ω𝜇 such that 𝑀𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑃)< and 𝐺𝑎Ω𝜇(𝑃)𝐺𝑎Ω𝑃0,𝑃(𝑃𝑄;𝑃𝐸).(2.10)

Theorem 2.4. Let 𝐸𝐶𝑛(Ω), 𝑄0𝐶𝑛(Ω) and a fixed point 𝑄ΔΔ0. 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 liminf𝑃𝑄,𝑃𝐸𝑢(𝑃)𝐺𝑎Ω𝑄0,𝑃>liminf𝑃𝑄𝑢(𝑃)𝐺𝑎Ω𝑄0.,𝑃(2.11)

The generalized Green energy 𝛾𝑎Ω(𝐸) of 𝜆𝑎𝐸 is defined by 𝛾𝑎Ω(𝐸)=𝐶𝑛(Ω)𝐺𝑎Ω𝜆𝑎𝐸𝑑𝜆𝑎𝐸.(2.12) Let 𝐸 be a subset of 𝐶𝑛(Ω) and 𝐸𝑘=𝐸𝐼𝑘(Ω), where 𝐼𝑘(Ω)={𝑃=(𝑟,Ω)𝐶𝑛(Ω)2𝑘𝑟2𝑘+1}. The previous theorems are concerned with the fixed boundary points. Next we will consider the case at infinity.

Theorem 2.5. A subset 𝐸 of 𝐶𝑛(Ω) is 𝑎-minimally thin at with respect to 𝐶𝑛(Ω) if and only if 𝑘=0𝛾𝑎Ω𝐸𝑘𝑊2𝑘𝑉2𝑘1<.(2.13)

A subset 𝐸 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 inf𝑃𝐶𝑛(Ω)𝜐(𝑃)𝑀𝑎Ω(𝑃,)0,𝐸𝐻𝜐,(2.14) where 𝐻𝜐=𝑃=(𝑟,Θ)𝐶𝑛(Ω)𝜐(𝑃)𝑉(𝑟).(2.15)

Theorem 2.6. A subset 𝐸 of 𝐶𝑛(Ω) is 𝑎-rarefied at with respect to 𝐶𝑛(Ω) if and only if 𝑘=0𝑊2𝑘𝜆𝑎Ω𝐸𝑘<.(2.16)

Remark 2.7. When 𝑎=0, Theorems 2.5 and 2.6 reduce to the results by Miyamoto and Yoshida [5]. When 𝑎=0 and Ω=𝐒+𝑛1, these are exactly due to Aikawa and Essén [22].
Set 𝑐(𝜐,𝑎)=inf𝑃𝐶𝑛(Ω)𝜐(𝑃)𝑀𝑎Ω(𝑃,)(2.17) 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 limsup𝑃𝑄,𝑃𝐶𝑛(Ω)𝑢(𝑃)0(2.18) for any 𝑄𝜕𝐶𝑛(Ω){𝑂} and sup𝑃=(𝑟,Θ)𝐶𝑛(Ω)𝑢(𝑃)𝑉(𝑟)𝜑(Θ)=(𝑎)<.(2.19) Then we see (𝑎)> (for 𝑎=0, see Yoshida [23]). If we apply this to 𝑢=𝜐, we may obtain 𝑐(𝜐,𝑎)<.

Theorem 2.8. Let 𝜐(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 of 𝑎-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)11 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.

Let 𝑢(𝑃) be a subfunction on 𝐶𝑛(Ω) satisfying (2.18) and (2.19). Then 𝜐(𝑃)=(𝑎)𝑉(𝑟)𝜑(Θ)𝑢(𝑃),𝑃=(𝑟,Θ)𝐶𝑛(Ω)(2.20) 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. Let 𝑢(𝑃) be a subfunction on 𝐶𝑛(Ω) satisfying (2.18) and (2.19) for 𝑃𝐶𝑛(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 (𝑎)𝜑(Θ) on 𝐶𝑛(Ω)𝐸 as 𝑟, where 𝑃=(𝑟,Θ)𝐶𝑛(Ω).

A cone 𝐶𝑛(Ω) is called a subcone of 𝐶𝑛(Ω) if ΩΩ, where Ω is the closure of Ω𝑆𝑛1.

Theorem 2.10. Let 𝐸 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. Let 𝐸1,𝐸2,,𝐸𝑚𝐶𝑛(Ω) and 𝑄Δ.(i)If 𝐸1𝐸2 and 𝐸2 is 𝑎-minimally thin at 𝑄, then 𝐸1 is 𝑎-minimally thin at 𝑄.(ii)If 𝐸1,𝐸2,,𝐸𝑚 are 𝑎-minimally thin at 𝑄, then 𝑚𝑘=1𝐸𝑘 is 𝑎-minimally thin at 𝑄.(iii)If 𝐸1 is 𝑎-minimally thin at 𝑄, then there is an open subset 𝐸 of 𝐶𝑛(Ω) such that 𝐸1𝐸 and 𝐸 is 𝑎-minimally thin at 𝑄.

Proof. Since 𝑅𝐸1𝑀𝑎Ω(,𝑄)𝑅𝐸2𝑀𝑎Ω(,𝑄), we see (i) holds. To prove (ii) we note that 𝑅𝐸𝑘𝑀𝑎Ω(,𝑄) is an 𝑎-potential for each 𝑘 and 𝑚𝑘=1𝑅𝐸𝑘𝑀𝑎Ω(,𝑄)𝑀𝑎Ω(,𝑄)quasieverywhereon𝑚𝑘=1𝐸𝑘,(3.1) so 𝑅𝑘𝐸𝑘𝑀𝑎Ω(,𝑄) is an 𝑎-potential. Finally, to prove (iii), let 𝑅𝑢=𝐸1𝑀𝑎Ω(,𝑄). Then 𝑢 is an 𝑎-potential and 𝑢𝑀𝑎Ω(,𝑄) on 𝐸1𝐹 for some 𝑎-polar set 𝐹. Let 𝜐 be a nonzero 𝑎-potential such that 𝜐= on 𝐹, and let 𝑍=𝑃𝐶𝑛(Ω)𝑢(𝑃)+𝜐(𝑃)𝑀𝑎Ω(𝑃,𝑄).(3.2) Then 𝑍 is open, 𝐸1𝑍 and 𝑅𝑍𝑀𝑎Ω(,𝑄)𝑢+𝜐, so 𝑅𝑍𝑀𝑎Ω(,𝑄) is an 𝑎-potential and 𝑍 is 𝑎-minimally thin at 𝑄.

Lemma 3.2 (see [24]). Consider 𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑡1𝑉(𝑡)𝑊(𝑟)𝜑(Θ)𝜕𝜑(Φ)𝜕𝑛Φ,(3.3)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑉(𝑟)𝑡1𝑊(𝑡)𝜑(Θ)𝜕𝜑(Φ)𝜕𝑛Φ(3.4) for any 𝑃=(𝑟,Θ)𝐶𝑛(Ω) and any 𝑄=(𝑡,Φ)𝑆𝑛(Ω) satisfying 0<𝑡/𝑟4/5(resp.,0<𝑟/𝑡4/5). In addition, 𝜕𝐺0Ω(𝑃,𝑄)𝜕𝑛𝑄𝜑(Θ)𝑡𝑛1𝜕𝜑(Φ)𝜕𝑛Φ+𝑟𝜑(Θ)||||𝑃𝑄𝑛𝜕𝜑(Φ)𝜕𝑛Φ(3.5) 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]). Let 𝜇 be a positive measure on 𝐶𝑛(Ω) such that there is a sequence of points 𝑃𝑖=(𝑟𝑖,Θ𝑖)𝐶𝑛(Ω), 𝑟𝑖(𝑖) satisfying 𝐺𝑎Ω𝜇𝑃𝑖=𝐶𝑛(Ω)𝐺𝑎Ω𝑃𝑖,𝑄𝑑𝜇(𝑡,Φ)<𝑖=1,2,3,;𝑄=(𝑡,Φ)𝐶𝑛(.Ω)(3.6) Then for a positive number , 𝐶𝑛(Ω;(,))𝑊(𝑡)𝜑(Φ)𝑑𝜇(𝑡,Φ)<,lim𝑅𝑊(𝑅)𝑉(𝑅)𝐶𝑛(Ω;(0,𝑅))𝑉(𝑡)𝜑(Φ)𝑑𝜇(𝑡,Φ)=0.(3.7)

Lemma 3.4 (see [24]). Let 𝜈 be a positive measure on 𝑆𝑛(Ω) such that there is a sequence of points 𝑃𝑖=(𝑟𝑖,Θ𝑖)𝐶𝑛(Ω), 𝑟𝑖(𝑖) satisfying 𝑆𝑛(Ω)𝜕𝐺𝑎Ω𝑃𝑖,𝑄𝜕𝑛𝑄𝑑𝜈(𝑄)<𝑖=1,2,3,;𝑄=(𝑡,Φ)𝐶𝑛.(Ω)(3.8) Then for a positive number , 𝑆𝑛(Ω;(,))𝑊(𝑡)𝑡1𝜕𝜑(Φ)𝜕𝑛Φ𝑑𝜈(𝑡,Φ)<,lim𝑅𝑊(𝑅)𝑉(𝑅)𝑆𝑛(Ω;(0,𝑅))𝑉(𝑡)𝑡1𝜕𝜑(Φ)𝜕𝑛Φ𝑑𝜈(𝑡,Φ)=0.(3.9)

Lemma 3.5. Let 𝜇 be a positive measure on 𝐶𝑛(Ω) for which 𝐺𝑎Ω𝜇(𝑃) is defined. Then for any positive number 𝐴 the set 𝑃=(𝑟,Θ)𝐶𝑛(Ω)𝐺𝑎Ω𝜇(𝑃)𝐴𝑉(𝑟)𝜑(Θ)(3.10) is 𝑎-minimally thin at with respect to 𝐶𝑛(Ω).

Lemma 3.6. Let 𝜐(𝑃) be a positive superfunction on 𝐶𝑛(Ω) and put 𝑐(𝜐,𝑎)=inf𝑃𝐶𝑛(Ω)𝜐(𝑃)𝑀𝑎Ω(𝑃,),𝑐𝑂(𝜐,𝑎)=inf𝑃𝐶𝑛(Ω)𝜐(𝑃)𝑀𝑎Ω(.𝑃,𝑂)(3.11) Then there are a unique positive measure 𝜇 on 𝐶𝑛(Ω) and a unique positive measure 𝜈 on 𝑆𝑛(Ω) such that 𝜐(𝑃)=𝑐(𝜐,𝑎)𝑀𝑎Ω(𝑃,)+𝑐𝑂(𝜐,𝑎)𝑀𝑎Ω(+𝑃,𝑂)𝐶𝑛(Ω)𝐺𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑄)+𝑆𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜈(𝑄),(3.12) where 𝜕/𝜕𝑛𝑄 denotes the differentiation at 𝑄 along the inward normal into 𝐶𝑛(Ω).

Proof. By the Riesz decomposition theorem, we have a unique measure 𝜇 on 𝐶𝑛(Ω) such that 𝜐(𝑃)=𝐶𝑛(Ω)𝐺𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑄)+(𝑃)𝑃𝐶𝑛(,Ω)(3.13) where 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 (𝑃)=𝜕𝐶𝑛(Ω){}𝑀𝑎Ω(𝑃,𝑄)𝑑𝜈(𝑄)=𝑀𝑎Ω(𝑃,)𝜈({})+𝑀𝑎Ω+(𝑃,𝑂)𝜈({𝑂})𝑆𝑛(Ω)𝑀𝑎Ω(𝑃,𝑄)𝑑𝜈(𝑄)𝑃𝐶𝑛.(Ω)(3.14) We know from (3.11) that 𝜈({})=𝑐(𝜐,𝑎) and 𝜈({𝑂})=𝑐𝑂(𝜐,𝑎).
Since 𝑀𝑎Ω(𝑃,𝑄)=lim𝑃1𝑄,𝑃1𝐶𝑛(Ω)𝐺𝑎Ω𝑃,𝑃1𝐺𝑎Ω𝑃0,𝑃1=𝜕𝐺𝑎Ω(𝑃,𝑄)/𝜕𝑛𝑄𝜕𝐺𝑎Ω𝑃0,𝑄/𝜕𝑛𝑄,(3.15) where 𝑃0 is a fixed reference point of the generalized Martin kernel, we also obtain (𝑃)=𝑐(𝜐,𝑎)𝑀𝑎Ω(𝑃,)+𝑐𝑂(𝜐,𝑎)𝑀𝑎Ω(𝑃,𝑂)+𝑆𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜈(𝑄)𝑃𝐶𝑛(Ω)(3.16) by taking 𝑑𝜈(𝑄)=𝜕𝐺𝑎Ω𝑃0,𝑄𝜕𝑛𝑄1𝑑𝜈(𝑄)𝑄𝑆𝑛.(Ω)(3.17) Hence by (3.13) and (3.16) we get the required.

Lemma 3.7. Let 𝐸 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 𝑢(𝑃)=𝐶𝑛(Ω)𝐺𝑎Ω(𝑃,𝑄)𝑑𝜇𝑢(𝑄)+𝑆𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜈𝑢(𝑄)(3.18) with two positive measures 𝜇𝑢(𝑄) and 𝜈𝑢(𝑄) on 𝐶𝑛(Ω) and 𝑆𝑛(Ω), respectively, and satisfies 𝑢(𝑃)1 for any 𝑃𝐸. Then 𝜆𝑎Ω(𝐸)𝐶𝑛(Ω)𝑉(𝑡)𝜑(Φ)𝑑𝜇𝑢(𝑡,Φ)+𝑆𝑛(Ω)𝑉(𝑡)𝑡1𝜑(Φ)𝑑𝜈𝑢(𝑡,Φ).(3.19) When 𝑅𝑢(𝑃)=𝐸1(𝑃)(𝑃𝐶𝑛(Ω)), the equality holds in (3.19).

Proof. Since 𝜆𝑎𝐸 is concentrated on 𝐵𝐸 and 𝑢(𝑃)1 for any 𝑃𝐵𝐸, we see that 𝜆𝑎Ω(𝐸)=𝐶𝑛(Ω)𝑑𝜆𝑎𝐸(𝑃)𝐶𝑛(Ω)𝑢(𝑃)𝑑𝜆𝑎𝐸(=𝑃)𝐶𝑛(Ω)𝑅𝐸𝑀𝑎Ω(,)𝑑𝜇𝑢(𝑄)+𝑆𝑛(Ω)𝐶𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜆𝑎𝐸(𝑃)𝑑𝜈𝑢(𝑄).(3.20) In addition, we have 𝑅𝐸𝑀𝑎Ω(,)(𝑄)𝑀𝑎Ω(𝑄,)=𝑉(𝑡)𝜑(Φ)𝑄=(𝑡,Φ)𝐶𝑛(.Ω)(3.21) Since 𝐶𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜆𝑎𝐸(𝑃)liminf𝜌01𝜌𝐶𝑛(Ω)𝐺𝑎Ω𝑃,𝑃𝜌𝑑𝜆𝑎𝐸(𝑃)(3.22) for any 𝑄𝑆𝑛(Ω), where 𝑃𝜌=(𝑟𝜌,Θ𝜌)=𝑄+𝜌𝑛𝑄𝐶𝑛(Ω) and 𝑛𝑄 is the inward normal unit vector at 𝑄, and 𝐶𝑛(Ω)𝐺𝑎Ω𝑃,𝑃𝜌𝑑𝜆𝑎𝐸(𝑅𝑃)=𝐸𝑀𝑎Ω(,)𝑃𝜌𝑀𝑎Ω𝑃𝜌𝑟,=𝑉𝜌𝜑Θ𝜌,(3.23) we have 𝐶𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜆𝑎𝐸(𝑃)𝑉(𝑡)𝑡1𝜕𝜑(Φ)𝜕𝑛Φ(3.24) for any 𝑄=(𝑡,Φ)𝑆𝑛(Ω). Thus (3.19) follows from (3.20), (3.21), and (3.24). Because 𝑅𝐸1(𝑃) is bounded on 𝐶𝑛(Ω), 𝑢(𝑃) has the expression (3.18) by Lemma 3.6 when 𝑅𝑢(𝑃)=𝐸1(𝑃). Then the equalities in (3.20) hold because 𝑅𝐸1(𝑃)=1 for any 𝑃𝐵𝐸 (Doob [21, page 169]). Hence we claim if 𝜇𝑢𝑃𝐶𝑛𝑅(Ω)𝐸𝑀𝑎Ω(,)(𝑃)<𝑀𝑎Ω𝜈(𝑃,)=0,(3.25)𝑢𝑄=(𝑡,Φ)𝑆𝑛(Ω)𝐶𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜆𝑎𝐸(𝑃)<𝑉(𝑡)𝑡1𝜕𝜑(Φ)𝜕𝑛Φ=0,(3.26) then the equality in (3.19) holds.
To see (3.25) we remark that 𝑃𝐶𝑛𝑅(Ω)𝐸𝑀𝑎Ω(,)(𝑃)<𝑀𝑎Ω𝐶(𝑃,)𝑛(Ω)𝐵𝐸,𝜇𝑢𝐶𝑛(Ω)𝐵𝐸=0.(3.27) To prove (3.26) we set 𝐵𝐸=𝑄𝑆𝑛,𝑅(Ω)𝐸isnot𝑎-minimallythinat𝑄𝑒=𝑃𝐸𝐸𝑀𝑎Ω(,)(𝑃)<𝑀𝑎Ω.(𝑃,)(3.28) Then 𝑒 is an 𝑎-polar set, and hence 𝑅𝐸𝑀𝑎Ω(,𝑄)=𝑅𝑀𝐸𝑒𝑎Ω(,𝑄)(3.29) for any 𝑄𝑆𝑛(Ω). Consequently, for any 𝑄𝐵𝐸, 𝐸𝑒 is not also 𝑎-minimally thin at 𝑄, and so 𝐶𝑛(Ω)𝑀𝑎Ω(𝑃,𝑄)𝑑𝜂(𝑃)=liminf𝑃𝑄,𝑃𝐸𝑒𝐶𝑛(Ω)𝑀𝑎Ω𝑃,𝑃𝑑𝜂(𝑃)(3.30) for any positive measure 𝜂 on 𝐶𝑛(Ω), where 𝑀𝑎Ω𝑃,𝑃=𝐺𝑎Ω𝑃,𝑃𝐺𝑎Ω𝑃0,𝑃𝑃,𝑃𝐶𝑛(Ω).(3.31) Take 𝜂=𝜆𝑎𝐸 in (3.30). Since lim𝑃𝑄,𝑃𝐶𝑛(Ω)𝑀𝑎Ω(𝑃,)𝐺𝑎Ω𝑃0,𝑃=𝑉(𝑡)𝑡1𝜕𝜑(Φ)𝜕𝑛Φ𝜕𝐺𝑎Ω𝑃0,𝑄𝜕𝑛𝑄1,𝑄=(𝑡,Φ)𝑆𝑛,(Ω)(3.32) we obtain from (3.15) 𝐶𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛Φ𝑑𝜆𝑎𝐸(𝑃)=𝑉(𝑡)𝑡1𝜕𝜑(Φ)𝜕𝑛Φliminf𝑃𝑄,𝑃𝐸𝑒𝐶𝑛(Ω)𝐺𝑎Ω𝑃,𝑃𝑀𝑎Ω𝑃,𝑑𝜆𝑎𝐸(𝑃)(3.33) for any 𝑄(𝑡,Φ)𝐵𝐸. Since 𝐶𝑛(Ω)𝐺𝑎Ω𝑃,𝑃𝑀𝑎Ω𝑃,𝑑𝜆𝑎𝐸1(𝑃)=𝑀𝑎Ω𝑃𝑅,𝐸𝑀𝑎Ω(,)𝑃=1(3.34) for any 𝑃𝐸𝑒, we have 𝐶𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜆𝑎𝐸(𝑃)=𝑉(𝑡)𝑡1𝜕𝜑(Φ)𝜕𝑛Φ(3.35) for any 𝑄=(𝑡,Φ)𝐵𝐸, which shows 𝑄=(𝑡,Φ)𝑆𝑛(Ω)𝐶𝑛(Ω)𝜕𝐺𝑎Ω(𝑃,𝑄)𝜕𝑛𝑄𝑑𝜆𝑎𝐸(𝑃)<𝑉(𝑡)𝑡1𝜕𝜑(Φ)𝜕𝑛Φ𝑆𝑛(Ω)𝐵𝐸.(3.36) Let be the greatest a-harmonic minorant of 𝑅𝑢(𝑃)=𝐸1(𝑃), and let 𝜈𝑢 be the generalized Martin representing measure of . We claim if 𝑅𝐸(𝑃)=(3.37) on 𝐶𝑛(Ω), then 𝜈𝑢(𝑆𝑛(Ω)𝐵𝐸)=0. Since 𝑑𝜈𝑢(𝑄)=𝜕𝐺𝑎Ω𝑃0,𝑄𝜕𝑛𝑄𝑑𝜈𝑢(𝑄)𝑄𝑆𝑛(Ω)(3.38) 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=𝑅𝐸1(𝑃). Then 𝑢𝑅+=𝐸1=𝑅𝐸𝑢+𝑅𝐸𝑢+𝑅𝐸,(3.39) and hence 𝑅𝐸𝑢𝑅𝐸𝑢0,(3.40) from which (3.37) follows.

Lemma 3.8 (the generalized Martin representation). If 𝑢 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 𝑢(𝑃)=Δ𝑀𝑎Ω(𝑃,𝑄)𝑑𝜇𝑢(𝑄)𝑃𝐶𝑛(,Ω)(3.41) 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 Theorem 2.1. First we assume that (b) holds, and let 𝑅𝑢=𝐸𝑀𝑎Ω(,𝑄). Since 𝑀𝑎Ω(,𝑄) is minimal, the Riesz decomposition of 𝑢 is of the form 𝜐+𝑀𝑎Ω(,𝑄), where 𝜐 is an 𝑎-potential associated with the stationary Schrödinger operator on 𝐶𝑛(Ω) and 0<<1. Since 𝑢=𝑀𝑎Ω(,𝑄) quasieverywhere on 𝐸 and 𝑅𝐸𝜐+𝑢=𝜐+𝑀𝑎Ω(,𝑄)=𝑢 quasieverywhere on 𝐸, 𝑅𝐸𝑀𝑎Ω(,𝑄)=𝑅𝐸𝑢𝑅𝐸𝜐+𝑢𝜐+𝑀𝑎Ω(𝑅,𝑄)=𝐸𝑀𝑎Ω(,𝑄).(4.1) Hence (𝑀𝑎Ω(,𝑄)𝑢)0, so =0 by the hypothesis and (a) holds.
Next we assume (a) holds, and let 𝜔𝑚 be a decreasing sequence of compact neighborhoods of 𝑄 in the Martin topology such that 𝑚𝜔𝑚={𝑄}. Then 𝑅𝐸𝜔𝑚𝑀𝑎Ω(,𝑄) is a-harmonic on 𝐶𝑛(Ω)𝜔𝑚, and the decreasing sequence {𝑅𝐸𝜔𝑚𝑀𝑎Ω(,𝑄)} has a limit which is a-harmonic on 𝐶𝑛(Ω). Since is majorized by 𝑅𝐸𝑀𝑎Ω(,𝑄), it follows that 0 and (c) holds.
Finally we assume (c) holds, then there is a Martin topology neighborhood 𝜔 of 𝑄 such that 𝑅𝐸𝜔𝑀𝑎Ω(,𝑄)𝑀𝑎Ω(,𝑄). Since (b) implies (a), the set 𝐸𝜔 is 𝑎-minimally thin at 𝑄 and so 𝑅𝐸𝜔𝑀𝑎Ω(,𝑄) is an 𝑎-potential. Then 𝑅𝐸𝑀𝑎Ω(,𝑄) is an 𝑎-potential and we yield (b).

Proof of Theorem 2.2. Obviously we see that (c) implies (b). If (b) holds, then there exist >𝜇𝑢({𝑄}) and a Martin topology neighborhood 𝜔 of 𝑄 such that 𝑢𝑀𝑎Ω(,𝑄) on 𝐸𝜔. If 𝑅𝐸𝜔𝑀𝑎Ω(,𝑄)=𝑀𝑎Ω(,𝑄), then 𝑅𝑢𝐸𝜔𝑢𝑀𝑎Ω(,𝑄), and this yields contradictory conclusion that 𝜇𝑢=𝛿𝑄+𝜇𝑢𝑀𝑎Ω(,𝑄)>𝜇𝑢({𝑄})𝛿𝑄, where 𝛿𝑄 is the unit measure with support {𝑄}. Hence 𝑅𝐸𝜔𝑀𝑎Ω(,𝑄)𝑀𝑎Ω(,𝑄). Thus 𝐸𝜔 is 𝑎-minimally thin at 𝑄, and so (a) holds.
Finally we assume (a) holds. By Lemma 3.1 there is an open subset 𝑈 of 𝐶𝑛(Ω) such that 𝐸𝑈 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 𝑅𝐸𝜔𝑚𝑀𝑎Ω(,𝑄)(𝑃0)<2𝑚. The function 𝑢1=𝑛𝑅𝑈𝜔𝑚𝑀𝑎Ω(,𝑄), being a sum of a-potentials, is an 𝑎-potential since 𝑢1(𝑃0)<. Further, since 𝑅𝐸𝜔𝑚𝑀𝑎Ω(,𝑄)=𝑀𝑎Ω(,𝑄) on the open set 𝐸𝜔𝑚, 𝑢1(𝑃)𝑀𝑎Ω(𝑃,𝑄)(𝑃𝑄;𝑃𝑈),(4.2) and so (c) holds.

Proof of Theorem 2.3. Clearly (c) implies (b). To prove that (b) implies (a), we suppose that (b) holds and choose A such that liminf𝑃𝑄,𝑃𝐸𝐺𝑎Ω𝜇(𝑃)𝐺𝑎Ω𝑃0𝑀,𝑃>𝐴>𝑎Ω(,𝑄)𝑑𝜇.(4.3) Then 𝐺𝑎Ω𝜇>𝐴𝐺𝑎Ω(𝑃0,) on 𝐸𝜔 for some Martin topology neighborhood 𝜔 of 𝑄. If 𝜈 denotes the swept measure of 𝛿𝑃0 onto 𝐸𝜔, where 𝛿𝑃0 is the unit measure with support {𝑃0}, then it follows that 𝐺𝑎Ω𝑅𝜇𝐴𝐺𝐸𝜔𝑎Ω𝑃0,=𝐴𝐺𝑎Ω𝜈(4.4) on 𝐶𝑛(Ω). Let {𝐾𝑛} be a sequence of compact subsets of 𝐶𝑛(Ω) such that 𝑛𝐾𝑛=𝐶𝑛(Ω), and let 𝐺𝑎Ω𝜇𝑛 denote the 𝑎-potential 𝑅𝐾𝑛𝑀𝑎Ω(,𝑄). Then 𝑅𝐾𝑛𝑀𝑎Ω(,𝑄)𝐺𝑑𝜈=𝑎Ω𝜈𝑑𝜇𝑛𝐴1𝐺𝑎Ω𝜇𝑑𝜇𝑛=𝐴1𝑅𝐾𝑛𝑀𝑎Ω(,𝑄)𝑑𝜇.(4.5) Letting 𝑛, we see from our choice of 𝐴 that 𝑅𝑀𝐸𝜔𝑎Ω(,𝑄)𝑃0=𝑀𝑎Ω(,𝑄)𝑑𝜈𝐴1𝑀𝑎Ω(,𝑄)𝑑𝜇<1=𝑀𝑎Ω𝑃0,𝑄,(4.6) then 𝐸𝜔 is 𝑎-minimally thin at 𝑄 by Theorem 2.1, and so (a) holds.
Next we suppose that (a) holds. By Lemma 3.1 there is an open subset 𝑈 of 𝐶𝑛(Ω) such that 𝐸𝑈 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 𝑛𝑅𝑈𝜔𝑛𝑀𝑎Ω(,𝑄)𝑃0<.(4.7) Let 𝜇=𝑛𝜈𝑛, where 𝜈𝑛 is swept measure of 𝛿𝑃0 onto 𝑈𝜔𝑛. Then 𝑀𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑃)=𝑛𝑀𝑎Ω(𝑃,𝑄)𝑑𝜈𝑛(𝑃)=𝑛𝑅𝑈𝜔𝑛𝑀𝑎Ω(,𝑄)𝑃0<,(4.8) and (2.10) holds since 𝐺𝑎Ω𝜈𝑛=𝑅𝑈𝜔𝑛𝐺𝑎Ω𝑃0,=𝐺𝑎Ω𝑃0,(4.9) on the open set 𝑈𝜔𝑛, so (c) holds.

Proof of Theorem 2.4. Since (2.11) is independent of the choice of 𝑄0, we may multiply across by 𝑀𝑎Ω(𝑄0,𝑄). Thus we may assume that 𝑄0=𝑃0 and claim that liminf𝑃𝑄𝐺𝑎Ω𝜇(𝑃)𝐺𝑎Ω𝑃0=𝑀,𝑃𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑃)(4.10) for any 𝑎-potential 𝐺𝑎Ω𝜇. According to Fatou's lemma, we may yield liminf𝑃𝑄𝐺𝑎Ω𝜇(𝑃)𝐺𝑎Ω𝑃0𝑀,𝑃𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑃).(4.11) Since 𝐶𝑛(Ω) is not 𝑎-minimally thin at 𝑄, we know that liminf𝑃𝑄𝐺𝑎Ω𝜇(𝑃)𝐺𝑎Ω𝑃0<𝑀,𝑃𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑃)(4.12) from Theorem 2.3. Hence the claim holds.
When 𝐸 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 liminf𝑃𝑄,𝑃𝐸𝑢(𝑃)𝐺𝑎Ω𝑃0,𝑃>𝐴>liminf𝑃𝑄𝑢(𝑃)𝐺𝑎Ω𝑃0,𝑃(4.13) and define 𝐺𝑎Ω𝜇 by min{𝑢,𝐴𝐺𝑎Ω(𝑃0,)}. Then by (4.10) liminf𝑃𝑄,𝑃𝐸𝐺𝑎Ω𝜇(𝑃)𝐺𝑎Ω𝑃0,𝑃=𝐴>liminf𝑃𝑄𝐺𝑎Ω𝜇(𝑃)𝐺𝑎Ω𝑃0=𝑀,𝑃𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑃),(4.14) and it follows from Theorem 2.3 that 𝐸 is 𝑎-minimally thin at 𝑄.

Proof of Theorem 2.5. By applying the Riesz decomposition theorem to the superfunction 𝑅𝐸𝑀𝑎Ω(,) on 𝐶𝑛(Ω), we have a positive measure 𝜇 on 𝐶𝑛(Ω) satisfying 𝐺𝑎Ω𝜇(𝑃)<(4.15) for any 𝑃𝐶𝑛(Ω) and a nonnegative greatest a-harmonic minorant 𝐻 of 𝑅𝐸𝑀𝑎Ω(,) such that 𝑅𝐸𝑀𝑎Ω(,)=𝐺𝑎Ω𝜇(𝑃)+𝐻.(4.16) We remark that 𝑀𝑎Ω(,)(𝑃𝐶𝑛(Ω)) is a minimal function at . If 𝐸 is 𝑎-minimally thin at with respect to 𝐶𝑛(Ω), then 𝑅𝐸𝑀𝑎Ω(,) is an 𝑎-potential, and hence 𝐻0 on 𝐶𝑛(Ω). Since 𝑅𝐸𝑀𝑎Ω(,)(𝑃)=𝑀𝑎Ω(𝑃,)(4.17) for any 𝑃𝐵𝐸, we see from (4.16) that 𝐺𝑎Ω𝜇(𝑃)=𝑀𝑎Ω(𝑃,)(4.18) for any 𝑃𝐵𝐸. Take a sufficiently large 𝑅 from Lemma 3.3 such that 𝐶2𝑊(𝑅)𝑉(𝑅)𝐶𝑛(Ω;(0,𝑅])1𝑉(𝑡)𝜑(Φ)𝑑𝜇(𝑡,Φ)<4.(4.19) Then from (1.16) or (1.17), 𝐶𝑛(Ω;(0,𝑅])𝐺𝑎Ω(1𝑃,𝑄)𝑑𝜇(𝑄)<4𝑀𝑎Ω(𝑃,)(4.20) for any 𝑃=(𝑟,Θ)𝐶𝑛(Ω) and 𝑟(5/4)𝑟, and hence from (4.18) 𝐶𝑛(Ω;[𝑅,))𝐺𝑎Ω(3𝑃,𝑄)𝑑𝜇(𝑄)4𝑀𝑎Ω(𝑃,)(4.21) for any 𝑃=(𝑟,Θ)𝐵𝐸 and 𝑟(5/4)𝑟. Divide 𝐺𝑎Ω𝜇 into three parts as follows: 𝐺𝑎Ω𝜇(𝑃)=𝐴1(𝑘)(𝑃)+𝐴2(𝑘)(𝑃)+𝐴3(𝑘)(𝑃)𝑃=(𝑟,Θ)𝐶𝑛,(Ω)(4.22) where 𝐴1(𝑘)(𝑃)=𝐶𝑛(Ω;(2𝑘1,2𝑘+2))𝐺𝑎Ω(𝐴𝑃,𝑄)𝑑𝜇(𝑄),2(𝑘)(𝑃)=𝐶𝑛(Ω;(0,2𝑘1])𝐺𝑎Ω𝐴(𝑃,𝑄)𝑑𝜇(𝑄),3(𝑘)(𝑃)=𝐶𝑛(Ω;[2𝑘+2,))𝐺𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑄).(4.23) Now we claim that there exists an integer 𝑁 such that 𝐵𝐸𝐼𝑘(Ω)𝑃=(𝑟,Θ)𝐶𝑛(Ω)𝐴1(𝑘)1(𝑃)4𝑉(𝑟)𝜑(Θ)(𝑘𝑁).(4.24) When we choose a sufficiently large integer 𝑁1 by Lemma 3.3 such that 𝑊2𝑘𝑉2𝑘𝐶𝑛(Ω;(0,2𝑘])1𝑉(𝑡)𝜑(Φ)𝑑𝜇(𝑡,Φ)<4𝐶2𝑘𝑁1,𝐶𝑛(Ω;[2𝑘+2,))1𝑊(𝑡)𝜑(Φ)𝑑𝜇(𝑡,Φ)<4𝐶2𝑘𝑁1(4.25) for any 𝑃=(𝑟,Θ)𝐼𝑘(Ω)𝐶𝑛(Ω), we have from (1.16) or (1.17) that 𝐴2(𝑘)1(𝑃)4𝑉(𝑟)𝜑(Θ)𝑘𝑁1,𝐴3(𝑘)1(𝑃)4𝑉(𝑟)𝜑(Θ)𝑘𝑁1.(4.26) Put 𝑁𝑁=max1,log𝑅log2+2.(4.27) For any 𝑃=(𝑟,Θ)𝐵𝐸𝐼𝑘(Ω)(𝑘𝑁), we have from (4.21), (4.22), and (4.26) that 𝐴1(𝑘)(𝑃)𝐶𝑛(Ω;[𝑅,))𝐺𝑎Ω(𝑃,𝑄)𝑑𝜇(𝑄)𝐴2(𝑘)(𝑃)𝐴3(𝑘)(1𝑃)4𝑉(𝑟)𝜑(Θ),(4.28) which shows (4.24).
Since the measure 𝜆𝑎𝐸𝑘 is concentrated on 𝐵𝐸𝑘 and 𝐵𝐸𝑘𝐵𝐸𝐼𝑘(Ω), finally we obtain by (4.24) that 𝛾𝑎Ω𝐸𝑘=𝐶𝑛(Ω)𝐺𝑎Ω𝜆𝑎𝐸𝑘𝑑𝜆𝑎𝐸𝑘(𝑃)𝐵𝐸𝑘𝑉(𝑟)𝜑(Θ)𝑑𝜆𝑎𝐸𝑘(𝑟,Θ)4𝐵𝐸𝑘𝐴1(𝑘)(𝑃)𝑑𝜆𝑎𝐸𝑘(𝑃)4𝐶𝑛(Ω;(2𝑘1,2𝑘+2))𝐶𝑛(Ω)𝐺𝑎Ω(𝑃,𝑄)𝑑𝜆𝑎𝐸𝑘(𝑃)𝑑𝜇(𝑄)4𝐶𝑛(Ω;(2𝑘1,2𝑘+2))𝑉(𝑡)𝜑(Φ)𝑑𝜇(𝑡,Φ)(𝑘𝑁),(4.29) and hence 𝑘=𝑁𝛾𝑎Ω𝐸𝑘𝑊2𝑘𝑉2𝑘1𝑘=𝑁𝐶𝑛(Ω;(2𝑘1,2𝑘+2))=𝑊(𝑡)𝜑(Φ)𝑑𝜇(𝑡,Φ)𝐶𝑛(Ω;(2𝑁1,))𝑊(𝑡)𝜑(Φ)𝑑𝜇(𝑡,Φ)<(4.30) 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 𝑅𝐸𝑘𝑀𝑎Ω(,)(𝑄)=𝑀𝑎Ω(𝑄,)(4.31) for any 𝑄𝐵𝐸𝑘 as in (4.17), we have 𝛾𝑎Ω𝐸𝑘=𝐵𝐸𝑘𝑀𝑎Ω(𝑄,)𝑑𝜆𝑎𝐸𝑘(2𝑄)𝑉𝑘𝐵𝐸𝑘𝜑(Φ)𝑑𝜆𝑎𝐸𝑘(𝑡,Φ)𝑄=(𝑡,Φ)𝐶