# Rarefied sets at infinity associated with the Schrödinger operator

## Abstract

This paper gives some criteria for a-rarefied sets at infinity associated with the Schrödinger operator in a cone. Our proofs are based on estimating Green a-potential with a positive measure by connecting with a kind of density of the modified measure. Meanwhile, the geometrical property of this a-rarefied sets at infinity is also considered. By giving an example, we show that the reverse of this property is not true.

## 1 Introduction and results

Let R and ${\mathbf{R}}_{+}$ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by ${\mathbf{R}}^{n}$ ($n\ge 2$) the n-dimensional Euclidean space. A point in ${\mathbf{R}}^{n}$ is denoted by $P=\left(X,{x}_{n}\right)$, $X=\left({x}_{1},{x}_{2},\dots ,{x}_{n-1}\right)$. The Euclidean distance between two points P and Q in ${\mathbf{R}}^{n}$ is denoted by $|P-Q|$. Also $|P-O|$ with the origin O of ${\mathbf{R}}^{n}$ is simply denoted by $|P|$. The boundary and the closure of a set S in ${\mathbf{R}}^{n}$ are denoted by ∂S and $\overline{S}$, respectively.

We introduce a system of spherical coordinates $\left(r,\mathrm{\Theta }\right)$, $\mathrm{\Theta }=\left({\theta }_{1},{\theta }_{2},\dots ,{\theta }_{n-1}\right)$, in ${\mathbf{R}}^{n}$ which are related to Cartesian coordinates $\left({x}_{1},{x}_{2},\dots ,{x}_{n-1},{x}_{n}\right)$ by ${x}_{n}=rcos{\theta }_{1}$.

Let D be an arbitrary domain in ${\mathbf{R}}^{n}$ and let ${\mathcal{A}}_{a}$ denote the class of non-negative radial potentials $a\left(P\right)$, i.e., $0\le a\left(P\right)=a\left(r\right)$, $P=\left(r,\mathrm{\Theta }\right)\in D$, such that $a\in {L}_{\mathrm{loc}}^{b}\left(D\right)$ with some $b>n/2$ if $n\ge 4$ and with $b=2$ if $n=2$ or $n=3$.

If $a\in {\mathcal{A}}_{a}$, then the Schrödinger operator

${\mathit{Sch}}_{a}=-\mathrm{\Delta }+a\left(P\right)I=0,$

where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space ${C}_{0}^{\mathrm{\infty }}\left(D\right)$ to an essentially self-adjoint operator on ${L}^{2}\left(D\right)$ (see [, Ch. 11]). We will denote it by ${\mathit{Sch}}_{a}$ as well. This last one has a Green a-function ${G}_{D}^{a}\left(P,Q\right)$. Here ${G}_{D}^{a}\left(P,Q\right)$ is positive on D and its inner normal derivative $\partial {G}_{D}^{a}\left(P,Q\right)/\partial {n}_{Q}\ge 0$, where $\partial /\partial {n}_{Q}$ denotes the differentiation at Q along the inward normal into D.

We call a function $u\not\equiv -\mathrm{\infty }$ that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator ${\mathit{Sch}}_{a}$ if its values belong to the interval $\left[-\mathrm{\infty },\mathrm{\infty }\right)$ and at each point $P\in D$ with $0 the generalized mean-value inequality (see )

$u\left(P\right)\le {\int }_{S\left(P,r\right)}u\left(Q\right)\frac{\partial {G}_{B\left(P,r\right)}^{a}\left(P,Q\right)}{\partial {n}_{Q}}\phantom{\rule{0.2em}{0ex}}d\sigma \left(Q\right)$

is satisfied, where ${G}_{B\left(P,r\right)}^{a}\left(P,Q\right)$ is the Green a-function of ${\mathit{Sch}}_{a}$ in $B\left(P,r\right)$ and $d\sigma \left(Q\right)$ is a surface measure on the sphere $S\left(P,r\right)=\partial B\left(P,r\right)$.

If −u is a subfunction, then we call u a superfunction. If a function u is both subfunction and superfunction, it is, clearly, continuous and is called an a-harmonic function (with respect to the Schrödinger operator ${\mathit{Sch}}_{a}$).

The unit sphere and the upper half unit sphere in ${\mathbf{R}}^{n}$ are denoted by ${\mathbf{S}}^{n-1}$ and ${\mathbf{S}}_{+}^{n-1}$, respectively. For simplicity, a point $\left(1,\mathrm{\Theta }\right)$ on ${\mathbf{S}}^{n-1}$ and the set $\left\{\mathrm{\Theta };\left(1,\mathrm{\Theta }\right)\in \mathrm{\Omega }\right\}$ for a set Ω, $\mathrm{\Omega }\subset {\mathbf{S}}^{n-1}$, are often identified with Θ and Ω, respectively. For two sets $\mathrm{\Xi }\subset {\mathbf{R}}_{+}$ and $\mathrm{\Omega }\subset {\mathbf{S}}^{n-1}$, the set $\left\{\left(r,\mathrm{\Theta }\right)\in {\mathbf{R}}^{n};r\in \mathrm{\Xi },\left(1,\mathrm{\Theta }\right)\in \mathrm{\Omega }\right\}$ in ${\mathbf{R}}^{n}$ is simply denoted by $\mathrm{\Xi }×\mathrm{\Omega }$. By ${C}_{n}\left(\mathrm{\Omega }\right)$ we denote the set ${\mathbf{R}}_{+}×\mathrm{\Omega }$ in ${\mathbf{R}}^{n}$ with the domain Ω on ${\mathbf{S}}^{n-1}$. We call it a cone. We denote the set $I×\mathrm{\Omega }$ with an interval on R by ${C}_{n}\left(\mathrm{\Omega };I\right)$.

We shall say that a set $H\subset {C}_{n}\left(\mathrm{\Omega }\right)$ has a covering $\left\{{r}_{j},{R}_{j}\right\}$ if there exists a sequence of balls $\left\{{B}_{j}\right\}$ with centers in ${C}_{n}\left(\mathrm{\Omega }\right)$ such that $H\subset {\bigcup }_{j=0}^{\mathrm{\infty }}{B}_{j}$, where ${r}_{j}$ is the radius of ${B}_{j}$ and ${R}_{j}$ is the distance from the origin to the center of ${B}_{j}$.

From now on, we always assume $D={C}_{n}\left(\mathrm{\Omega }\right)$. For the sake of brevity, we shall write ${G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)$ instead of ${G}_{{C}_{n}\left(\mathrm{\Omega }\right)}^{a}\left(P,Q\right)$. Throughout this paper, let c denote various positive constants, because we do not need to specify them. Moreover, ϵ appearing in the expression in the following sections will be a sufficiently small positive number.

Let Ω be a domain on ${\mathbf{S}}^{n-1}$ with smooth boundary. Consider the Dirichlet problem

where ${\mathrm{\Lambda }}_{n}$ is the spherical part of the Laplace operator ${\mathrm{\Delta }}_{n}$

${\mathrm{\Delta }}_{n}=\frac{n-1}{r}\frac{\partial }{\partial r}+\frac{{\partial }^{2}}{\partial {r}^{2}}+\frac{{\mathrm{\Lambda }}_{n}}{{r}^{2}}.$

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by $\phi \left(\mathrm{\Theta }\right)$. In order to ensure the existence of λ and a smooth $\phi \left(\mathrm{\Theta }\right)$, we put a rather strong assumption on Ω: if $n\ge 3$, then Ω is a ${C}^{2,\alpha }$-domain ($0<\alpha <1$) on ${\mathbf{S}}^{n-1}$ surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see [, pp.88-89] for the definition of ${C}^{2,\alpha }$-domain).

For any $\left(1,\mathrm{\Theta }\right)\in \mathrm{\Omega }$, we have (see [, pp.7-8])

${c}^{-1}r\phi \left(\mathrm{\Theta }\right)\le \delta \left(P\right)\le cr\phi \left(\mathrm{\Theta }\right),$
(1)

where $P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega }\right)$ and $\delta \left(P\right)=dist\left(P,\partial {C}_{n}\left(\mathrm{\Omega }\right)\right)$.

Solutions of an ordinary differential equation

$-{Q}^{″}\left(r\right)-\frac{n-1}{r}{Q}^{\prime }\left(r\right)+\left(\frac{\lambda }{{r}^{2}}+a\left(r\right)\right)Q\left(r\right)=0,\phantom{\rule{1em}{0ex}}0
(2)

It is known (see, for example, ) that if the potential $a\in {\mathcal{A}}_{a}$, then equation (2) has a fundamental system of positive solutions $\left\{V,W\right\}$ such that V and W are increasing and decreasing, respectively.

We will also consider the class ${\mathcal{B}}_{a}$, consisting of the potentials $a\in {\mathcal{A}}_{a}$, such that there exists the finite limit ${lim}_{r\to \mathrm{\infty }}{r}^{2}a\left(r\right)=k\in \left[0,\mathrm{\infty }\right)$ and, moreover, ${r}^{-1}|{r}^{2}a\left(r\right)-k|\in L\left(1,\mathrm{\infty }\right)$. If $a\in {\mathcal{B}}_{a}$, then the (sub)superfunctions are continuous (see ).

In the rest of paper, we assume that $a\in {\mathcal{B}}_{a}$ and we shall suppress this assumption for simplicity.

Denote

${\iota }_{k}^{±}=\frac{2-n±\sqrt{{\left(n-2\right)}^{2}+4\left(k+\lambda \right)}}{2},$

then the solutions to equation (2) have the asymptotic (see )

(3)

Let ν be any positive measure on ${C}_{n}\left(\mathrm{\Omega }\right)$ such that the Green a-potential

${G}_{\mathrm{\Omega }}^{a}\nu \left(P\right)={\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right)\not\equiv +\mathrm{\infty }$

for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$. Then the positive measure $m\left(\nu \right)$ on ${\mathbf{R}}^{n}$ is defined by

$dm\left(\nu \right)\left(Q\right)=\left\{\begin{array}{ll}W\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right),& Q=\left(t,\mathrm{\Phi }\right)\in {C}_{n}\left(\mathrm{\Omega };\left(1,+\mathrm{\infty }\right)\right),\\ 0,& Q\in {\mathbf{R}}^{n}-{C}_{n}\left(\mathrm{\Omega };\left(1,+\mathrm{\infty }\right)\right).\end{array}$

Remark 1 We remark that the total mass $m\left(\nu \right)$ is finite (see [, Lemma 5]).

For each $P=\left(r,\mathrm{\Theta }\right)\in {\mathbf{R}}^{n}-\left\{O\right\}$, the maximal function $M\left(P;\lambda ,\beta \right)$ is defined by

$M\left(P;\lambda ,\beta \right)=\underset{0<\rho <\frac{r}{2}}{sup}\frac{\lambda \left(B\left(P,\rho \right)\right)}{{\rho }^{\beta }},$

where $\beta \ge 0$ and λ is a positive measure on ${\mathbf{R}}^{n}$. The set

$\left\{P=\left(r,\mathrm{\Theta }\right)\in {\mathbf{R}}^{n}-\left\{O\right\};M\left(P;\lambda ,\beta \right){r}^{\beta }>ϵ\right\}$

is denoted by $E\left(ϵ;\lambda ,\beta \right)$.

It is known that the Martin boundary of ${C}_{n}\left(\mathrm{\Omega }\right)$ is the set $\partial {C}_{n}\left(\mathrm{\Omega }\right)\cup \left\{\mathrm{\infty }\right\}$, each of which is a minimal Martin boundary point. For $P\in {C}_{n}\left(\mathrm{\Omega }\right)$ and $Q\in \partial {C}_{n}\left(\mathrm{\Omega }\right)\cup \left\{\mathrm{\infty }\right\}$, the Martin kernel can be defined by ${M}_{\mathrm{\Omega }}^{a}\left(P,Q\right)$. If the reference point P is chosen suitably, then we have

${M}_{\mathrm{\Omega }}^{a}\left(P,\mathrm{\infty }\right)=V\left(r\right)\phi \left(\mathrm{\Theta }\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{M}_{\mathrm{\Omega }}^{a}\left(P,O\right)=cW\left(r\right)\phi \left(\mathrm{\Theta }\right)$
(4)

for any $P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega }\right)$.

In , Long et al. introduced the notations of a-thin (with respect to the Schrödinger operator ${\mathit{Sch}}_{a}$) at a point, a-polar set (with respect to the Schrödinger operator ${\mathit{Sch}}_{a}$) and a-rarefied sets at infinity (with respect to the Schrödinger operator ${\mathit{Sch}}_{a}$), which generalized earlier notations obtained by Brelot and Miyamoto (see [8, 9]). A set H in ${\mathbf{R}}^{n}$ is said to be a-thin at a point Q if there is a fine neighborhood E of Q which does not intersect $H\mathrm{\setminus }\left\{Q\right\}$. Otherwise H is said to be not a-thin at Q on ${C}_{n}\left(\mathrm{\Omega }\right)$. A set H in ${\mathbf{R}}^{n}$ is called a polar set if there is a superfunction u on some open set E such that $H\subset \left\{P\in E;u\left(P\right)=\mathrm{\infty }\right\}$. A subset H of ${C}_{n}\left(\mathrm{\Omega }\right)$ is said to be a-rarefied at infinity on ${C}_{n}\left(\mathrm{\Omega }\right)$ if there exists a positive superfunction $v\left(P\right)$ on ${C}_{n}\left(\mathrm{\Omega }\right)$ such that

$\underset{P\in {C}_{n}\left(\mathrm{\Omega }\right)}{inf}\frac{v\left(P\right)}{{M}_{\mathrm{\Omega }}^{a}\left(P,\mathrm{\infty }\right)}\equiv 0$

and

$H\subset \left\{P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega }\right);v\left(P\right)\ge V\left(r\right)\right\}.$

Let H be a bounded subset of ${C}_{n}\left(\mathrm{\Omega }\right)$. Then ${\stackrel{ˆ}{R}}_{{M}_{\mathrm{\Omega }}^{a}\left(\cdot ,\mathrm{\infty }\right)}^{H}$ is bounded on ${C}_{n}\left(\mathrm{\Omega }\right)$ and the greatest a-harmonic minorant of ${\stackrel{ˆ}{R}}_{{M}_{\mathrm{\Omega }}^{a}\left(\cdot ,\mathrm{\infty }\right)}^{H}$ is zero. We see from the Riesz decomposition theorem (see [, Theorem 2]) that there exists a unique positive measure ${\lambda }_{H}^{a}$ on ${C}_{n}\left(\mathrm{\Omega }\right)$ such that (see [, p.6])

${\stackrel{ˆ}{R}}_{{M}_{\mathrm{\Omega }}^{a}\left(\cdot ,\mathrm{\infty }\right)}^{H}\left(P\right)={G}_{\mathrm{\Omega }}^{a}{\lambda }_{H}^{a}\left(P\right)$
(5)

for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$ and ${\lambda }_{H}^{a}$ is concentrated on ${I}_{H}$, where

We denote the total mass ${\lambda }_{H}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)$ of ${\lambda }_{H}^{a}$ by ${\lambda }_{\mathrm{\Omega }}^{a}\left(H\right)$.

By using this positive measure ${\lambda }_{H}^{a}$ (with respect to the Schrödinger operator ${\mathit{Sch}}_{a}$), we can further define another measure ${\eta }_{H}^{a}$ on ${C}_{n}\left(\mathrm{\Omega }\right)$ by

$d{\eta }_{H}^{a}\left(P\right)={M}_{\mathrm{\Omega }}^{a}\left(P,\mathrm{\infty }\right)\phantom{\rule{0.2em}{0ex}}d{\lambda }_{H}^{a}\left(P\right)$

for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$. It is easy to see that ${\eta }_{H}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)<+\mathrm{\infty }$.

Recently, Long et al. (see [, Theorem 2.5]) gave a criterion for a subset H of ${C}_{n}\left(\mathrm{\Omega }\right)$ to be a-rarefied set at infinity.

Theorem A A subset H of ${C}_{n}\left(\mathrm{\Omega }\right)$ is a-rarefied at infinity on ${C}_{n}\left(\mathrm{\Omega }\right)$ if and only if

$\sum _{j=0}^{\mathrm{\infty }}{\lambda }_{\mathrm{\Omega }}^{a}\left({H}_{j}\right)W\left({2}^{j}\right)<\mathrm{\infty },$

where ${H}_{j}=H\cap {C}_{n}\left(\mathrm{\Omega };\left[{2}^{j},{2}^{j+1}\right)\right)$ and $j=0,1,2,\dots$ .

In this paper, we shall obtain a series of new criteria for a-rarefied sets at infinity on ${C}_{n}\left(\mathrm{\Omega }\right)$, which complement Theorem A. Our results are essentially based on Qiao and Deng, Ren and Zhao, Xue (see [2, 1114]). In order to avoid complexity of our proofs, we shall assume $n\ge 3$. But our results in this paper are also true for $n=2$.

First we shall state Theorem 1, which is the main result in this paper.

Theorem 1 A subset H of ${C}_{n}\left(\mathrm{\Omega }\right)$ is a-rarefied at infinity on ${C}_{n}\left(\mathrm{\Omega }\right)$ if and only if there exists a positive measure ${\xi }_{H}^{a}$ on ${C}_{n}\left(\mathrm{\Omega }\right)$ such that

${G}_{\mathrm{\Omega }}^{a}{\xi }_{H}^{a}\left(P\right)\not\equiv +\mathrm{\infty }$
(6)

for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$ and

$H\subset \left\{P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega }\right);{G}_{\mathrm{\Omega }}^{a}{\xi }_{H}^{a}\left(P\right)\ge V\left(r\right)\right\}.$
(7)

Next we give the geometrical property of a-rarefied sets at infinity.

Theorem 2 If a subset H of ${C}_{n}\left(\mathrm{\Omega }\right)$ is a-rarefied at infinity on ${C}_{n}\left(\mathrm{\Omega }\right)$, then H has a covering $\left\{{r}_{j},{R}_{j}\right\}$ ($j=0,1,2,\dots$) satisfying

$\sum _{j=0}^{\mathrm{\infty }}\left(\frac{{r}_{j}}{{R}_{j}}\right)V\left(\frac{{R}_{j}}{{r}_{j}}\right)W\left(\frac{{R}_{j}}{{r}_{j}}\right)<\mathrm{\infty }.$
(8)

Finally, by an example we show that the reverse of Theorem 2 is not true.

Example Put

${r}_{j}=3\cdot {2}^{j-1}\cdot {j}^{\frac{1}{2-n}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{R}_{j}=3\cdot {2}^{j-1}\phantom{\rule{1em}{0ex}}\left(j=1,2,3,\dots \right).$

A covering $\left\{{r}_{j},{R}_{j}\right\}$ satisfies

$\sum _{j=1}^{\mathrm{\infty }}\left(\frac{{r}_{j}}{{R}_{j}}\right)V\left(\frac{{R}_{j}}{{r}_{j}}\right)W\left(\frac{{R}_{j}}{{r}_{j}}\right)\le c\sum _{j=1}^{\mathrm{\infty }}{\left(\frac{{r}_{j}}{{R}_{j}}\right)}^{n-1}=c\sum _{j=1}^{\mathrm{\infty }}{j}^{\frac{n-1}{2-n}}<+\mathrm{\infty }$

from equation (3).

Let ${C}_{n}\left({\mathrm{\Omega }}^{\prime }\right)$ be a subset of ${C}_{n}\left(\mathrm{\Omega }\right)$, i.e., ${\overline{\mathrm{\Omega }}}^{\prime }\subset \mathrm{\Omega }$. Suppose that this covering is located as follows: there is an integer ${j}_{0}$ such that ${B}_{j}\subset {C}_{n}\left({\mathrm{\Omega }}^{\prime }\right)$ and ${R}_{j}>2{r}_{j}$ for $j\ge {j}_{0}$. Then the set $H={\bigcup }_{j={j}_{0}}^{\mathrm{\infty }}{B}_{j}$ is not a-rarefied at infinity on ${C}_{n}\left(\mathrm{\Omega }\right)$. This fact will be proved in Section 5.

## 2 Lemmas

Lemma 1 (see [, Ch. 11] and [, Lemma 4])

for any $P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega }\right)$ and any $Q=\left(t,\mathrm{\Phi }\right)\in {C}_{n}\left(\mathrm{\Omega }\right)$ satisfying $r\ge 2t$ (resp. $t\ge 2r$).

Lemma 2 (see [, Lemma 5])

Let ν be a positive measure on ${C}_{n}\left(\mathrm{\Omega }\right)$ such that there is a sequence of points ${P}_{i}=\left({r}_{i},{\mathrm{\Theta }}_{i}\right)\in {C}_{n}\left(\mathrm{\Omega }\right)$, ${r}_{i}\to +\mathrm{\infty }$ ($i\to +\mathrm{\infty }$) satisfying ${G}_{\mathrm{\Omega }}^{a}\nu \left({P}_{i}\right)<+\mathrm{\infty }$ ($i=1,2,\dots$ ; $Q\in {C}_{n}\left(\mathrm{\Omega }\right)$). Then, for a positive number L,

${\int }_{{C}_{n}\left(\mathrm{\Omega };\left(L,+\mathrm{\infty }\right)\right)}W\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right)<+\mathrm{\infty }$

and

$\underset{R\to +\mathrm{\infty }}{lim}\frac{W\left(R\right)}{V\left(R\right)}{\int }_{{C}_{n}\left(\mathrm{\Omega };\left(0,R\right)\right)}V\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right)=0.$

Lemma 3 (see [, Theorem 3])

Let ν be any positive measure on ${C}_{n}\left(\mathrm{\Omega }\right)$ such that ${G}_{\mathrm{\Omega }}^{a}\nu \left(P\right)\not\equiv +\mathrm{\infty }$ for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$. Then, for a sufficiently large L,

$\left\{P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega };\left(L,+\mathrm{\infty }\right)\right);{G}_{\mathrm{\Omega }}^{a}\nu \left(P\right)\ge V\left(r\right)\phi \left(\mathrm{\Theta }\right)\right\}\subset E\left(ϵ;m\left(\nu \right),n-1\right).$

Lemma 4 (see [, Lemma 6])

Let λ be any positive measure on ${\mathbf{R}}^{n}$ having finite total mass. Then $E\left(ϵ;\lambda ,n-1\right)$ has a covering $\left\{{r}_{j},{R}_{j}\right\}$ ($j=1,2,\dots$) satisfying

$\sum _{j=1}^{\mathrm{\infty }}\left(\frac{{r}_{j}}{{R}_{j}}\right)V\left(\frac{{R}_{j}}{{r}_{j}}\right)W\left(\frac{{R}_{j}}{{r}_{j}}\right)<\mathrm{\infty }.$
(9)

## 3 Proof of Theorem 1

Suppose that

$H\subset \mathrm{\Pi }\left({\xi }_{H}^{a}\right)=\left\{P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega }\right);{G}_{\mathrm{\Omega }}^{a}{\xi }_{H}^{a}\left(P\right)\ge V\left(r\right)\right\}$
(10)

for a positive measure ${\xi }_{H}^{a}$ on ${C}_{n}\left(\mathrm{\Omega }\right)$ satisfying equation (6).

We write

${G}_{\mathrm{\Omega }}^{a}\nu \left(P\right)={G}_{\mathrm{\Omega }}^{a}\left(1,j\right)\left(P\right)+{G}_{\mathrm{\Omega }}^{a}\left(2,j\right)\left(P\right)+{G}_{\mathrm{\Omega }}^{a}\left(3,j\right)\left(P\right),$

where

$\begin{array}{c}{G}_{\mathrm{\Omega }}^{a}\left(1,j\right)\left(P\right)={\int }_{{C}_{n}\left(\mathrm{\Omega };\left(0,{2}^{j-1}\right)\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right),\hfill \\ {G}_{\mathrm{\Omega }}^{a}\left(2,j\right)\left(P\right)={\int }_{{C}_{n}\left(\mathrm{\Omega };\left[{2}^{j-1},{2}^{j+2}\right)\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right)\hfill \end{array}$

and

${G}_{\mathrm{\Omega }}^{a}\left(3,j\right)\left(P\right)={\int }_{{C}_{n}\left(\mathrm{\Omega };\left[{2}^{j+2},\mathrm{\infty }\right)\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right).$

Now we shall show the existence of an integer N such that for any integer j (≥N), we have

$\mathrm{\Pi }\left({\xi }_{H}^{a}\right)\left(j\right)\subset \left\{P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega };\left[{2}^{j},{2}^{j+1}\right)\right);2{G}_{\mathrm{\Omega }}^{a}\left(2,j\right)\left(P\right)\ge V\left(r\right)\right\}$
(11)

for any integer j (≥N).

For any $P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega };\left[{2}^{j},{2}^{j+1}\right)\right)$, we have

${G}_{\mathrm{\Omega }}^{a}\left(1,j\right)\left(P\right)\le cW\left(r\right)\phi \left(\mathrm{\Theta }\right){\int }_{{C}_{n}\left(\mathrm{\Omega };\left(0,{2}^{j-1}\right)\right)}V\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right)$

and

${G}_{\mathrm{\Omega }}^{a}\left(3,j\right)\left(P\right)\le cV\left(r\right)\phi \left(\mathrm{\Theta }\right){\int }_{{C}_{n}\left(\mathrm{\Omega };\left[{2}^{j+2},\mathrm{\infty }\right)\right)}dm\left(\nu \right)\left(Q\right)$

from Lemma 1.

By applying Lemma 2, we can take an integer N such that for any j (≥N),

$W\left({2}^{j}\right){V}^{-1}\left({2}^{j}\right){\int }_{{C}_{n}\left(\mathrm{\Omega };\left(0,{2}^{j-1}\right)\right)}V\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d\nu \left(Q\right)\le \frac{1}{4c}$

and

${\int }_{{C}_{n}\left(\mathrm{\Omega };\left[{2}^{j+2},\mathrm{\infty }\right)\right)}dm\left(\nu \right)\left(Q\right)\le \frac{1}{4c}.$

Thus we obtain

$4{G}_{\mathrm{\Omega }}^{a}\left(1,j\right)\left(P\right)\le V\left(r\right)\phi \left(\mathrm{\Theta }\right)$
(12)

and

$4{G}_{\mathrm{\Omega }}^{a}\left(3,j\right)\left(P\right)\le V\left(r\right)\phi \left(\mathrm{\Theta }\right)$
(13)

for any $P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega };\left[{2}^{j},{2}^{j+1}\right)\right)$, where $j\ge N$.

Thus, if $P=\left(r,\mathrm{\Theta }\right)\in \mathrm{\Pi }\left(\nu \right)\left(j\right)$ ($j\ge N$), then we obtain

$2{G}_{\mathrm{\Omega }}^{a}\left(1,j\right)\left(P\right)\ge V\left(r\right)\phi \left(\mathrm{\Theta }\right)$

from equations (12) and (13), which gives equation (11).

From equations (4), (7) and (11), we have

${G}_{\mathrm{\Omega }}^{a}\left(2,j\right)\left(P\right)={\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d{\tau }_{j}^{a}\left(Q\right)\ge {M}_{\mathrm{\Omega }}^{a}\left(P,\mathrm{\infty }\right),$

where $P\in {I}_{j}$ ($j\ge N$) and

$d{\tau }_{j}^{a}\left(Q\right)=\left\{\begin{array}{ll}{2}^{1-j}\phantom{\rule{0.2em}{0ex}}d{\xi }_{H}^{a}\left(Q\right),& Q\in {C}_{n}\left(\mathrm{\Omega };\left[{2}^{j-1},{2}^{j+2}\right)\right),\\ 0,& Q\in {C}_{n}\left(\mathrm{\Omega };\left(0,{2}^{j-1}\right)\right)\cup {C}_{n}\left(\mathrm{\Omega };\left[{2}^{j+2},\mathrm{\infty }\right)\right).\end{array}$

And then we obtain

${\eta }_{{H}_{j}}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)\le {\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}V\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d{\tau }_{j}^{a}\left(Q\right)={\int }_{{C}_{n}\left(\mathrm{\Omega };\left[{2}^{j-1},{2}^{j+2}\right)\right)}V\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d{\xi }_{H}^{a}\left(Q\right)$

for $j\ge N$. Then we have

$\sum _{j=N}^{\mathrm{\infty }}{\lambda }_{\mathrm{\Omega }}^{a}\left({H}_{j}\right)W\left({2}^{j}\right)=\sum _{j=N}^{\mathrm{\infty }}{\eta }_{{H}_{j}}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)W\left({2}^{j}\right)\le c{\int }_{{C}_{n}\left(\mathrm{\Omega };\left[{2}^{N-1},\mathrm{\infty }\right)\right)}dm\left({\xi }_{H}^{a}\right),$

in which the last integral is finite by Remark 1. And hence H is a-rarefied set at infinity from Theorem A.

Suppose that

$\sum _{j=0}^{\mathrm{\infty }}{\lambda }_{\mathrm{\Omega }}^{a}\left({H}_{j}\right)W\left({2}^{j}\right)<\mathrm{\infty }.$

Consider a function ${f}_{H}^{a}\left(P\right)$ on ${C}_{n}\left(\mathrm{\Omega }\right)$ defined by

${f}_{H}^{a}\left(P\right)=\sum _{j=-1}^{\mathrm{\infty }}{\stackrel{ˆ}{R}}_{{M}_{\mathrm{\Omega }}^{a}\left(\cdot ,\mathrm{\infty }\right)}^{{H}_{j}}\left(P\right)$

for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$, where ${H}_{-1}=H\cap {C}_{n}\left(\mathrm{\Omega };\left(0,1\right)\right)$.

If we put ${\mu }_{H}^{a}\left(1\right)\left(P\right)={\sum }_{j=-1}^{\mathrm{\infty }}{\lambda }_{{H}_{j}}^{a}\left(P\right)$, then from equation (5) we have that

${f}_{H}^{a}\left(P\right)={\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{H}^{a}\left(1\right)\left(Q\right)$

for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$.

Next we shall show that ${f}_{H}^{a}\left(P\right)$ is always finite on ${C}_{n}\left(\mathrm{\Omega }\right)$. Take any point $P=\left(r,\mathrm{\Theta }\right)\in {C}_{n}\left(\mathrm{\Omega }\right)$ and a positive integer $j\left(P\right)$ satisfying $r\le {2}^{j\left(P\right)+1}$. We write

${f}_{H}^{a}\left(P\right)={f}_{H}^{a}\left(1\right)\left(P\right)+{f}_{H}^{a}\left(2\right)\left(P\right),$

where

$\begin{array}{c}{f}_{H}^{a}\left(1\right)\left(P\right)=\sum _{j=-1}^{j\left(P\right)+1}{\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d{\lambda }_{{H}_{j}}^{a}\left(Q\right)\phantom{\rule{1em}{0ex}}\text{and}\hfill \\ {f}_{H}^{a}\left(2\right)\left(P\right)=\sum _{j=j\left(P\right)+2}^{\mathrm{\infty }}{\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d{\lambda }_{{H}_{j}}^{a}\left(Q\right).\hfill \end{array}$

Since ${\lambda }_{{H}_{j}}^{a}$ is concentrated on ${I}_{{H}_{j}}\subset {\overline{H}}_{j}\cap {C}_{n}\left(\mathrm{\Omega }\right)$, we have that

$\begin{array}{rcl}{\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}{G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d{\lambda }_{{H}_{j}}^{a}\left(Q\right)& \le & cV\left(r\right)\phi \left(\mathrm{\Theta }\right){\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}W\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d{\lambda }_{{H}_{j}}^{a}\left(t,\mathrm{\Phi }\right)\\ \le & cV\left(r\right)\phi \left(\mathrm{\Theta }\right)W\left({2}^{j}\right){V}^{-1}\left({2}^{j}\right){\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}V\left(t\right)\phi \left(\mathrm{\Phi }\right)\phantom{\rule{0.2em}{0ex}}d{\lambda }_{{H}_{j}}^{a}\left(t,\mathrm{\Phi }\right)\end{array}$

for $j\ge j\left(P\right)+2$. Hence we have

${f}_{H}^{a}\left(2\right)\left(P\right)\le cV\left(r\right)\phi \left(\mathrm{\Theta }\right)\sum _{j=j\left(P\right)+2}^{\mathrm{\infty }}{\eta }_{{H}_{j}}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)W\left({2}^{j}\right){V}^{-1}\left({2}^{j}\right),$
(14)

which, together with Theorem A, shows that ${f}_{H}^{a}\left(2\right)\left(P\right)$ is finite and hence ${f}_{H}^{a}\left(P\right)$ is also finite for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$.

Since

${\stackrel{ˆ}{R}}_{{M}_{\mathrm{\Omega }}^{a}\left(\cdot ,\mathrm{\infty }\right)}^{{H}_{j}}\left(P\right)={M}_{\mathrm{\Omega }}^{a}\left(P,\mathrm{\infty }\right)$

holds on ${I}_{{H}_{j}}$ and ${I}_{{H}_{j}}\subset {\overline{H}}_{j}\cap {C}_{n}\left(\mathrm{\Omega }\right)$, we see that for any $P=\left(r,\mathrm{\Theta }\right)\in {I}_{{H}_{j}}$ ($j=-1,0,1,2,3,\dots$)

${f}_{H}^{a}\left(P\right)\ge c{\stackrel{ˆ}{R}}_{{M}_{\mathrm{\Omega }}^{a}\left(\cdot ,\mathrm{\infty }\right)}^{{H}_{j}}\left(P\right)\ge V\left(r\right)\phi \left(\mathrm{\Theta }\right).$
(15)

And hence equation (15) also holds for any $P=\left(r,\mathrm{\Theta }\right)\in {H}^{\prime }={\bigcup }_{j=-1}^{\mathrm{\infty }}{I}_{{H}_{j}}$. Since ${H}^{\prime }$ is equal to H except a polar set ${H}^{0}$, we can take another positive superfunction ${f}_{H}^{a}\left(3\right)\left(P\right)$ on ${C}_{n}\left(\mathrm{\Omega }\right)$ such that ${f}_{H}^{a}\left(3\right)\left(P\right)={G}_{\mathrm{\Omega }}^{a}{\mu }_{H}^{a}\left(2\right)\left(P\right)$ with a positive measure ${\mu }_{H}^{a}\left(2\right)\left(P\right)$ on ${C}_{n}\left(\mathrm{\Omega }\right)$ and ${f}_{H}^{a}\left(3\right)\left(P\right)$ is identically +∞ on ${H}^{0}$.

Finally, we can define a positive superfunction g on ${C}_{n}\left(\mathrm{\Omega }\right)$ by $g\left(P\right)={f}_{H}^{a}\left(P\right)+{f}_{H}^{a}\left(3\right)\left(P\right)={G}_{\mathrm{\Omega }}^{a}{\xi }_{H}^{a}\left(P\right)$ for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$ with ${\xi }_{H}^{a}={\mu }_{H}^{a}\left(1\right)+{\mu }_{H}^{a}\left(2\right)$. Also we see from equation (15) that equations (6) and (7) hold.

Thus we complete the proof of Theorem 1.

## 4 Proof of Theorem 2

From Theorem 1 and Lemma 3, we have a positive number L such that

$H\cap {C}_{n}\left(\mathrm{\Omega };\left(L,+\mathrm{\infty }\right)\right)\subset E\left(ϵ;m\left({\xi }_{H}^{a}\right),n-1\right).$

Hence by Remark 1 and Lemma 4, $E\left(ϵ;m\left({\xi }_{H}^{a}\right),n-1\right)$ has a covering $\left\{{r}_{j},{R}_{j}\right\}$ ($j=1,2,3,\dots$) satisfying equation (9) and hence H has also a covering $\left\{{r}_{j},{R}_{j}\right\}$ ($j=0,1,2,3,\dots$) with an additional finite ${B}_{0}$ covering ${C}_{n}\left(\mathrm{\Omega };\left(0,L\right]\right)$, satisfying equation (8), which is the conclusion of Theorem 2.

## 5 Proof of an example

Since $\phi \left(\mathrm{\Theta }\right)\ge c$ for any $\mathrm{\Theta }\in {\mathrm{\Omega }}^{\prime }$, we have ${M}_{\mathrm{\Omega }}^{a}\left(P,\mathrm{\infty }\right)\ge cV\left({R}_{j}\right)$ for any $P\in {\overline{B}}_{j}$, where $j\ge {j}_{0}$. Hence we have

${\stackrel{ˆ}{R}}_{{M}_{\mathrm{\Omega }}^{a}\left(\cdot ,\mathrm{\infty }\right)}^{{B}_{j}}\left(P\right)\ge cV\left({R}_{j}\right)$
(16)

for any $P\in {\overline{B}}_{j}$, where $j\ge {j}_{0}$.

Take a measure δ on ${C}_{n}\left(\mathrm{\Omega }\right)$, $supp\delta \subset {\overline{B}}_{j}$, $\delta \left({\overline{B}}_{j}\right)=1$ such that

${\int }_{{C}_{n}\left(\mathrm{\Omega }\right)}{|P-Q|}^{2-n}\phantom{\rule{0.2em}{0ex}}d\delta \left(P\right)={\left\{Cap\left({\overline{B}}_{j}\right)\right\}}^{-1}$
(17)

for any $Q\in {\overline{B}}_{j}$, where Cap denotes the Newton capacity. Since

${G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\le {|P-Q|}^{2-n}$

for any $P\in {C}_{n}\left(\mathrm{\Omega }\right)$ and $Q\in {C}_{n}\left(\mathrm{\Omega }\right)$ (see , the case $n=2$ is implicitly contained in ),

$\begin{array}{rl}{\left\{Cap\left({\overline{B}}_{j}\right)\right\}}^{-1}{\lambda }_{{B}_{j}}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)& =\int \left(\int {|P-Q|}^{2-n}\phantom{\rule{0.2em}{0ex}}d\delta \left(P\right)\right)\phantom{\rule{0.2em}{0ex}}d{\lambda }_{{B}_{j}}^{a}\left(Q\right)\\ \ge \int \left(\int {G}_{\mathrm{\Omega }}^{a}\left(P,Q\right)\phantom{\rule{0.2em}{0ex}}d{\lambda }_{{B}_{j}}^{a}\left(Q\right)\right)\phantom{\rule{0.2em}{0ex}}d\delta \left(P\right)\\ =\int {\stackrel{ˆ}{R}}_{{M}_{\mathrm{\Omega }}^{a}\left(\cdot ,\mathrm{\infty }\right)}^{{B}_{j}}\phantom{\rule{0.2em}{0ex}}d\delta \left(P\right)\\ \ge cV\left({R}_{j}\right)\delta \left({\overline{B}}_{j}\right)=cV\left({R}_{j}\right)\end{array}$

from equations (16) and (17). Hence we have

${\lambda }_{{B}_{j}}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)\ge cCap\left({\overline{B}}_{j}\right)V\left({R}_{j}\right)\ge c{r}_{j}^{n-2}V\left({R}_{j}\right).$
(18)

If we observe ${\lambda }_{{H}_{j}}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)={\lambda }_{{B}_{j}}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)$, then we have by equation (3)

$\sum _{j={j}_{0}}^{\mathrm{\infty }}W\left({2}^{j}\right){\lambda }_{{H}_{j}}^{a}\left({C}_{n}\left(\mathrm{\Omega }\right)\right)\ge c\sum _{j={j}_{0}}^{\mathrm{\infty }}{\left(\frac{{r}_{j}}{{R}_{j}}\right)}^{n-2}=c\sum _{j={j}_{0}}^{\mathrm{\infty }}\frac{1}{j}=+\mathrm{\infty },$

from which it follows by Theorem A that H is not a-rarefied at infinity on ${C}_{n}\left(\mathrm{\Omega }\right)$.

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## Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102.

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Correspondence to Gaixian Xue. 