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 $P=(X,x_n)$, $X=(x_1,x_2,\dots ,x_{n-1})$. 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 $(r,\mathrm{\Theta })$, $\mathrm{\Theta }=({\theta }_1,{\theta }_2,\dots ,{\theta }_{n-1})$, in ${\mathbf{R}}^n$ which are related to Cartesian coordinates $(x_1,x_2,\dots ,x_{n-1},x_n)$ 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 $a(P)$, i.e., $0\le a(P)=a(r)$, $P=(r,\mathrm{\Theta })\in D$, such that $a\in L_{\mathrm{loc}}^b(D)$ 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

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

where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space $C_0^{\mathrm{\infty }}(D)$ to an essentially self-adjoint operator on $L^2(D)$ (see [[1], Ch. 11]). We will denote it by ${\mathit{Sch}}_a$ 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 $\partial G_D^a(P,Q)/\partial n_Q\ge 0$, 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 $[-\mathrm{\infty },\mathrm{\infty })$ and at each point $P\in D$ with $0<r<r(P)$ the generalized mean-value inequality (see [2])

$$u(P)\le {\int }_{S(P,r)}u(Q)\frac{\partial G_{B(P,r)}^a(P,Q)}{\partial n_Q}d\sigma (Q)$$

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

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 ${\mathit{Sch}}_a$).

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 $(1,\mathrm{\Theta })$ on ${\mathbf{S}}^{n-1}$ and the set $\{\mathrm{\Theta };(1,\mathrm{\Theta })\in \mathrm{\Omega }\}$ 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 $\{(r,\mathrm{\Theta })\in {\mathbf{R}}^n;r\in \mathrm{\Xi },(1,\mathrm{\Theta })\in \mathrm{\Omega }\}$ in ${\mathbf{R}}^n$ is simply denoted by $\mathrm{\Xi }\times \mathrm{\Omega }$. By $C_n(\mathrm{\Omega })$ we denote the set ${\mathbf{R}}_{+}\times \mathrm{\Omega }$ in ${\mathbf{R}}^n$ with the domain Ω on ${\mathbf{S}}^{n-1}$. We call it a cone. We denote the set $I\times \mathrm{\Omega }$ with an interval on R by $C_n(\mathrm{\Omega };I)$.

We shall say that a set $H\subset C_n(\mathrm{\Omega })$ has a covering $\{r_j,R_j\}$ if there exists a sequence of balls $\{B_j\}$ with centers in $C_n(\mathrm{\Omega })$ 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 $D=C_n(\mathrm{\Omega })$. For the sake of brevity, we shall write $G_{\mathrm{\Omega }}^a(P,Q)$ instead of $G_{C_n(\mathrm{\Omega })}^a(P,Q)$. 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 ${\mathbf{S}}^{n-1}$ with smooth boundary. Consider the Dirichlet problem

$$\begin{array}{c}({\mathrm{\Lambda }}_n+\lambda )\phi =0\text{on }\mathrm{\Omega },\hfill \\ \phi =0\text{on }\partial \mathrm{\Omega },\hfill \end{array}$$

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 $\phi (\mathrm{\Theta })$. In order to ensure the existence of λ and a smooth $\phi (\mathrm{\Theta })$, 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 [[3], pp.88-89] for the definition of $C^{2,\alpha }$-domain).

For any $(1,\mathrm{\Theta })\in \mathrm{\Omega }$, we have (see [[4], pp.7-8])

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

(1)

where $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$ and $\delta (P)=dist(P,\partial C_n(\mathrm{\Omega }))$.

Solutions of an ordinary differential equation

$$-Q^{″}(r)-\frac{n-1}{r}{Q}^{\prime }(r)+(\frac{\lambda }{r^2}+a(r))Q(r)=0,0<r<\mathrm{\infty }.$$

(2)

It is known (see, for example, [5]) that if the potential $a\in {\mathcal{A}}_a$, 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 ${\mathcal{B}}_a$, consisting of the potentials $a\in {\mathcal{A}}_a$, such that there exists the finite limit ${lim}_{r\to \mathrm{\infty }}{r}^{2}a(r)=k\in [0,\mathrm{\infty })$ and, moreover, $r^{-1}|r^{2}a(r)-k|\in L(1,\mathrm{\infty })$. If $a\in {\mathcal{B}}_a$, then the (sub)superfunctions are continuous (see [6]).

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

Denote

$${\iota }_k^{\pm }=\frac{2-n\pm \sqrt{{(n-2)}^2+4(k+\lambda )}}{2},$$

then the solutions to equation (2) have the asymptotic (see [3])

$$c^{-1}{r}^{{\iota }_k^{+}}\le V(r)\le c{r}^{{\iota }_k^{+}},c^{-1}{r}^{{\iota }_k^{-}}\le W(r)\le c{r}^{{\iota }_k^{-}},\text{as }r\to \mathrm{\infty }.$$

(3)

Let ν be any positive measure on $C_n(\mathrm{\Omega })$ such that the Green a-potential

$$G_{\mathrm{\Omega }}^a\nu (P)={\int }_{C_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d\nu (Q)\not\equiv +\mathrm{\infty }$$

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

$$dm(\nu )(Q)=\{\begin{array}{ll}W(t)\phi (\mathrm{\Phi })d\nu (Q),& Q=(t,\mathrm{\Phi })\in C_n(\mathrm{\Omega };(1,+\mathrm{\infty })),\\ 0,& Q\in {\mathbf{R}}^n-C_n(\mathrm{\Omega };(1,+\mathrm{\infty })).\end{array}$$

Remark 1 We remark that the total mass $m(\nu )$ is finite (see [[2], Lemma 5]).

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

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

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

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

is denoted by $E(ϵ;\lambda ,\beta )$.

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

$$M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })=V(r)\phi (\mathrm{\Theta })\text{and}{M}_{\mathrm{\Omega }}^a(P,O)=cW(r)\phi (\mathrm{\Theta })$$

(4)

for any $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$.

In [7], 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 $H\mathrm{\setminus }\{Q\}$. Otherwise H is said to be not a-thin at Q on $C_n(\mathrm{\Omega })$. 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 $H\subset \{P\in E;u(P)=\mathrm{\infty }\}$. A subset H of $C_n(\mathrm{\Omega })$ is said to be a-rarefied at infinity on $C_n(\mathrm{\Omega })$ if there exists a positive superfunction $v(P)$ on $C_n(\mathrm{\Omega })$ such that

$$\underset{P\in C_n(\mathrm{\Omega })}{inf}\frac{v(P)}{M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })}\equiv 0$$

and

$$H\subset \{P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega });v(P)\ge V(r)\}.$$

Let H be a bounded subset of $C_n(\mathrm{\Omega })$. Then ${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^H$ is bounded on $C_n(\mathrm{\Omega })$ and the greatest a-harmonic minorant of ${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^H$ is zero. We see from the Riesz decomposition theorem (see [[10], Theorem 2]) that there exists a unique positive measure ${\lambda }_H^a$ on $C_n(\mathrm{\Omega })$ such that (see [[7], p.6])

$${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^H(P)=G_{\mathrm{\Omega }}^a{\lambda }_H^a(P)$$

(5)

for any $P\in C_n(\mathrm{\Omega })$ and ${\lambda }_H^a$ is concentrated on $I_H$, where

$$I_H=\{P\in C_n(\mathrm{\Omega });H\text{ is not }a\text{-thin at }P\}.$$

We denote the total mass ${\lambda }_H^a(C_n(\mathrm{\Omega }))$ of ${\lambda }_H^a$ by ${\lambda }_{\mathrm{\Omega }}^a(H)$.

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 $C_n(\mathrm{\Omega })$ by

$$d{\eta }_H^a(P)=M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })d{\lambda }_H^a(P)$$

for any $P\in C_n(\mathrm{\Omega })$. It is easy to see that ${\eta }_H^a(C_n(\mathrm{\Omega }))<+\mathrm{\infty }$.

Recently, Long et al. (see [[7], Theorem 2.5]) gave a criterion for a subset H of $C_n(\mathrm{\Omega })$ to be a-rarefied set at infinity.

Theorem A A subset H of $C_n(\mathrm{\Omega })$ is a-rarefied at infinity on $C_n(\mathrm{\Omega })$ if and only if

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

where $H_j=H\cap C_n(\mathrm{\Omega };[2^j,2^{j+1}))$ and $j=0,1,2,\dots$ .

In this paper, we shall obtain a series of new criteria for a-rarefied sets at infinity on $C_n(\mathrm{\Omega })$, which complement Theorem A. Our results are essentially based on Qiao and Deng, Ren and Zhao, Xue (see [2, 11–14]). 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 $C_n(\mathrm{\Omega })$ is a-rarefied at infinity on $C_n(\mathrm{\Omega })$ if and only if there exists a positive measure ${\xi }_H^a$ on $C_n(\mathrm{\Omega })$ such that

$$G_{\mathrm{\Omega }}^a{\xi }_H^a(P)\not\equiv +\mathrm{\infty }$$

(6)

for any $P\in C_n(\mathrm{\Omega })$ and

$$H\subset \{P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega });G_{\mathrm{\Omega }}^a{\xi }_H^a(P)\ge V(r)\}.$$

(7)

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

Theorem 2 If a subset H of $C_n(\mathrm{\Omega })$ is a-rarefied at infinity on $C_n(\mathrm{\Omega })$, then H has a covering $\{r_j,R_j\}$ ($j=0,1,2,\dots$) 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

$$r_j=3\cdot 2^{j-1}\cdot j^{\frac{1}{2-n}}\text{and}{R}_j=3\cdot 2^{j-1}(j=1,2,3,\dots ).$$

A covering $\{r_j,R_j\}$ 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 $C_n({\mathrm{\Omega }}^{\prime })$ be a subset of $C_n(\mathrm{\Omega })$, 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 $B_j\subset C_n({\mathrm{\Omega }}^{\prime })$ and $R_j>2r_j$ for $j\ge j_0$. Then the set $H={\bigcup }_{j=j_0}^{\mathrm{\infty }}{B}_j$ is not a-rarefied at infinity on $C_n(\mathrm{\Omega })$. This fact will be proved in Section 5.

2 Lemmas

Lemma 1 (see [[1], Ch. 11] and [[15], Lemma 4])

$$\begin{array}{c}{G}_{\mathrm{\Omega }}^a(P,Q)\le cV(t)W(r)\phi (\mathrm{\Theta })\phi (\mathrm{\Phi })\hfill \\ (\mathit{\text{resp. }}{G}_{\mathrm{\Omega }}^a(P,Q)\le cV(r)W(t)\phi (\mathrm{\Theta })\phi (\mathrm{\Phi }))\hfill \end{array}$$

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

Lemma 2 (see [[2], Lemma 5])

Let ν be a positive measure on $C_n(\mathrm{\Omega })$ such that there is a sequence of points $P_i=(r_i,{\mathrm{\Theta }}_i)\in C_n(\mathrm{\Omega })$, $r_i\to +\mathrm{\infty }$ ($i\to +\mathrm{\infty }$) satisfying $G_{\mathrm{\Omega }}^a\nu (P_i)<+\mathrm{\infty }$ ($i=1,2,\dots$ ; $Q\in C_n(\mathrm{\Omega })$). Then, for a positive number L,

$${\int }_{C_n(\mathrm{\Omega };(L,+\mathrm{\infty }))}W(t)\phi (\mathrm{\Phi })d\nu (Q)<+\mathrm{\infty }$$

and

$$\underset{R\to +\mathrm{\infty }}{lim}\frac{W(R)}{V(R)}{\int }_{C_n(\mathrm{\Omega };(0,R))}V(t)\phi (\mathrm{\Phi })d\nu (Q)=0.$$

Lemma 3 (see [[2], Theorem 3])

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

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

Lemma 4 (see [[2], Lemma 6])

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

$$H\subset \mathrm{\Pi }\left({\xi }_H^a\right)=\{P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega });G_{\mathrm{\Omega }}^a{\xi }_H^a(P)\ge V(r)\}$$

(10)

for a positive measure ${\xi }_H^a$ on $C_n(\mathrm{\Omega })$ satisfying equation (6).

We write

$$G_{\mathrm{\Omega }}^a\nu (P)=G_{\mathrm{\Omega }}^a(1,j)(P)+G_{\mathrm{\Omega }}^a(2,j)(P)+G_{\mathrm{\Omega }}^a(3,j)(P),$$

where

$$\begin{array}{c}{G}_{\mathrm{\Omega }}^a(1,j)(P)={\int }_{C_n(\mathrm{\Omega };(0,2^{j-1}))}{G}_{\mathrm{\Omega }}^a(P,Q)d\nu (Q),\hfill \\ G_{\mathrm{\Omega }}^a(2,j)(P)={\int }_{C_n(\mathrm{\Omega };[2^{j-1},2^{j+2}))}{G}_{\mathrm{\Omega }}^a(P,Q)d\nu (Q)\hfill \end{array}$$

and

$$G_{\mathrm{\Omega }}^a(3,j)(P)={\int }_{C_n(\mathrm{\Omega };[2^{j+2},\mathrm{\infty }))}{G}_{\mathrm{\Omega }}^a(P,Q)d\nu (Q).$$

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

$$\mathrm{\Pi }\left({\xi }_H^a\right)(j)\subset \{P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega };[2^j,2^{j+1}));2G_{\mathrm{\Omega }}^a(2,j)(P)\ge V(r)\}$$

(11)

for any integer j (≥N).

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

$$G_{\mathrm{\Omega }}^a(1,j)(P)\le cW(r)\phi (\mathrm{\Theta }){\int }_{C_n(\mathrm{\Omega };(0,2^{j-1}))}V(t)\phi (\mathrm{\Phi })d\nu (Q)$$

and

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

from Lemma 1.

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

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

and

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

Thus we obtain

$$4G_{\mathrm{\Omega }}^a(1,j)(P)\le V(r)\phi (\mathrm{\Theta })$$

(12)

and

$$4G_{\mathrm{\Omega }}^a(3,j)(P)\le V(r)\phi (\mathrm{\Theta })$$

(13)

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

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

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

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

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

$$G_{\mathrm{\Omega }}^a(2,j)(P)={\int }_{C_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d{\tau }_j^a(Q)\ge M_{\mathrm{\Omega }}^a(P,\mathrm{\infty }),$$

where $P\in I_j$ ($j\ge N$) and

$$d{\tau }_j^a(Q)=\{\begin{array}{ll}{2}^{1-j}d{\xi }_H^a(Q),& Q\in C_n(\mathrm{\Omega };[2^{j-1},2^{j+2})),\\ 0,& Q\in C_n(\mathrm{\Omega };(0,2^{j-1}))\cup C_n(\mathrm{\Omega };[2^{j+2},\mathrm{\infty })).\end{array}$$

And then we obtain

$${\eta }_{H_j}^a(C_n(\mathrm{\Omega }))\le {\int }_{C_n(\mathrm{\Omega })}V(t)\phi (\mathrm{\Phi })d{\tau }_j^a(Q)={\int }_{C_n(\mathrm{\Omega };[2^{j-1},2^{j+2}))}V(t)\phi (\mathrm{\Phi })d{\xi }_H^a(Q)$$

for $j\ge N$. Then we have

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

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

Suppose that

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

Consider a function $f_H^a(P)$ on $C_n(\mathrm{\Omega })$ defined by

$$f_H^a(P)=\sum _{j=-1}^{\mathrm{\infty }}{\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^{H_j}(P)$$

for any $P\in C_n(\mathrm{\Omega })$, where $H_{-1}=H\cap C_n(\mathrm{\Omega };(0,1))$.

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

$$f_H^a(P)={\int }_{C_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d{\mu }_H^a(1)(Q)$$

for any $P\in C_n(\mathrm{\Omega })$.

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

$$f_H^a(P)=f_H^a(1)(P)+f_H^a(2)(P),$$

where

$$\begin{array}{c}{f}_H^a(1)(P)=\sum _{j=-1}^{j(P)+1}{\int }_{C_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d{\lambda }_{H_j}^a(Q)\text{and}\hfill \\ f_H^a(2)(P)=\sum _{j=j(P)+2}^{\mathrm{\infty }}{\int }_{C_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d{\lambda }_{H_j}^a(Q).\hfill \end{array}$$

Since ${\lambda }_{H_j}^a$ is concentrated on $I_{H_j}\subset {\overline{H}}_j\cap C_n(\mathrm{\Omega })$, we have that

$$\begin{array}{rcl}{\int }_{C_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d{\lambda }_{H_j}^a(Q)& \le & cV(r)\phi (\mathrm{\Theta }){\int }_{C_n(\mathrm{\Omega })}W(t)\phi (\mathrm{\Phi })d{\lambda }_{H_j}^a(t,\mathrm{\Phi })\\ \le & cV(r)\phi (\mathrm{\Theta })W\left(2^j\right)V^{-1}\left(2^j\right){\int }_{C_n(\mathrm{\Omega })}V(t)\phi (\mathrm{\Phi })d{\lambda }_{H_j}^a(t,\mathrm{\Phi })\end{array}$$

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

$$f_H^a(2)(P)\le cV(r)\phi (\mathrm{\Theta })\sum _{j=j(P)+2}^{\mathrm{\infty }}{\eta }_{H_j}^a(C_n(\mathrm{\Omega }))W\left(2^j\right)V^{-1}\left(2^j\right),$$

(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 $P\in C_n(\mathrm{\Omega })$.

Since

$${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^{H_j}(P)=M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })$$

holds on $I_{H_j}$ and $I_{H_j}\subset {\overline{H}}_j\cap C_n(\mathrm{\Omega })$, we see that for any $P=(r,\mathrm{\Theta })\in I_{H_j}$ ($j=-1,0,1,2,3,\dots$)

$$f_H^a(P)\ge c{\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^{H_j}(P)\ge V(r)\phi (\mathrm{\Theta }).$$

(15)

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

Finally, we can define a positive superfunction g on $C_n(\mathrm{\Omega })$ by $g(P)=f_H^a(P)+f_H^a(3)(P)=G_{\mathrm{\Omega }}^a{\xi }_H^a(P)$ for any $P\in C_n(\mathrm{\Omega })$ with ${\xi }_H^a={\mu }_H^a(1)+{\mu }_H^a(2)$. 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

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

Hence by Remark 1 and Lemma 4, $E(ϵ;m({\xi }_H^a),n-1)$ has a covering $\{r_j,R_j\}$ ($j=1,2,3,\dots$) satisfying equation (9) and hence H has also a covering $\{r_j,R_j\}$ ($j=0,1,2,3,\dots$) with an additional finite $B_0$ covering $C_n(\mathrm{\Omega };(0,L])$, satisfying equation (8), which is the conclusion of Theorem 2.

5 Proof of an example

Since $\phi (\mathrm{\Theta })\ge c$ for any $\mathrm{\Theta }\in {\mathrm{\Omega }}^{\prime }$, we have $M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })\ge cV(R_j)$ for any $P\in {\overline{B}}_j$, where $j\ge j_0$. Hence we have

$${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^{B_j}(P)\ge cV(R_j)$$

(16)

for any $P\in {\overline{B}}_j$, where $j\ge j_0$.

Take a measure δ on $C_n(\mathrm{\Omega })$, $supp\delta \subset {\overline{B}}_j$, $\delta ({\overline{B}}_j)=1$ such that

$${\int }_{C_n(\mathrm{\Omega })}{|P-Q|}^{2-n}d\delta (P)={\{Cap({\overline{B}}_j)\}}^{-1}$$

(17)

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

$$G_{\mathrm{\Omega }}^a(P,Q)\le {|P-Q|}^{2-n}$$

for any $P\in C_n(\mathrm{\Omega })$ and $Q\in C_n(\mathrm{\Omega })$ (see [16], the case $n=2$ is implicitly contained in [17]),

$$\begin{array}{rl}{\{Cap({\overline{B}}_j)\}}^{-1}{\lambda }_{B_j}^a(C_n(\mathrm{\Omega }))& =\int (\int {|P-Q|}^{2-n}d\delta (P))d{\lambda }_{B_j}^a(Q)\\ \ge \int (\int G_{\mathrm{\Omega }}^a(P,Q)d{\lambda }_{B_j}^a(Q))d\delta (P)\\ =\int {\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^{B_j}d\delta (P)\\ \ge cV(R_j)\delta ({\overline{B}}_j)=cV(R_j)\end{array}$$

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

$${\lambda }_{B_j}^a(C_n(\mathrm{\Omega }))\ge cCap({\overline{B}}_j)V(R_j)\ge c{r}_j^{n-2}V(R_j).$$

(18)

If we observe ${\lambda }_{H_j}^a(C_n(\mathrm{\Omega }))={\lambda }_{B_j}^a(C_n(\mathrm{\Omega }))$, then we have by equation (3)

$$\sum _{j=j_0}^{\mathrm{\infty }}W\left(2^j\right){\lambda }_{H_j}^a(C_n(\mathrm{\Omega }))\ge c\sum _{j=j_0}^{\mathrm{\infty }}{\left(\frac{r_j}{R_j}\right)}^{n-2}=c\sum _{j=j_0}^{\mathrm{\infty }}\frac{1}{j}=+\mathrm{\infty },$$

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