# 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 $R +$ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by $R n$ ($n≥2$) the n-dimensional Euclidean space. A point in $R n$ is denoted by $P=(X, x n )$, $X=( x 1 , x 2 ,…, x n − 1 )$. The Euclidean distance between two points P and Q in $R n$ is denoted by $|P−Q|$. Also $|P−O|$ with the origin O of $R n$ is simply denoted by $|P|$. The boundary and the closure of a set S in $R n$ are denoted by ∂S and $S ¯$, respectively.

We introduce a system of spherical coordinates $(r,Θ)$, $Θ=( θ 1 , θ 2 ,…, θ n − 1 )$, in $R n$ which are related to Cartesian coordinates $( x 1 , x 2 ,…, x n − 1 , x n )$ by $x n =rcos θ 1$.

Let D be an arbitrary domain in $R n$ and let $A a$ denote the class of non-negative radial potentials $a(P)$, i.e., $0≤a(P)=a(r)$, $P=(r,Θ)∈D$, such that $a∈ L loc b (D)$ with some $b>n/2$ if $n≥4$ and with $b=2$ if $n=2$ or $n=3$.

If $a∈ A a$, then the Schrödinger operator

$Sch a =−Δ+a(P)I=0,$

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

We call a function $u≢−∞$ that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator $Sch a$ if its values belong to the interval $[−∞,∞)$ and at each point $P∈D$ with $0 the generalized mean-value inequality (see )

$u(P)≤ ∫ S ( P , r ) u(Q) ∂ G B ( P , r ) a ( P , Q ) ∂ n Q dσ(Q)$

is satisfied, where $G B ( P , r ) a (P,Q)$ is the Green a-function of $Sch a$ in $B(P,r)$ and $dσ(Q)$ is a surface measure on the sphere $S(P,r)=∂B(P,r)$.

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 $Sch a$).

The unit sphere and the upper half unit sphere in $R n$ are denoted by $S n − 1$ and $S + n − 1$, respectively. For simplicity, a point $(1,Θ)$ on $S n − 1$ and the set ${Θ;(1,Θ)∈Ω}$ for a set Ω, $Ω⊂ S n − 1$, are often identified with Θ and Ω, respectively. For two sets $Ξ⊂ R +$ and $Ω⊂ S n − 1$, the set ${(r,Θ)∈ R n ;r∈Ξ,(1,Θ)∈Ω}$ in $R n$ is simply denoted by $Ξ×Ω$. By $C n (Ω)$ we denote the set $R + ×Ω$ in $R n$ with the domain Ω on $S n − 1$. We call it a cone. We denote the set $I×Ω$ with an interval on R by $C n (Ω;I)$.

We shall say that a set $H⊂ C n (Ω)$ has a covering ${ r j , R j }$ if there exists a sequence of balls ${ B j }$ with centers in $C n (Ω)$ such that $H⊂ ⋃ j = 0 ∞ B j$, 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 $D= C n (Ω)$. For the sake of brevity, we shall write $G Ω a (P,Q)$ instead of $G C n ( Ω ) a (P,Q)$. 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 $S n − 1$ with smooth boundary. Consider the Dirichlet problem

where $Λ n$ is the spherical part of the Laplace operator $Δ n$

$Δ n = n − 1 r ∂ ∂ r + ∂ 2 ∂ r 2 + Λ n r 2 .$

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by $φ(Θ)$. In order to ensure the existence of λ and a smooth $φ(Θ)$, we put a rather strong assumption on Ω: if $n≥3$, then Ω is a $C 2 , α$-domain ($0<α<1$) on $S n − 1$ surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see [, pp.88-89] for the definition of $C 2 , α$-domain).

For any $(1,Θ)∈Ω$, we have (see [, pp.7-8])

$c − 1 rφ(Θ)≤δ(P)≤crφ(Θ),$
(1)

where $P=(r,Θ)∈ C n (Ω)$ and $δ(P)=dist(P,∂ C n (Ω))$.

Solutions of an ordinary differential equation

$− Q ″ (r)− n − 1 r Q ′ (r)+ ( λ r 2 + a ( r ) ) Q(r)=0,0
(2)

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

We will also consider the class $B a$, consisting of the potentials $a∈ A a$, such that there exists the finite limit $lim r → ∞ r 2 a(r)=k∈[0,∞)$ and, moreover, $r − 1 | r 2 a(r)−k|∈L(1,∞)$. If $a∈ B a$, then the (sub)superfunctions are continuous (see ).

In the rest of paper, we assume that $a∈ B a$ and we shall suppress this assumption for simplicity.

Denote

$ι k ± = 2 − n ± ( n − 2 ) 2 + 4 ( k + λ ) 2 ,$

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

(3)

Let ν be any positive measure on $C n (Ω)$ such that the Green a-potential

$G Ω a ν(P)= ∫ C n ( Ω ) G Ω a (P,Q)dν(Q)≢+∞$

for any $P∈ C n (Ω)$. Then the positive measure $m(ν)$ on $R n$ is defined by

$dm(ν)(Q)= { W ( t ) φ ( Φ ) d ν ( Q ) , Q = ( t , Φ ) ∈ C n ( Ω ; ( 1 , + ∞ ) ) , 0 , Q ∈ R n − C n ( Ω ; ( 1 , + ∞ ) ) .$

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

For each $P=(r,Θ)∈ R n −{O}$, the maximal function $M(P;λ,β)$ is defined by

$M(P;λ,β)= sup 0 < ρ < r 2 λ ( B ( P , ρ ) ) ρ β ,$

where $β≥0$ and λ is a positive measure on $R n$. The set

${ P = ( r , Θ ) ∈ R n − { O } ; M ( P ; λ , β ) r β > ϵ }$

is denoted by $E(ϵ;λ,β)$.

It is known that the Martin boundary of $C n (Ω)$ is the set $∂ C n (Ω)∪{∞}$, each of which is a minimal Martin boundary point. For $P∈ C n (Ω)$ and $Q∈∂ C n (Ω)∪{∞}$, the Martin kernel can be defined by $M Ω a (P,Q)$. If the reference point P is chosen suitably, then we have

$M Ω a (P,∞)=V(r)φ(Θ)and M Ω a (P,O)=cW(r)φ(Θ)$
(4)

for any $P=(r,Θ)∈ C n (Ω)$.

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

$inf P ∈ C n ( Ω ) v ( P ) M Ω a ( P , ∞ ) ≡0$

and

$H⊂ { P = ( r , Θ ) ∈ C n ( Ω ) ; v ( P ) ≥ V ( r ) } .$

Let H be a bounded subset of $C n (Ω)$. Then $R ˆ M Ω a ( ⋅ , ∞ ) H$ is bounded on $C n (Ω)$ and the greatest a-harmonic minorant of $R ˆ M Ω a ( ⋅ , ∞ ) H$ is zero. We see from the Riesz decomposition theorem (see [, Theorem 2]) that there exists a unique positive measure $λ H a$ on $C n (Ω)$ such that (see [, p.6])

$R ˆ M Ω a ( ⋅ , ∞ ) H (P)= G Ω a λ H a (P)$
(5)

for any $P∈ C n (Ω)$ and $λ H a$ is concentrated on $I H$, where

We denote the total mass $λ H a ( C n (Ω))$ of $λ H a$ by $λ Ω a (H)$.

By using this positive measure $λ H a$ (with respect to the Schrödinger operator $Sch a$), we can further define another measure $η H a$ on $C n (Ω)$ by

$d η H a (P)= M Ω a (P,∞)d λ H a (P)$

for any $P∈ C n (Ω)$. It is easy to see that $η H a ( C n (Ω))<+∞$.

Recently, Long et al. (see [, Theorem 2.5]) gave a criterion for a subset H of $C n (Ω)$ to be a-rarefied set at infinity.

Theorem A A subset H of $C n (Ω)$ is a-rarefied at infinity on $C n (Ω)$ if and only if

$∑ j = 0 ∞ λ Ω a ( H j )W ( 2 j ) <∞,$

where $H j =H∩ C n (Ω;[ 2 j , 2 j + 1 ))$ and $j=0,1,2,…$ .

In this paper, we shall obtain a series of new criteria for a-rarefied sets at infinity on $C n (Ω)$, 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 $n≥3$. 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 $C n (Ω)$ is a-rarefied at infinity on $C n (Ω)$ if and only if there exists a positive measure $ξ H a$ on $C n (Ω)$ such that

$G Ω a ξ H a (P)≢+∞$
(6)

for any $P∈ C n (Ω)$ and

$H⊂ { P = ( r , Θ ) ∈ C n ( Ω ) ; G Ω a ξ H a ( P ) ≥ V ( r ) } .$
(7)

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

Theorem 2 If a subset H of $C n (Ω)$ is a-rarefied at infinity on $C n (Ω)$, then H has a covering ${ r j , R j }$ ($j=0,1,2,…$) satisfying

$∑ j = 0 ∞ ( r j R j ) V ( R j r j ) W ( R j r j ) <∞.$
(8)

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

Example Put

$r j =3⋅ 2 j − 1 ⋅ j 1 2 − n and R j =3⋅ 2 j − 1 (j=1,2,3,…).$

A covering ${ r j , R j }$ satisfies

$∑ j = 1 ∞ ( r j R j ) V ( R j r j ) W ( R j r j ) ≤c ∑ j = 1 ∞ ( r j R j ) n − 1 =c ∑ j = 1 ∞ j n − 1 2 − n <+∞$

from equation (3).

Let $C n ( Ω ′ )$ be a subset of $C n (Ω)$, i.e., $Ω ¯ ′ ⊂Ω$. Suppose that this covering is located as follows: there is an integer $j 0$ such that $B j ⊂ C n ( Ω ′ )$ and $R j >2 r j$ for $j≥ j 0$. Then the set $H= ⋃ j = j 0 ∞ B j$ is not a-rarefied at infinity on $C n (Ω)$. This fact will be proved in Section 5.

## 2 Lemmas

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

for any $P=(r,Θ)∈ C n (Ω)$ and any $Q=(t,Φ)∈ C n (Ω)$ satisfying $r≥2t$ (resp. $t≥2r$).

Lemma 2 (see [, Lemma 5])

Let ν be a positive measure on $C n (Ω)$ such that there is a sequence of points $P i =( r i , Θ i )∈ C n (Ω)$, $r i →+∞$ ($i→+∞$) satisfying $G Ω a ν( P i )<+∞$ ($i=1,2,…$ ; $Q∈ C n (Ω)$). Then, for a positive number L,

$∫ C n ( Ω ; ( L , + ∞ ) ) W(t)φ(Φ)dν(Q)<+∞$

and

$lim R → + ∞ W ( R ) V ( R ) ∫ C n ( Ω ; ( 0 , R ) ) V(t)φ(Φ)dν(Q)=0.$

Lemma 3 (see [, Theorem 3])

Let ν be any positive measure on $C n (Ω)$ such that $G Ω a ν(P)≢+∞$ for any $P∈ C n (Ω)$. Then, for a sufficiently large L,

${ P = ( r , Θ ) ∈ C n ( Ω ; ( L , + ∞ ) ) ; G Ω a ν ( P ) ≥ V ( r ) φ ( Θ ) } ⊂E ( ϵ ; m ( ν ) , n − 1 ) .$

Lemma 4 (see [, Lemma 6])

Let λ be any positive measure on $R n$ having finite total mass. Then $E(ϵ;λ,n−1)$ has a covering ${ r j , R j }$ ($j=1,2,…$) satisfying

$∑ j = 1 ∞ ( r j R j ) V ( R j r j ) W ( R j r j ) <∞.$
(9)

## 3 Proof of Theorem 1

Suppose that

$H⊂Π ( ξ H a ) = { P = ( r , Θ ) ∈ C n ( Ω ) ; G Ω a ξ H a ( P ) ≥ V ( r ) }$
(10)

for a positive measure $ξ H a$ on $C n (Ω)$ satisfying equation (6).

We write

$G Ω a ν(P)= G Ω a (1,j)(P)+ G Ω a (2,j)(P)+ G Ω a (3,j)(P),$

where

$G Ω a ( 1 , j ) ( P ) = ∫ C n ( Ω ; ( 0 , 2 j − 1 ) ) G Ω a ( P , Q ) d ν ( Q ) , G Ω a ( 2 , j ) ( P ) = ∫ C n ( Ω ; [ 2 j − 1 , 2 j + 2 ) ) G Ω a ( P , Q ) d ν ( Q )$

and

$G Ω a (3,j)(P)= ∫ C n ( Ω ; [ 2 j + 2 , ∞ ) ) G Ω a (P,Q)dν(Q).$

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

$Π ( ξ H a ) (j)⊂ { P = ( r , Θ ) ∈ C n ( Ω ; [ 2 j , 2 j + 1 ) ) ; 2 G Ω a ( 2 , j ) ( P ) ≥ V ( r ) }$
(11)

for any integer j (≥N).

For any $P=(r,Θ)∈ C n (Ω;[ 2 j , 2 j + 1 ))$, we have

$G Ω a (1,j)(P)≤cW(r)φ(Θ) ∫ C n ( Ω ; ( 0 , 2 j − 1 ) ) V(t)φ(Φ)dν(Q)$

and

$G Ω a (3,j)(P)≤cV(r)φ(Θ) ∫ C n ( Ω ; [ 2 j + 2 , ∞ ) ) dm(ν)(Q)$

from Lemma 1.

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

$W ( 2 j ) V − 1 ( 2 j ) ∫ C n ( Ω ; ( 0 , 2 j − 1 ) ) V(t)φ(Φ)dν(Q)≤ 1 4 c$

and

$∫ C n ( Ω ; [ 2 j + 2 , ∞ ) ) dm(ν)(Q)≤ 1 4 c .$

Thus we obtain

$4 G Ω a (1,j)(P)≤V(r)φ(Θ)$
(12)

and

$4 G Ω a (3,j)(P)≤V(r)φ(Θ)$
(13)

for any $P=(r,Θ)∈ C n (Ω;[ 2 j , 2 j + 1 ))$, where $j≥N$.

Thus, if $P=(r,Θ)∈Π(ν)(j)$ ($j≥N$), then we obtain

$2 G Ω a (1,j)(P)≥V(r)φ(Θ)$

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

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

$G Ω a (2,j)(P)= ∫ C n ( Ω ) G Ω a (P,Q)d τ j a (Q)≥ M Ω a (P,∞),$

where $P∈ I j$ ($j≥N$) and

$d τ j a (Q)= { 2 1 − j d ξ H a ( Q ) , Q ∈ C n ( Ω ; [ 2 j − 1 , 2 j + 2 ) ) , 0 , Q ∈ C n ( Ω ; ( 0 , 2 j − 1 ) ) ∪ C n ( Ω ; [ 2 j + 2 , ∞ ) ) .$

And then we obtain

$η H j a ( C n ( Ω ) ) ≤ ∫ C n ( Ω ) V(t)φ(Φ)d τ j a (Q)= ∫ C n ( Ω ; [ 2 j − 1 , 2 j + 2 ) ) V(t)φ(Φ)d ξ H a (Q)$

for $j≥N$. Then we have

$∑ j = N ∞ λ Ω a ( H j )W ( 2 j ) = ∑ j = N ∞ η H j a ( C n ( Ω ) ) W ( 2 j ) ≤c ∫ C n ( Ω ; [ 2 N − 1 , ∞ ) ) dm ( ξ H a ) ,$

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

Suppose that

$∑ j = 0 ∞ λ Ω a ( H j )W ( 2 j ) <∞.$

Consider a function $f H a (P)$ on $C n (Ω)$ defined by

$f H a (P)= ∑ j = − 1 ∞ R ˆ M Ω a ( ⋅ , ∞ ) H j (P)$

for any $P∈ C n (Ω)$, where $H − 1 =H∩ C n (Ω;(0,1))$.

If we put $μ H a (1)(P)= ∑ j = − 1 ∞ λ H j a (P)$, then from equation (5) we have that

$f H a (P)= ∫ C n ( Ω ) G Ω a (P,Q)d μ H a (1)(Q)$

for any $P∈ C n (Ω)$.

Next we shall show that $f H a (P)$ is always finite on $C n (Ω)$. Take any point $P=(r,Θ)∈ C n (Ω)$ and a positive integer $j(P)$ satisfying $r≤ 2 j ( P ) + 1$. We write

$f H a (P)= f H a (1)(P)+ f H a (2)(P),$

where

$f H a ( 1 ) ( P ) = ∑ j = − 1 j ( P ) + 1 ∫ C n ( Ω ) G Ω a ( P , Q ) d λ H j a ( Q ) and f H a ( 2 ) ( P ) = ∑ j = j ( P ) + 2 ∞ ∫ C n ( Ω ) G Ω a ( P , Q ) d λ H j a ( Q ) .$

Since $λ H j a$ is concentrated on $I H j ⊂ H ¯ j ∩ C n (Ω)$, we have that

$∫ C n ( Ω ) G Ω a ( P , Q ) d λ H j a ( Q ) ≤ c V ( r ) φ ( Θ ) ∫ C n ( Ω ) W ( t ) φ ( Φ ) d λ H j a ( t , Φ ) ≤ c V ( r ) φ ( Θ ) W ( 2 j ) V − 1 ( 2 j ) ∫ C n ( Ω ) V ( t ) φ ( Φ ) d λ H j a ( t , Φ )$

for $j≥j(P)+2$. Hence we have

$f H a (2)(P)≤cV(r)φ(Θ) ∑ j = j ( P ) + 2 ∞ η H j a ( C n ( Ω ) ) W ( 2 j ) V − 1 ( 2 j ) ,$
(14)

which, together with Theorem A, shows that $f H a (2)(P)$ is finite and hence $f H a (P)$ is also finite for any $P∈ C n (Ω)$.

Since

$R ˆ M Ω a ( ⋅ , ∞ ) H j (P)= M Ω a (P,∞)$

holds on $I H j$ and $I H j ⊂ H ¯ j ∩ C n (Ω)$, we see that for any $P=(r,Θ)∈ I H j$ ($j=−1,0,1,2,3,…$)

$f H a (P)≥c R ˆ M Ω a ( ⋅ , ∞ ) H j (P)≥V(r)φ(Θ).$
(15)

And hence equation (15) also holds for any $P=(r,Θ)∈ H ′ = ⋃ j = − 1 ∞ I H j$. Since $H ′$ is equal to H except a polar set $H 0$, we can take another positive superfunction $f H a (3)(P)$ on $C n (Ω)$ such that $f H a (3)(P)= G Ω a μ H a (2)(P)$ with a positive measure $μ H a (2)(P)$ on $C n (Ω)$ and $f H a (3)(P)$ is identically +∞ on $H 0$.

Finally, we can define a positive superfunction g on $C n (Ω)$ by $g(P)= f H a (P)+ f H a (3)(P)= G Ω a ξ H a (P)$ for any $P∈ C n (Ω)$ with $ξ H a = μ H a (1)+ μ H a (2)$. 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

$H∩ C n ( Ω ; ( L , + ∞ ) ) ⊂E ( ϵ ; m ( ξ H a ) , n − 1 ) .$

Hence by Remark 1 and Lemma 4, $E(ϵ;m( ξ H a ),n−1)$ has a covering ${ r j , R j }$ ($j=1,2,3,…$) satisfying equation (9) and hence H has also a covering ${ r j , R j }$ ($j=0,1,2,3,…$) with an additional finite $B 0$ covering $C n (Ω;(0,L])$, satisfying equation (8), which is the conclusion of Theorem 2.

## 5 Proof of an example

Since $φ(Θ)≥c$ for any $Θ∈ Ω ′$, we have $M Ω a (P,∞)≥cV( R j )$ for any $P∈ B ¯ j$, where $j≥ j 0$. Hence we have

$R ˆ M Ω a ( ⋅ , ∞ ) B j (P)≥cV( R j )$
(16)

for any $P∈ B ¯ j$, where $j≥ j 0$.

Take a measure δ on $C n (Ω)$, $suppδ⊂ B ¯ j$, $δ( B ¯ j )=1$ such that

$∫ C n ( Ω ) | P − Q | 2 − n dδ(P)= { Cap ( B ¯ j ) } − 1$
(17)

for any $Q∈ B ¯ j$, where Cap denotes the Newton capacity. Since

$G Ω a (P,Q)≤ | P − Q | 2 − n$

for any $P∈ C n (Ω)$ and $Q∈ C n (Ω)$ (see , the case $n=2$ is implicitly contained in ),

${ Cap ( B ¯ j ) } − 1 λ B j a ( C n ( Ω ) ) = ∫ ( ∫ | P − Q | 2 − n d δ ( P ) ) d λ B j a ( Q ) ≥ ∫ ( ∫ G Ω a ( P , Q ) d λ B j a ( Q ) ) d δ ( P ) = ∫ R ˆ M Ω a ( ⋅ , ∞ ) B j d δ ( P ) ≥ c V ( R j ) δ ( B ¯ j ) = c V ( R j )$

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

$λ B j a ( C n ( Ω ) ) ≥cCap( B ¯ j )V( R j )≥c r j n − 2 V( R j ).$
(18)

If we observe $λ H j a ( C n (Ω))= λ B j a ( C n (Ω))$, then we have by equation (3)

$∑ j = j 0 ∞ W ( 2 j ) λ H j a ( C n ( Ω ) ) ≥c ∑ j = j 0 ∞ ( r j R j ) n − 2 =c ∑ j = j 0 ∞ 1 j =+∞,$

from which it follows by Theorem A that H is not a-rarefied at infinity on $C n (Ω)$.

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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. 