RETRACTED ARTICLE: Sharp geometrical properties of a-rarefied sets via fixed point index for the Schrödinger operator equations

Abstract

In this paper, we use the theory of fixed point index for the Schrödinger operator equations to obtain a geometrical property of a-rarefied sets at infinity on cones. Meanwhile, we give an example to show that the reverse of this property is not true.

1 Introduction and main theorem

Let R and ${\mathbf{R}}_{+}$ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by ${\mathbf{R}}^{n}$ ($n\geq2$) the n-dimensional Euclidean space. A point in ${\mathbf{R}}^{n}$ is denoted by $P=(X,x_{n})$, $X=(x_{1},x_{2},\ldots,x_{n-1})$. 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. For $P\in{\mathbf{R}}^{n}$ and $r>0$, let $B(P,r)$ denote the open ball with center at P and radius r in ${\mathbf{R}}^{n}$.

We introduce a system of spherical coordinates $(r,\Theta)$, $\Theta=(\theta_{1},\theta_{2},\ldots, \theta_{n-1})$, in ${\mathbf{R}}^{n}$ which are related to Cartesian coordinates $(x_{1},x_{2},\ldots,x_{n-1},x_{n})$ by $x_{n}=r\cos\theta_{1}$.

Let D be an arbitrary domain in ${\mathbf{R}}^{n}$ and $\mathscr{A}_{a}$ denote the class of nonnegative radial potentials $a(P)$, i.e. $0\leq a(P)=a(r)$, $P=(r,\Theta)\in D$, such that $a\in L_{\mathrm{loc}}^{b}(D)$ with some $b> {n}/{2}$ if $n\geq4$ and with $b=2$ if $n=2$ or $n=3$.

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

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

where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space $C_{0}^{\infty}(D)$ to an essentially self-adjoint operator on $L^{2}(D)$ (see [1], Chapter 11). We will denote it by $\mathit{Sch}_{a}$ as well. This last one has a Green-Sch function $G_{D}^{a}(P,Q)$. Here $G_{D}^{a}(P,Q)$ is positive on D and its inner normal derivative $\partial G_{D}^{a}(P,Q)/{\partial n_{Q}}\geq0$, where ${\partial}/{\partial n_{Q}}$ denotes the differentiation at Q along the inward normal into D.

We call a function $u\not\equiv-\infty$ 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 $[-\infty,\infty)$ and at each point $P\in D$ with $0< r< r(P)$ the generalized mean-value inequality (see [2])

$$u(P)\leq\int_{\partial{B(P,r)}}u(Q)\frac{\partial G_{B(P,r)}^{a}(P,Q)}{\partial n_{Q}}\, d\sigma(Q) $$

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

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 a generalized 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 $(1,\Theta)$ on ${\mathbf{S}}^{n-1}$ and the set $\{\Theta; (1,\Theta)\in\Omega\}$ for a set Ω, $\Omega\subset{\mathbf{S}}^{n-1}$, are often identified with Θ and Ω, respectively. For two sets $\Xi\subset{\mathbf{R}}_{+}$ and $\Omega\subset{\mathbf{S}}^{n-1}$, the set $\{(r,\Theta)\in{\mathbf{R}}^{n}; r\in\Xi,(1,\Theta)\in\Omega\}$ in ${\mathbf{R}}^{n}$ is simply denoted by $\Xi\times\Omega$. By $C_{n}(\Omega)$, we denote the set ${\mathbf{R}}_{+}\times\Omega$ in ${\mathbf{R}}^{n}$ with the domain Ω on ${\mathbf{S}}^{n-1}$. We call it a cone. We denote the set $I\times\Omega$ with an interval on R by $C_{n}(\Omega;I)$.

We shall say that a set $H\subset C_{n}(\Omega)$ has a covering $\{r_{j}, R_{j}\}$ if there exists a sequence of balls $\{B_{j}\}$ with centers in $C_{n}(\Omega)$ such that $H\subset\bigcup_{j=0}^{\infty} B_{j}$, where $r_{j}$ is the radius of $B_{j}$ and $R_{j}$ is the distance from the origin to the center of $B_{j}$. For positive functions $h_{1}$ and $h_{2}$, we say that $h_{1}\lesssim h_{2}$ if $h_{1}\leq Mh_{2}$ for some constant $M>0$. If $h_{1}\lesssim h_{2}$ and $h_{2}\lesssim h_{1}$, we say that $h_{1}\approx h_{2}$.

From now on, we always assume $D=C_{n}(\Omega)$. For the sake of brevity, we shall write $G_{\Omega}^{a}(P,Q)$ instead of $G_{C_{n}(\Omega)}^{a}(P,Q)$. 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 all 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

$$\begin{aligned}& (\Lambda_{n}+\lambda)\varphi=0 \quad \text{on } \Omega, \\& \varphi=0 \quad \text{on } \partial{\Omega}, \end{aligned}$$

where $\Lambda_{n}$ is the spherical part of the Laplace operator $\Delta_{n}$:

$$\Delta_{n}=\frac{n-1}{r}\frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial r^{2}}+ \frac{\Lambda_{n}}{r^{2}}. $$

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by $\varphi(\Theta)$. In order to ensure the existence of λ and a smooth $\varphi(\Theta)$. We put a rather strong assumption on Ω: if $n\geq3$, 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.

Solutions of an ordinary differential equation

$$ -Q''(r)-\frac{n-1}{r}Q'(r)+ \biggl( \frac{\lambda}{r^{2}}+a(r) \biggr)Q(r)=0,\quad 0< r< \infty. $$

(1.1)

It is well known (see, for example, [3]) that if the potential $a\in \mathscr{A}_{a}$, then (1.1) has a fundamental system of positive solutions $\{V,W\}$ such that V and W are increasing and decreasing, respectively (see [4–7]).

We will also consider the class $\mathscr{B}_{a}$, consisting of the potentials $a\in\mathscr{A}_{a}$ such that there exists the finite limit $\lim_{r\rightarrow\infty}r^{2} a(r)=k\in[0,\infty)$, and moreover, $r^{-1}|r^{2} a(r)-k|\in L(1,\infty)$. If $a\in \mathscr{B}_{a}$, then the (sub)superfunctions are continuous (see [8]).

In the rest of paper, we assume that $a\in\mathscr{B}_{a}$ 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 (1.1) have the asymptotics (see [9])

$$ V(r)\approx r^{\iota_{k}^{+}}, \qquad W(r)\approx r^{\iota _{k}^{-}}, \quad \text{as } r\rightarrow\infty. $$

(1.2)

Let ν be any positive measure on cones such that the Green-Sch potential

$$G_{\Omega}^{a} \nu(P)=\int_{C_{n}(\Omega)}G_{\Omega}^{a}(P,Q) \, d\nu(Q)\not\equiv +\infty $$

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

$$d\nu'(Q)=\left \{ \textstyle\begin{array}{l@{\quad}l} W(t) \varphi(\Phi)\, d\nu(Q), & Q=(t,\Phi)\in C_{n}(\Omega; (1,+\infty)) , \\ 0,& Q\in{\mathbf{R}}^{n}-C_{n}(\Omega; (1,+\infty)). \end{array}\displaystyle \right . $$

The Poisson-Sch integral $PI_{\Omega}^{a} \mu(P)\not\equiv+\infty$ ($P\in C_{n}(\Omega)$) of μ on cones is defined as follows:

$$PI_{\Omega}^{a} \mu(P)=\frac{1}{c_{n}}\int _{S_{n}(\Omega)}PI_{\Omega}^{a}(P,Q)\, d\mu(Q), $$

where

$$PI_{\Omega}^{a}(P,Q)=\frac{\partial G_{\Omega}^{a}(P,Q)}{\partial n_{Q}},\qquad c_{n}= \left \{ \textstyle\begin{array}{l@{\quad}l} 2\pi, & n=2, \\ (n-2)s_{n}, & n\geq3, \end{array}\displaystyle \right . $$

μ is a positive measure on $\partial{C_{n}(\Omega)}$ and ${\partial}/{\partial n_{Q}}$ denotes the differentiation at Q along the inward normal into cones. Then the positive measure $\mu'$ on ${\mathbf{R}}^{n}$ is defined by

$$d\mu'(Q)=\left \{ \textstyle\begin{array}{ll} t^{-1}W(t)\frac{\partial\varphi(\Phi)}{\partial n_{\Phi}}\, d\mu(Q), & Q=(t,\Phi)\in S_{n}(\Omega; (1,+\infty)) , \\ 0,&Q;\in{\mathbf{R}}^{n}-S_{n}(\Omega; (1,+\infty)). \end{array}\displaystyle \right . $$

Remark

We remark that the total masses of $\mu'$ and $\nu'$ are finite (see [2], Lemma 5 and [6], Lemma 4).

Let $0\leq\alpha\leq n$ and λ be any positive measure on ${\mathbf{R}}^{n}$ having finite total mass. For each $P=(r,\Theta)\in {\mathbf{R}}^{n}-\{O\}$, the maximal function $M(P;\lambda,\alpha)$ with respect to $\mathit{Sch}_{a}$ is defined by

$$M(P;\lambda,\alpha)=\sup_{ 0< \rho< \frac{r}{2}}\lambda\bigl(B(P,\rho)\bigr)V( \rho)W(\rho)\rho^{\alpha-2}. $$

The set

$$\bigl\{ P=(r,\Theta)\in{\mathbf{R}}^{n}-\{O\}; M(P;\lambda, \alpha)V^{-1}(r)W^{-1}(r)r^{2-\alpha}>\epsilon\bigr\} $$

is denoted by $E(\epsilon; \lambda, \alpha)$.

The following Theorems A and B give a way to estimate the Green-Sch potential and the Poisson-Sch integrals with measures on $C_{n}(\Omega)$ and $S_{n}(\Omega)$, respectively.

Theorem A

Letνbe a positive measure on $C_{n}(\Omega)$such that $G_{\Omega}^{a} \nu(P)\not\equiv +\infty$ ($P=(r,\Theta)\in C_{n}(\Omega)$) holds. Then for a sufficiently largeLwe have

$$\bigl\{ P\in C_{n}\bigl(\Omega; (L, +\infty)\bigr); G_{\Omega}^{a}\nu(P)\geq V(r)\bigr\} \subset E\bigl(\epsilon; \mu',1\bigr). $$

Theorem B

Letμbe a positive measure on $S_{n}(\Omega)$such that $PI_{\Omega}^{a} \mu(P)\not\equiv+\infty$ ($P=(r,\Theta)\in C_{n}(\Omega)$). Then for a sufficiently largeLwe have

$$\bigl\{ P\in C_{n}\bigl(\Omega; (L, +\infty)\bigr); PI_{\Omega}^{a} \mu(P)\geq V(r)\bigr\} \subset E\bigl(\epsilon; \mu',1\bigr). $$

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

$$ M_{\Omega}^{a}(P,\infty)=V(r)\varphi(\Theta)\quad \text {and} \quad M_{\Omega}^{a}(P,O)=cW(r)\varphi(\Theta) $$

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

In [7, 10], Xue and Zhao-Yamada introduce the notations of a-thin (with respect to the Schrödinger operator $\mathit{Sch}_{a}$) at a point and a-rarefied sets at infinity (with respect to the Schrödinger operator $\mathit{Sch}_{a}$), which generalized the earlier notations obtained by Miyamoto, Hoshida, Brelot (see [11–14]).

Definition 1

(see [7])

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 $H\backslash\{Q\}$. Otherwise H is said to be not a-thin at Q on cones.

Definition 2

(see [10])

A subset H of $C_{n}(\Omega)$ is said to be a-rarefied at infinity on cones, if there exists a positive superfunction $v(P)$ on cones such that

$$ \inf_{P\in C_{n}(\Omega)}\frac{v(P)}{M_{\Omega}^{a}(P,\infty)}\equiv0 $$

(1.3)

and

$$ H\subset\bigl\{ P=(r,\Theta)\in C_{n}(\Omega); v(P)\geq V(r)\bigr\} . $$

(1.4)

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

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

for any $P\in C_{n}(\Omega)$ and $\lambda_{H}^{a}$ is concentrated on $I_{H}$, where

$$I_{H}=\bigl\{ P\in C_{n}(\Omega); H \text{ is not } a \text{-thin at } P\bigr\} . $$

We denote the total mass $\lambda_{H}^{a}(C_{n}(\Omega))$ of $\lambda_{H}^{a}$ by $\lambda_{\Omega}^{a}(H)$.

Recently, GX Xue (see [7], Theorem 2.5) gave a criterion for a subset H of $C_{n}(\Omega)$ to be a-rarefied set at infinity.

Theorem C

A subsetHof $C_{n}(\Omega)$isa-rarefied at infinity on cones if and only if

$$ \sum_{j=0}^{\infty}W\bigl(2^{j}\bigr) \lambda_{H_{j}}^{a}\bigl(C_{n}(\Omega)\bigr)< \infty, $$

where $H_{j}=H\cap C_{n}(\Omega;[2^{j},2^{j+1}))$and $j=0,1,2,\ldots$ .

Our aim in this paper is to characterize the geometrical property of a-rarefied sets at infinity.

Theorem 1

If a subsetHof $C_{n}(\Omega)$isa-rarefied at infinity on cones, thenHhas a covering $\{r_{j},R_{j}\}$ ($j=0,1,2,\ldots$) satisfying

$$ \sum_{j=0}^{\infty} \biggl( \frac{r_{j}}{R_{j}} \biggr)\frac {V(R_{j})}{V(r_{j})}\frac{W(R_{j})}{W(r_{j})}< \infty. $$

(1.5)

Next, we immediately have the following result from Theorem 1.

Corollary 1

Let $v(P)$be positive superfunction on cones. Then $v(P)V^{-1}(r)$uniformly converges to $c_{\infty}(v,a)\varphi(\Theta)$as $r\rightarrow\infty$outside a set which has a covering $\{r_{j},R_{j}\}$ ($j=0,1,2,\ldots$) satisfying (1.5), where

$$c_{\infty}(v, a)=\inf_{P\in C_{n}(\Omega)}\frac{v(P)}{M^{a}_{\Omega}(P, \infty)}. $$

Finally, we prove the following result.

Theorem 2

If a subsetHof $C_{n}(\Omega)$has a covering $\{r_{j},R_{j}\}$ ($j=0,1,2,\ldots$) satisfying (1.5), then it is possible thatHis nota-rarefied at infinity on cones.

2 Main lemmas

Lemma 1

Letλbe any positive measure on ${\mathbf{R}}^{n}$having finite total mass. Then $E(\epsilon; \lambda, 1)$has a covering $\{r_{j},R_{j}\}$ ($j=1,2,\ldots$) satisfying

$$\sum_{j=1}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)\frac {V(R_{j})W(R_{j})}{V(r_{j})W(r_{j})}< \infty. $$

Proof

Set

$$E_{j}(\epsilon;\lambda, 1)= \bigl\{ P=(r,\Theta)\in E(\epsilon; \lambda, 1):2^{j}\leq r< 2^{j+1}\bigr\} \quad (j=2,3,4,\ldots). $$

If $P=(r,\Theta)\in E_{j}(\epsilon; \lambda, 1)$, then there exists a positive number $\rho(P)$ such that

$$\biggl(\frac{\rho(P)}{r} \biggr)\frac{V(r)W(R)}{V(\rho(P))W(\rho (P))}\approx \biggl( \frac{\rho(P)}{r} \biggr)^{n-1}\leq \frac{\lambda(B(P,\rho(P)))}{\epsilon}. $$

Since $E_{j}(\epsilon; \lambda, 1)$ can be covered by the union of a family of balls $\{B(P_{j,i},\rho_{j,i}):P_{j,i}\in E_{k}(\epsilon; \lambda, 1)\}$ ($\rho_{j,i}=\rho(P_{j,i})$). By the Vitali lemma (see [15]), there exists $\Lambda_{j} \subset E_{j}(\epsilon; \lambda, 1)$, which is at most countable, such that $\{B(P_{j,i},\rho_{j,i}):P_{j,i}\in\Lambda_{j} \}$ are disjoint and $E_{j}(\epsilon; \lambda, 1) \subset \bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},5\rho_{j,i})$. So

$$\bigcup_{j=2}^{\infty}E_{j}( \epsilon; \lambda, 1) \subset \bigcup_{j=2}^{\infty}\bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},5 \rho_{j,i}). $$

On the other hand, note that

$$\bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i}, \rho_{j,i}) \subset\bigl\{ P=(r,\Theta):2^{j-1}\leq r< 2^{j+2}\bigr\} , $$

so that

$$\begin{aligned} \sum_{P_{j,i} \in \Lambda_{j}} \biggl(\frac{5\rho_{j,i}}{|P_{j,i}|} \biggr) \frac{ V(|P_{j,i}|)W(|P_{j,i}|) }{V(\rho_{j,i})W(\rho_{j,i}) } \approx& \sum_{P_{j,i} \in\Lambda_{j}} \biggl( \frac{5\rho _{j,i}}{|P_{j,i}|} \biggr)^{n-1} \\ \leq&5;^{n-1}\sum_{P_{j,i}\in\Lambda_{j}}\frac{\lambda (B(P_{j,i},\rho_{j,i}))}{\epsilon} \\ \leq& \frac{5^{n-1}}{\epsilon} \lambda\bigl(C_{n}\bigl(\Omega; \bigl[2^{j-1},2^{j+2} \bigr)\bigr)\bigr). \end{aligned}$$

Hence we obtain

$$\begin{aligned} \sum_{j=1}^{\infty}\sum _{P_{j,i} \in \Lambda_{j}} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr)\frac{ V(|P_{j,i}|)W(|P_{j,i}|) }{V(\rho_{j,i})W(\rho_{j,i}) } \approx& \sum_{j=1}^{\infty}\sum _{P_{j,i} \in\Lambda_{j}} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr)^{n-1} \\ \leq&\sum_{j=1}^{\infty}\frac{ \lambda(C_{n}(\Omega ; [2^{j-1},2^{j+2} )))}{\epsilon} \\ \leq& \frac{3\lambda({\mathbf{R}}^{n})}{\epsilon}. \end{aligned}$$

Since $E(\epsilon; \lambda, 1)\cap\{P=(r,\Theta)\in{\mathbf{R}}^{n}; r\geq4\}=\bigcup_{j=2}^{\infty}E_{j}(\epsilon;\lambda, 1)$. Then $E(\epsilon; \lambda, 1)$ is finally covered by a sequence of balls $\{B(P_{j,i},\rho_{j,i}), B(P_{1},6)\}$ ($j=2,3,\ldots$ ; $i=1,2,\ldots$) satisfying

$$\sum_{j,i} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr) \frac{ V(|P_{j,i}|)W(|P_{j,i}|) }{V(\rho_{j,i})W(\rho_{j,i}) }\approx\sum_{j,i} \biggl( \frac{\rho_{j,i}}{|P_{j,i}|} \biggr)^{n-1}\leq\frac {3\lambda({\mathbf{R}}^{n})}{\epsilon}+6^{n-\alpha}< + \infty, $$

where $B(P_{1},6)$ ($P_{1}=(1,0,\ldots,0)\in{\mathbf{R}}^{n}$) is the ball which covers $\{P=(r,\Theta)\in{\mathbf{R}}^{n}; r<4\}$. □

3 Proof of Theorem 1

Since H is a-rarefied at infinity on cones, by Definition 2 there exists a positive superfunction $v(P)$ on cones such that (1.3) and (1.4) hold.

For this $v(P)$ there exists a unique positive measure $\mu''$ on $S_{n}(\Omega)$ and a unique positive measure $\nu''$ on cones such that (see [2], Theorem 3)

$$ v(P)=c_{0}(v, a)M_{\Omega}^{a}(P,O)+G_{\Omega}^{a} \nu''(P)+PI_{\Omega}^{a} \mu''(P), $$

(3.1)

where

$$c_{0}(v, a)=\inf_{P\in C_{n}(\Omega)}\frac{v(P)}{M^{a}_{\Omega}(P, O)}. $$

Let us denote

$$\begin{aligned}& H_{1}=\biggl\{ P=(r,\Theta)\in C_{n}(\Omega); c_{0}(v, a)M_{\Omega}^{a}(P,O)\geq\frac{V(r)}{3} \biggr\} , \\& H_{2}=\biggl\{ P=(r,\Theta)\in C_{n}(\Omega); G_{\Omega}^{a}\nu''(P)\geq \frac {V(r)}{3}\biggr\} \end{aligned}$$

and

$$H_{3}=\biggl\{ P=(r,\Theta)\in C_{n}(\Omega); PI_{\Omega}^{a}\mu''(P)\geq \frac{V(r)}{3}\biggr\} , $$

respectively.

Then we see from (1.4) that

$$ H\subset H_{1}\cup H_{2} \cup H_{3}. $$

(3.2)

For each $H_{i}$ ($i=1,2,3$), we know that it has a covering. It is evident from the boundedness of $H_{1}$ that $H_{1}$ has a covering $\{r_{1},R_{1}\}$ satisfying

$$ \frac{r_{1}}{R_{1}}< +\infty. $$

(3.3)

When we apply Theorems A and B with the measures μ and ν defined by $\mu=3\mu''$ and $\nu=3\nu''$, respectively, we can find two positive constants L and ϵ such that

$$H_{2}\cap C_{n}\bigl(\Omega; (L, +\infty)\bigr)\subset E \bigl(\epsilon; \mu',1\bigr) $$

and

$$H_{3}\cap C_{n}\bigl(\Omega; (L, +\infty)\bigr)\subset E \bigl(\epsilon; \nu',1\bigr), $$

respectively.

By Lemma 1, these sets $E(\epsilon; \mu',1)$ and $E(\epsilon; \nu',1)$ have coverings $\{r_{j}^{(2)},R_{j}^{(2)}\}$ ($j=1,2,\ldots$) and $\{r_{j}^{(3)},R_{j}^{(3)}\}$ ($j=1,2,\ldots$) satisfying

$$ \sum_{j=1}^{\infty} \biggl( \frac{r_{j}^{(2)}}{R_{j}^{(2)}} \biggr)\frac {V(R_{j}^{(2)})W(R_{j}^{(2)})}{V(r_{j}^{(2)})W(r_{j}^{(2)})}< +\infty $$

(3.4)

and

$$ \sum_{j=1}^{\infty} \biggl( \frac{r_{j}^{(3)}}{R_{j}^{(3)}} \biggr)\frac {V(R_{j}^{(3)})W(R_{j}^{(3)})}{V(r_{j}^{(3)})W(r_{j}^{(3)})}< +\infty , $$

(3.5)

respectively.

Then $H_{2}$ and $H_{3}$ also have coverings $\{r_{j}^{(2)},R_{j}^{(2)}\}$ ($j=1,2,\ldots$) and $\{r_{j}^{(3)},R_{j}^{(3)}\}$ ($j=1,2,\ldots$) satisfying (3.4) and (3.5), respectively.

Thus by rearranging coverings $\{r_{1},R_{1}\}$, $\{r_{j}^{(2)},R_{j}^{(2)}\}$ ($j=1,2,\ldots$) and $\{r_{j}^{(3)},R_{j}^{(3)}\}$ ($j=1,2,\ldots$), we know that the set H has a covering $\{r_{j},R_{j}\}$ ($j=0,1,2,\ldots$) from (3.2) and satisfies (1.5) from (3.3), (3.4), and (3.5).

Thus we complete the proof of Theorem 1.

4 Proof of Theorem 2

Put

$$r_{j}=3\cdot2^{j-1}\cdot j^{\frac{1}{2-n}}\quad \text{and} \quad R_{j}=3\cdot 2^{j-1} \quad (j=1,2,3,\ldots). $$

A covering $\{r_{j}, R_{j}\}$ satisfies

$$\sum_{j=1}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)\frac {V(R_{j})}{V(r_{j})}\frac{W(R_{j})}{W(r_{j})}\leq c\sum _{j=1}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)^{n-1}=c \sum_{j=1}^{\infty}j^{\frac{n-1}{2-n}}< + \infty $$

from (1.2).

Let $C_{n}(\Omega')$ be a subset of $C_{n}(\Omega)$, i.e. $\overline{\Omega}'\subset\Omega$. Suppose that this covering is so located: there is an integer $j_{0}$ such that $B_{j}\subset C_{n}(\Omega')$ and $R_{j}>2r_{j}$ for $j\geq j_{0}$.

Next we shall prove that the set $H=\bigcup_{j=j_{0}}^{\infty}B_{j}$ is not a-rarefied at infinity on $C_{n}(\Omega)$. Since $\varphi(\Theta)\geq c$ for any $\Theta\in\Omega'$, we have $M_{\Omega}^{a}(P,\infty)\geq cV(R_{j})$ for any $P\in \overline{B}_{j}$, where $j\geq j_{0}$. Hence we have

$$ \hat{R}_{M_{\Omega}^{a}(\cdot,\infty)}^{B_{j}}(P)\geq cV(R_{j}) $$

(4.1)

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

Take a measure δ on cones, $\operatorname{supp} \delta\subset \overline{B}_{j}$, $\delta(\overline{B}_{j})=1$ such that

$$ \int_{C_{n}(\Omega)}|P-Q|^{2-n}\, d\delta(P)=\bigl\{ \operatorname{Cap}(\overline{B}_{j})\bigr\} ^{-1} $$

(4.2)

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

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

for any $P\in C_{n}(\Omega)$ and $Q\in C_{n}(\Omega)$,

$$\begin{aligned} \bigl\{ \operatorname{Cap}(\overline{B}_{j})\bigr\} ^{-1} \lambda_{B_{j}}^{a}\bigl(C_{n}(\Omega)\bigr) =& \int \biggl(\int|P-Q|^{2-n}\, d\delta(P) \biggr)\, d\lambda _{B_{j}}^{a}(Q) \\ \geq& \int \biggl(\int G_{\Omega}^{a}(P,Q)\, d \lambda_{B_{j}}^{a}(Q) \biggr)\, d\delta(P) \\ =& \int\hat{R}_{M_{\Omega}^{a}(\cdot,\infty)}^{B_{j}}\, d\delta(P) \\ \geq& c V(R_{j})\delta(\overline{B}_{j})=c V(R_{j}) \end{aligned}$$

from (4.1) and (4.2). Hence we have (see [5], p.1517)

$$ \lambda_{B_{j}}^{a}\bigl(C_{n}(\Omega) \bigr)\geq c \operatorname{Cap}(\overline{B}_{j}) V(R_{j})\geq c r_{j}^{n-2}V(R_{j}). $$

(4.3)

If we observe $\lambda_{H_{j}}^{a}(C_{n}(\Omega))=\lambda_{B_{j}}^{a}(C_{n}(\Omega))$, then we have by (1.2)

$$\sum_{j=j_{0}}^{\infty}W\bigl(2^{j}\bigr) \lambda_{H_{j}}^{a}\bigl(C_{n}(\Omega)\bigr)\geq c \sum_{j=j_{0}}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)^{n-2}= c \sum_{j=j_{0}}^{\infty} \frac{1}{j}=+\infty, $$

from which it follows by Theorem C that H is not a-rarefied at infinity on cones.