RETRACTED ARTICLE: Growth properties of Green-Sch potentials at infinity

Abstract

This paper gives growth properties of Green-Sch potentials at infinity in a cone, which generalizes results obtained by Qiao-Deng. The proof is based on the fact that the estimations of Green-Sch potentials with measures are connected with a kind of densities of the measures modified by the measures.

MSC: 35J10, 35J25.

1 Introduction and main 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 of 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, the closure and the complement of a set S in ${\mathbf{R}}^n$ are denoted by ∂ S, $\overline{\mathbf{S}}$, and ${\mathbf{S}}^c$, 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 $(r,\mathrm{\Theta })$, $\mathrm{\Theta }=({\theta }_1,{\theta }_2,\dots ,{\theta }_{n-1})$, in ${\mathbf{R}}^n$ which are related to cartesian coordinates $(x_1,x_2,\dots ,x_{n-1},x_n)$ by

$$x_1=r(\prod _{j=1}^{n-1}sin{\theta }_j)(n\ge 2),x_n=rcos{\theta }_1,$$

and if $n\ge 3$, then

$$x_{n-m+1}=r(\prod _{j=1}^{m-1}sin{\theta }_j)cos{\theta }_m(2\le m\le n-1),$$

where $0\le r<+\mathrm{\infty }$, $-\frac{1}{2}\pi \le {\theta }_{n-1}<\frac{3}{2}\pi$, and if $n\ge 3$, then $0\le {\theta }_j\le \pi$ ($1\le j\le n-2$).

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 $\mathrm{\Xi }\times \mathrm{\Omega }$. In particular, the half space ${\mathbf{R}}_{+}\times {\mathbf{S}}_{+}^{n-1}=\{(X,x_n)\in {\mathbf{R}}^n;x_n>0\}$ will be denoted by ${\mathbf{T}}_n$.

By $C_n(\mathrm{\Omega })$, we denote the set ${\mathbf{R}}_{+}\times \mathrm{\Omega }$ in ${\mathbf{R}}^n$ with the domain Ω on ${\mathbf{S}}^{n-1}$ ($n\ge 2$). We call it a cone. Then $T_n$ is a special cone obtained by putting $\mathrm{\Omega }={\mathbf{S}}_{+}^{n-1}$. We denote the sets $I\times \mathrm{\Omega }$ and $I\times \partial \mathrm{\Omega }$ with an interval on R by $C_n(\mathrm{\Omega };I)$ and $S_n(\mathrm{\Omega };I)$. By $S_n(\mathrm{\Omega };r)$ we denote $C_n(\mathrm{\Omega })\cap S_r$. By $S_n(\mathrm{\Omega })$ we denote $S_n(\mathrm{\Omega };(0,+\mathrm{\infty }))$, which is $\partial C_n(\mathrm{\Omega })-\{O\}$.

Let $C_n(\mathrm{\Omega })$ be an arbitrary domain in ${\mathbf{R}}^n$ and ${\mathcal{A}}_a$ denote the class of nonnegative radial potentials $a(P)$, i.e.$0\le a(P)=a(r)$, $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$, such that $a\in L_{\mathrm{loc}}^b(C_n(\mathrm{\Omega }))$ 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 stationary Schrödinger operator

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

where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space $C_0^{\mathrm{\infty }}(C_n(\mathrm{\Omega }))$ to an essentially self-adjoint operator on $L^2(C_n(\mathrm{\Omega }))$ (see [1], Ch. 13]). We will denote it $Sc{h}_a$ as well. This last one has a Green-Sch function $G_{\mathrm{\Omega }}^a(P,Q)$. Here $G_{\mathrm{\Omega }}^a(P,Q)$ is positive on $C_n(\mathrm{\Omega })$ and its inner normal derivative $\partial G_{\mathrm{\Omega }}^a(P,Q)/\partial n_Q\ge 0$, where $\partial /\partial n_Q$ denotes the differentiation at Q along the inward normal into $C_n(\mathrm{\Omega })$. We denote this derivative by $P{I}_{\mathrm{\Omega }}^a(P,Q)$, which is called the Poisson-Sch kernel with respect to $C_n(\mathrm{\Omega })$.

We shall say that a set $E\subset C_n(\mathrm{\Omega })$ 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 $E\subset {\bigcup }_{j=0}^{\mathrm{\infty }}{B}_j$, where $r_j$ is the radius of $B_j$ and $R_j$ is the distance from the origin to the center of $B_j$.

For positive functions $h_1$ and $h_2$, we say that $h_1\lesssim h_2$ if $h_1\le M{h}_2$ for some constant $M>0$. If $h_1\lesssim h_2$ and $h_2\lesssim h_1$, we say that $h_1\approx h_2$.

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 opera ${\mathrm{\Delta }}_n$

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

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by $\phi (\mathrm{\Theta })$, ${\int }_{\mathrm{\Omega }}{\phi }^2(\mathrm{\Theta })d{S}_1=1$. 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 [2], pp.88-89] for the definition of $C^{2,\alpha }$-domain).

For any $(1,\mathrm{\Theta })\in \mathrm{\Omega }$, we have (see [3], pp.7-8])

$$\phi (\mathrm{\Theta })\approx dist((1,\mathrm{\Theta }),\partial C_n(\mathrm{\Omega })),$$

which yields

$$\delta (P)\approx r\phi (\mathrm{\Theta }),$$

(1.1)

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

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

(1.2)

It is well known (see, for example, [4]) that if the potential $a\in {\mathcal{A}}_a$, then (1.2) has a fundamental system of positive solutions $\{V,W\}$ such that V is nondecreasing with (see [5]–[8])

$$0\le V(0+)\le V(r)\text{as }r\to +\mathrm{\infty },$$

and W is monotonically decreasing with

$$+\mathrm{\infty }=W(0+)>W(r)\searrow 0\text{as }r\to +\mathrm{\infty }.$$

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 [9]).

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 (1.2) have the asymptotic (see [10])

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

(1.3)

We denote the Green-Sch potential with a positive measure v on $C_n(\mathrm{\Omega })$ by

$$G_{\mathrm{\Omega }}^a\nu (P)={\int }_{C_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d\nu (Q).$$

Let ν be any positive measure $C_n(\mathrm{\Omega })$ such that $G_{\mathrm{\Omega }}^a\nu (P)\not\equiv +\mathrm{\infty }$ (resp. $G_{\mathrm{\Omega }}^0\nu (P)\not\equiv +\mathrm{\infty }$) for $P\in C_n(\mathrm{\Omega })$. The positive measure ${\nu }^{\prime }$ (rep. ${\nu }^{″}$) on ${\mathbf{R}}^n$ is defined by

$$\begin{array}{c}d{\nu }^{\prime }(Q)=\{\begin{array}{ll}W(t)\phi (\mathrm{\Phi })d\nu (Q),& Q=(t,\mathrm{\Phi })\in C_n(\mathrm{\Omega };(1,+\mathrm{\infty })),\\ 0,& Q\in {\mathbf{R}}^n-C_n(\mathrm{\Omega };(1,+\mathrm{\infty })).\end{array}\hfill \\ (d{\nu }^{\prime }(Q)=\{\begin{array}{ll}{t}^{{\iota }_0^{-}}\phi (\mathrm{\Phi })d\nu (Q),& Q=(t,\mathrm{\Phi })\in C_n(\mathrm{\Omega };(1,+\mathrm{\infty })),\\ 0,& Q\in {\mathbf{R}}^n-C_n(\mathrm{\Omega };(1,+\mathrm{\infty })).\end{array})\hfill \end{array}$$

Let $ϵ>0$, $0\le \alpha <n$, and λ be any positive measure on ${\mathbf{R}}^n$ having finite total mass. For each $P=(r,\mathrm{\Theta })\in {\mathbf{R}}^n-\{O\}$, the maximal function $M(P;\lambda ,\alpha )$ is defined by (see [11])

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

The set

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

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

Remark 1

If $\lambda (\{P\})>0$ ($P\ne O$), then $M(P;\lambda ,\alpha )=+\mathrm{\infty }$ for any positive number β. So we can find $\{P\in {\mathbf{R}}^n-\{O\};\lambda (\{P\})>0\}\subset E(ϵ;\lambda ,\alpha )$.

About the growth properties of Green potentials at infinity in a cone, Qiao-Deng (see [12], Theorem 1]) has proved the following result.

Theorem A

Let ν be a positive measure on$C_n(\mathrm{\Omega })$such that$G_{\mathrm{\Omega }}^0\nu (P)\not\equiv +\mathrm{\infty }$for any$P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$. Then there exists a covering$\{r_j,R_j\}$of$F(ϵ;{\nu }^{″},\alpha )$ ($\subset C_n(\mathrm{\Omega })$) satisfying

$$\sum _{j=0}^{\mathrm{\infty }}{\left(\frac{r_j}{R_j}\right)}^{n-\alpha }<\mathrm{\infty },$$

such that

$$\underset{r\to \mathrm{\infty },P\in C_n(\mathrm{\Omega })-F(ϵ;{\nu }^{″},\alpha )}{lim}{r}^{-{\iota }_0^{+}}{\phi }^{\alpha -1}(\mathrm{\Theta })G_{\mathrm{\Omega }}^0\nu (P)=0,$$

where

$$H(P;{\nu }^{″},\alpha )=\underset{0<\rho <\frac{r}{2}}{sup}\frac{{\nu }^{″}(B(P,\rho ))}{{\rho }^{n-\alpha }}$$

and

$$F(ϵ;{\nu }^{″},\alpha )=\{P=(r,\mathrm{\Theta })\in {\mathbf{R}}^n-\{O\};H(P;{\nu }^{″},\alpha )r^{n-\alpha }>ϵ\}.$$

Now we state our first result.

Theorem 1

Let ν be a positive measure on $C_n(\mathrm{\Omega })$ such that

$$G_{\mathrm{\Omega }}^a\nu (P)\not\equiv +\mathrm{\infty }(P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })).$$

(1.4)

Then there exists a covering$\{r_j,R_j\}$of$E(ϵ;{\nu }^{\prime },\alpha )$ ($\subset C_n(\mathrm{\Omega })$) satisfying

$$\sum _{j=0}^{\mathrm{\infty }}{\left(\frac{r_j}{R_j}\right)}^{2-\alpha }\frac{V(R_j)W(R_j)}{V(r_j)W(r_j)}<\mathrm{\infty },$$

(1.5)

such that

$$\underset{r\to \mathrm{\infty },P\in C_n(\mathrm{\Omega })-E(ϵ;{\nu }^{\prime },\alpha )}{lim}{V}^{-1}(r){\phi }^{\alpha -1}(\mathrm{\Theta })G_{\mathrm{\Omega }}^a\nu (P)=0.$$

(1.6)

Remark 2

By comparison the condition (1.4) is fairly briefer and easily applied. Moreover, $E(ϵ;{\nu }^{\prime },1)$ is a set of 1-finite view in the sense of [13], [14] (see [13], Definition 2.1] for the definition of 1-finite view). In the case $a=0$, Theorem 1 (1.6) is just the result of Theorem A.

Corollary 1

Let ν be a positive measure on$C_n(\mathrm{\Omega })$such that (1.4) holds. Then for a sufficiently large L and a sufficiently small ϵ we have

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

2 Some lemmas

Lemma 1

(see [15], [16])

$$G_{\mathrm{\Omega }}^a(P,Q)\approx V(t)W(r)\phi (\mathrm{\Theta })\phi (\mathrm{\Phi })$$

(2.1)

$$(\mathit{\text{resp. }}{G}_{\mathrm{\Omega }}^a(P,Q)\approx V(r)W(t)\phi (\mathrm{\Theta })\phi (\mathrm{\Phi })),$$

(2.2)

for any$P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$and any$Q=(t,\mathrm{\Phi })\in C_n(\mathrm{\Omega })$satisfying$0<\frac{t}{r}\le \frac{4}{5}$ (resp. $0<\frac{r}{t}\le \frac{4}{5}$);

Further, for any$P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$and any$Q=(t,\mathrm{\Phi })\in C_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))$, we have

$$G_{\mathrm{\Omega }}^0(P,Q)\lesssim \frac{\phi (\mathrm{\Theta })\phi (\mathrm{\Phi })}{t^{n-2}}+{\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q),$$

(2.3)

where

$${\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q)=min\{\frac{1}{{|P-Q|}^{n-2}},\frac{rt\phi (\mathrm{\Theta })\phi (\mathrm{\Phi })}{{|P-Q|}^n}\}.$$

Lemma 2

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,

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

(2.4)

and

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

(2.5)

Proof

Take a positive number l satisfying $P_1=(r_1,{\mathrm{\Theta }}_1)\in C_n(\mathrm{\Omega })$, $r_1\le \frac{4}{5}l$. Then from (2.2), we have

$$V(r_1)\phi ({\mathrm{\Theta }}_1){\int }_{S_n(\mathrm{\Omega };(l,+\mathrm{\infty }))}W(t)\phi (\mathrm{\Phi })d\mu (Q)\lesssim {\int }_{S_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d\mu (Q)<+\mathrm{\infty },$$

which gives (2.4). For any positive number ϵ, from (2.4), we can take a number $R_{ϵ}$ such that

$${\int }_{S_n(\mathrm{\Omega };(R_{ϵ},+\mathrm{\infty }))}W(t)\phi (\mathrm{\Phi })d\mu (Q)<\frac{ϵ}{2}.$$

If we take a point $P_i=(r_i,{\mathrm{\Theta }}_i)\in C_n(\mathrm{\Omega })$, $r_i\ge \frac{5}{4}{R}_{ϵ}$, then we have from (2.1)

$$W(r_i)\phi ({\mathrm{\Theta }}_i){\int }_{S_n(\mathrm{\Omega };(0,R_{ϵ}])}V(t)\phi (\mathrm{\Phi })d\mu (Q)\lesssim {\int }_{S_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a(P,Q)d\mu (Q)<+\mathrm{\infty }.$$

If R ($R>R_{ϵ}$) is sufficiently large, then

$$\begin{array}{r}\frac{W(R)}{V(R)}{\int }_{S_n(\mathrm{\Omega };(0,R))}V(t)\phi (\mathrm{\Phi })d\mu (Q)\\ \lesssim \frac{W(R)}{V(R)}{\int }_{S_n(\mathrm{\Omega };(0,R_{ϵ}])}V(t)\phi (\mathrm{\Phi })d\mu (Q)+{\int }_{S_n(\mathrm{\Omega };(R_{ϵ},R))}W(t)\phi (\mathrm{\Phi })d\mu (Q)\\ \lesssim \frac{W(R)}{V(R)}{\int }_{S_n(\mathrm{\Omega };(0,R_{ϵ}])}V(t)\phi (\mathrm{\Phi })d\mu (Q)+{\int }_{S_n(\mathrm{\Omega };(R_{ϵ},+\mathrm{\infty }))}W(t)\phi (\mathrm{\Phi })d\mu (Q)\\ \lesssim ϵ,\end{array}$$

which gives (2.5). □

Lemma 3

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

$$\sum _{j=1}^{\mathrm{\infty }}{\left(\frac{r_j}{R_j}\right)}^{2-\alpha }\frac{V(R_j)W(R_j)}{V(r_j)W(r_j)}<\mathrm{\infty }.$$

Proof

Set

$$E_j(ϵ;\lambda ,\beta )=\{P=(r,\mathrm{\Theta })\in E(ϵ;\lambda ,\beta ):2^j\le r<2^{j+1}\}(j=2,3,4,\dots ).$$

If $P=(r,\mathrm{\Theta })\in E_j(ϵ;\lambda ,\beta )$, then there exists a positive number $\rho (P)$ such that

$${\left(\frac{\rho (P)}{r}\right)}^{2-\alpha }\frac{V(r)W(R)}{V(\rho (P))W(\rho (P))}\approx {\left(\frac{\rho (P)}{r}\right)}^{n-\alpha }\le \frac{\lambda (B(P,\rho (P)))}{ϵ}.$$

Since $E_j(ϵ;\lambda ,\beta )$ can be covered by the union of a family of balls $\{B(P_{j,i},{\rho }_{j,i}):P_{j,i}\in E_k(ϵ;\lambda ,\beta )\}$ (${\rho }_{j,i}=\rho (P_{j,i})$). By the Vitali lemma (see [17]), there exists ${\mathrm{\Lambda }}_j\subset E_j(ϵ;\lambda ,\beta )$, which is at most countable, such that $\{B(P_{j,i},{\rho }_{j,i}):P_{j,i}\in {\mathrm{\Lambda }}_j\}$ are disjoint and $E_j(ϵ;\lambda ,\beta )\subset {\bigcup }_{P_{j,i}\in {\mathrm{\Lambda }}_j}B(P_{j,i},5{\rho }_{j,i})$.

So

$$\bigcup _{j=2}^{\mathrm{\infty }}{E}_j(ϵ;\lambda ,\beta )\subset \bigcup _{j=2}^{\mathrm{\infty }}\bigcup _{P_{j,i}\in {\mathrm{\Lambda }}_j}B(P_{j,i},5{\rho }_{j,i}).$$

On the other hand, note that

$$\bigcup _{P_{j,i}\in {\mathrm{\Lambda }}_j}B(P_{j,i},{\rho }_{j,i})\subset \{P=(r,\mathrm{\Theta }):2^{j-1}\le r<2^{j+2}\},$$

so that

$$\begin{array}{rcl}\sum _{P_{j,i}\in {\mathrm{\Lambda }}_j}{\left(\frac{5{\rho }_{j,i}}{|P_{j,i}|}\right)}^{2-\alpha }\frac{V(|P_{j,i}|)W(|P_{j,i}|)}{V({\rho }_{j,i})W({\rho }_{j,i})}& \approx & \sum _{P_{j,i}\in {\mathrm{\Lambda }}_j}{\left(\frac{5{\rho }_{j,i}}{|P_{j,i}|}\right)}^{n-\alpha }\le 5^{n-\alpha }\sum _{P_{j,i}\in {\mathrm{\Lambda }}_j}\frac{\lambda (B(P_{j,i},{\rho }_{j,i}))}{ϵ}\\ \le & \frac{5^{n-\alpha }}{ϵ}\lambda \left(C_n(\mathrm{\Omega };[2^{j-1},2^{j+2})\right)).\end{array}$$

Hence we obtain

$$\begin{array}{rcl}\sum _{j=1}^{\mathrm{\infty }}\sum _{P_{j,i}\in {\mathrm{\Lambda }}_j}{\left(\frac{{\rho }_{j,i}}{|P_{j,i}|}\right)}^{2-\alpha }\frac{V(|P_{j,i}|)W(|P_{j,i}|)}{V({\rho }_{j,i})W({\rho }_{j,i})}& \approx & \sum _{j=1}^{\mathrm{\infty }}\sum _{P_{j,i}\in {\mathrm{\Lambda }}_j}{\left(\frac{{\rho }_{j,i}}{|P_{j,i}|}\right)}^{n-\alpha }\\ \le & \sum _{j=1}^{\mathrm{\infty }}\frac{\lambda (C_n(\mathrm{\Omega };[2^{j-1},2^{j+2})))}{ϵ}\\ \le & \frac{3\lambda ({\mathbf{R}}^n)}{ϵ}.\end{array}$$

Since $E(ϵ;\lambda ,\beta )\cap \{P=(r,\mathrm{\Theta })\in {\mathbf{R}}^n;r\ge 4\}={\bigcup }_{j=2}^{\mathrm{\infty }}{E}_j(ϵ;\lambda ,\beta )$. Then $E(ϵ;\lambda ,\beta )$ is finally covered by a sequence of balls $\{B(P_{j,i},{\rho }_{j,i}),B(P_1,6)\}$ ($j=2,3,\dots$ ; $i=1,2,\dots$) satisfying

$$\sum _{j,i}{\left(\frac{{\rho }_{j,i}}{|P_{j,i}|}\right)}^{2-\alpha }\frac{V(|P_{j,i}|)W(|P_{j,i}|)}{V({\rho }_{j,i})W({\rho }_{j,i})}\approx \sum _{j,i}{\left(\frac{{\rho }_{j,i}}{|P_{j,i}|}\right)}^{n-\alpha }\le \frac{3\lambda ({\mathbf{R}}^n)}{ϵ}+6^{n-\alpha }<+\mathrm{\infty },$$

where $B(P_1,6)$ ($P_1=(1,0,\dots ,0)\in {\mathbf{R}}^n$) is the ball which covers $\{P=(r,\mathrm{\Theta })\in {\mathbf{R}}^n;r<4\}$. □

3 Proof of Theorem 1

For any point $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega };(R,+\mathrm{\infty }))-E(ϵ;{\nu }^{\prime },\alpha )$, where R ($\le \frac{4}{5}r$) is a sufficiently large number and ϵ is a sufficiently small positive number.

Write

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

where

$$\begin{array}{c}{G}_{\mathrm{\Omega }}^a\nu (1)(P)={\int }_{C_n(\mathrm{\Omega };(0,\frac{4}{5}r])}{G}_{\mathrm{\Omega }}^a(P,Q)d\nu (Q),\hfill \\ G_{\mathrm{\Omega }}^a\nu (2)(P)={\int }_{C_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))}{G}_{\mathrm{\Omega }}^a(P,Q)d\nu (Q),\hfill \end{array}$$

and

$$G_{\mathrm{\Omega }}^a\nu (3)(P)={\int }_{C_n(\mathrm{\Omega };[\frac{5}{4}r,\mathrm{\infty }))}{G}_{\mathrm{\Omega }}^a(P,Q)d\nu (Q).$$

From (2.1) and (2.2) we obtain the following growth estimates:

$$G_{\mathrm{\Omega }}^a\nu (1)(P)\lesssim ϵV(r)\phi (\mathrm{\Theta }),$$

(3.1)

$$G_{\mathrm{\Omega }}^a\nu (3)(P)\lesssim ϵV(r)\phi (\mathrm{\Theta }).$$

(3.2)

By (2.3) and (3.1), we have

$$G_{\mathrm{\Omega }}^a\nu (2)(P)\le G_{\mathrm{\Omega }}^a\nu (21)(P)+G_{\mathrm{\Omega }}^a\nu (22)(P),$$

where

$$G_{\mathrm{\Omega }}^a\nu (21)(P)=\phi (\mathrm{\Theta }){\int }_{C_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))}V(t)d{\nu }^{\prime }(Q)$$

and

$$G_{\mathrm{\Omega }}^a\nu (22)(P)={\int }_{C_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))}{\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q)d\nu (Q).$$

Then by Lemma 2, we immediately get

$$G_{\mathrm{\Omega }}^a\nu (21)(P)\lesssim ϵV(r)\phi (\mathrm{\Theta }).$$

(3.3)

To estimate $G_{\mathrm{\Omega }}^a\nu (22)(P)$, take a sufficiently small positive number c independent of P such that

$$\mathrm{\Lambda }(P)=\{(t,\mathrm{\Phi })\in C_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r));|(1,\mathrm{\Phi })-(1,\mathrm{\Theta })|<c\}\subset B(P,\frac{r}{2})$$

(3.4)

and divide $C_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))$ into two sets $\mathrm{\Lambda }(P)$ and $\mathrm{\Lambda }(P)$, where

$$\mathrm{\Lambda }(P)=C_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))-\mathrm{\Lambda }(P).$$

Write

$$G_{\mathrm{\Omega }}^a\nu (22)(P)=G_{\mathrm{\Omega }}^a\nu (221)(P)+G_{\mathrm{\Omega }}^a\nu (222)(P),$$

where

$$G_{\mathrm{\Omega }}^a\nu (221)(P)={\int }_{\mathrm{\Lambda }(P)}{\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q)d\nu (Q)$$

and

$$G_{\mathrm{\Omega }}^a\nu (222)(P)={\int }_{\mathrm{\Lambda }(P)}{\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q)d\nu (Q).$$

There exists a positive $c^{\prime }$ such that $|P-Q|\ge c^{\prime }r$ for any $Q\in \mathrm{\Lambda }(P)$, and hence

$$\begin{array}{rl}{G}_{\mathrm{\Omega }}^a\nu (222)(P)& \lesssim {\int }_{C_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))}\frac{rt\phi (\mathrm{\Theta })\phi (\mathrm{\Phi })}{{|P-Q|}^n}d\nu (Q)\\ \lesssim V(r)\phi (\mathrm{\Theta }){\int }_{C_n(\mathrm{\Omega };(\frac{4}{5}r,\mathrm{\infty }))}d{\nu }^{\prime }(Q)\\ \lesssim ϵV(r)\phi (\mathrm{\Theta })\end{array}$$

(3.5)

from Lemma 2.

Now we estimate $G_{\mathrm{\Omega }}^a\nu (221)(P)$. Set

$$I_i(P)=\{Q\in \mathrm{\Lambda }(P);2^{i-1}\delta (P)\le |P-Q|<2^i\delta (P)\},$$

where $i=0,\pm 1,\pm 2,\dots$ .

Since $P=(r,\mathrm{\Theta })\notin E(ϵ;{\nu }^{\prime },\alpha )$ and hence ${\nu }^{\prime }(\{P\})=0$ from Remark 1, we can divide $G_{\mathrm{\Omega }}^a\nu (221)(P)$ into

$$G_{\mathrm{\Omega }}^a\nu (221)(P)=G_{\mathrm{\Omega }}^A\nu (2211)(P)+G_{\mathrm{\Omega }}^a\nu (2212)(P),$$

where

$$G_{\mathrm{\Omega }}^A\nu (2211)(P)=\sum _{i=-\mathrm{\infty }}^{-1}{\int }_{I_i(P)}{\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q)d\nu (Q)$$

and

$$G_{\mathrm{\Omega }}^a\nu (2212)(P)=\sum _{i=0}^{\mathrm{\infty }}{\int }_{I_i(P)}{\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q)d\nu (Q).$$

Since $\delta (Q)+|P-Q|\ge \delta (P)$, we have

$$t{f}_{\mathrm{\Omega }}(\mathrm{\Phi })\gtrsim \delta (Q)\gtrsim 2^{-1}\delta (P)$$

for any $Q=(t,\mathrm{\Phi })\in I_i(p)$ ($i=-1,-2,\dots$). Then by (1.1)

$$\begin{array}{rcl}{\int }_{I_i(P)}{\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q)d\nu (Q)& \lesssim & {\int }_{I_i(P)}\frac{1}{{|P-Q|}^{n-2}W(t)\phi (\mathrm{\Phi })}d{\nu }^{\prime }(Q)\\ \lesssim & \frac{r^{2-\alpha }}{W(r)}{\phi }^{1-\alpha }(\mathrm{\Theta })\frac{{\nu }^{\prime }(B(P,2^i\delta (P)))}{{\{2^i\delta (P)\}}^{n-\alpha }}\\ \lesssim & \frac{r^{2-\alpha }}{W(r)}{\phi }^{1-\alpha }(\mathrm{\Theta })M(P;{\nu }^{\prime },\alpha )(i=-1,-2,\dots ).\end{array}$$

Since $P=(r,\mathrm{\Theta })\notin E(ϵ;{\nu }^{\prime },\alpha )$, we obtain

$$G_{\mathrm{\Omega }}^a\nu (2211)(P)\lesssim ϵV(r){\phi }^{1-\alpha }(\mathrm{\Theta }).$$

(3.6)

By (3.4), we can take a positive integer $i(P)$ satisfying

$$2^{i(P)-1}\delta (P)\le \frac{r}{2}<2^{i(P)}\delta (P)$$

and $I_i(P)=\mathrm{\varnothing }$ ($i=i(P)+1,i(P)+2,\dots$).

Since $r{f}_{\mathrm{\Omega }}(\mathrm{\Theta })\lesssim \delta (P)$ ($P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$), we have

$$\begin{array}{rcl}{\int }_{I_i(P)}{\mathrm{\Pi }}_{\mathrm{\Omega }}(P,Q)d{\nu }^{\prime }(Q)& \lesssim & r\phi (\mathrm{\Theta }){\int }_{I_i(P)}\frac{t}{{|P-Q|}^{n}W(t)}d{\nu }^{\prime }(Q)\\ \lesssim & \frac{r^{2-\alpha }}{W(r)}{\phi }^{1-\alpha }(\mathrm{\Theta })\frac{{\nu }^{\prime }(I_i(P))}{{\{2^i\delta (P)\}}^{n-\alpha }}(i=0,1,2,\dots ,i(P)).\end{array}$$

Since $P=(r,\mathrm{\Theta })\notin E(ϵ;{\nu }^{\prime },\alpha )$, we have

$$\begin{array}{rcl}\frac{{\nu }^{\prime }(I_i(P))}{{\{2^i\delta (P)\}}^{n-\alpha }}& \lesssim & {\nu }^{\prime }\left(B(P,2^i\delta (P))\right)V(2^i\delta (P))W(2^i\delta (P)){\{2^i\delta (P)\}}^{\alpha -2}\\ \lesssim & M(P;{\nu }^{\prime },\alpha )\\ \le & ϵV(r)W(r)r^{\alpha -2}(i=0,1,2,\dots ,i(P)-1)\end{array}$$

and

$$\frac{{\nu }^{\prime }(I_i(P))}{{\{2^i\delta (P)\}}^{n-\alpha }}\lesssim {\nu }^{\prime }(\mathrm{\Lambda }(P))V\left(\frac{r}{2}\right)W\left(\frac{r}{2}\right){\left(\frac{r}{2}\right)}^{\alpha -2}\le ϵV(r)W(r)r^{\alpha -2}.$$

Hence we obtain

$$G_{\mathrm{\Omega }}^a\nu (2212)(P)\lesssim ϵV(r){\phi }^{1-\alpha }(\mathrm{\Theta }).$$

(3.7)

Combining (3.1)-(3.3) and (3.5)-(3.7), we finally obtain the result that if R is sufficiently large and ϵ is a sufficiently small, then $G_{\mathrm{\Omega }}^a\nu (P)=o(V(r){\phi }^{1-\alpha }(\mathrm{\Theta }))$ as $r\to \mathrm{\infty }$, where $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega };(R,+\mathrm{\infty }))-E(ϵ;{\nu }^{\prime },\alpha )$. Finally, there exists an additional finite ball $B_0$ covering $C_n(\mathrm{\Omega };(0,R])$, which together with Lemma 3, gives the conclusion of Theorem 1.