RETRACTED ARTICLE: New applications of Schrödingerean Green potential to boundary behaviors of superharmonic functions

Abstract

By using the Schrödingerean continuation theorem due to Li (Bound. Value Probl. 2015:242, 2015), we obtain some new results for boundary value problems of Schrödingerean Green potentials. As new applications, the boundary behaviors of superharmonic functions at infinity are also obtained.

Introduction

Cartesian coordinates of a point G of $\mathbf{R}^{n}$, $n\geq2$, are denoted by $(X,x_{n})$, where $\mathbf{R}^{n}$ is the n-dimensional Euclidean space and $X=(x_{1},x_{2},\ldots,x_{n-1})$. We introduce spherical coordinates for $G=(r,\Xi)$ $(\Xi=(\theta _{1},\theta_{2},\ldots,\theta_{n-1}))$ by $\vert x\vert =r$,

$$\textstyle\begin{cases} x_{n}=r\cos\theta_{1}, \qquad x_{1}=r(\prod_{j=1}^{n-1}\sin\theta_{j})& n= 2, \\ x_{n-m+1}=r(\prod_{j=1}^{m-1}\sin\theta_{j})\cos\theta_{m} & n\geq3, \end{cases} $$

where $0\leq r<+\infty$, $-\frac{1}{2}\pi\leq\theta_{n-1}< \frac{3}{2}\pi$ and $0\leq\theta_{j}\leq\pi$ for $1\leq j\leq n-2$ ($n\geq3$).

We denote the unit sphere and the upper half unit sphere by $\mathbf{S}^{n-1}$ and $\mathbf{S}_{+}^{n-1}$, respectively. Let $\Sigma \subset\mathbf{S}^{n-1}$. The point $(1,\Xi)$ and the set $\{\Xi; (1, \Xi)\in\Sigma\}$ are identified with Ξ and Σ, respectively. Let $\Xi\times\Sigma$ denote the set $\{(r,\Xi) \in\mathbf{R}^{n}; r\in\Xi,(1,\Xi)\in\Sigma\}$, where $\Xi\subset \mathbf{R}_{+}$. The set $\mathbf{R}_{+}\times\Sigma$ is denoted by $\beth_{n}(\Sigma)$, which is called a cone. Especially, the set $\mathbf{R}_{+}\times\mathbf{S}_{+}^{n-1}$ is called the upper-half space, which is denoted by $\mathcal{T}_{n}$. Let $I\subset\mathbf{R}$. Two sets $I\times\Sigma$ and $I\times\partial {\Sigma}$ are denoted by $\beth_{n}(\Sigma;I)$ and $\daleth_{n}( \Sigma;I)$, respectively. We denote $\daleth_{n}(\Sigma; \mathbf{R} ^{+})$ by $\daleth_{n}(\Sigma)$, which is $\partial{\beth_{n}( \Sigma)}-\{O\}$.

Let $B(G,l)$ denote the open ball, where $G\in\mathbf{R}^{n}$ is the center and $l>0$ is the radius.

Definition 1

Let $E\subset\beth_{n}(\Sigma)$. If there exists a sequence of countable balls $\{B_{k}\}$ ($k=1,2,3,\ldots$) with centers in $\beth_{n}(\Sigma)$ satisfying

$$E\subset\bigcup_{k=0}^{\infty} B_{k}, $$

then we say that E has a covering $\{r_{k},R_{k}\}$, where $r_{k}$ is the radius of $B_{k}$ and $R_{k}$ is the distance from the origin to the center of $B_{k}$.

In spherical coordinates the Laplace operator is

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

where $\Lambda_{n}$ is the Beltrami operator. Now we consider the boundary value problem

$$\begin{aligned}& (\Lambda_{n}+\tau)h=0 \quad \text{on } \Sigma, \\& h=0 \quad \text{on } \partial{\Sigma}. \end{aligned}$$

If the least positive eigenvalue of it is denoted by $\tau_{\Sigma}$, then we can denote by $h_{\Sigma}(\Xi)$ the normalized positive eigenfunction corresponding to it.

We denote by $\iota_{\Sigma}$ (>0) and $-\kappa_{\Sigma}$ (<0) two solutions of the problem $t^{2}+(n-2)t-\tau_{\Sigma}=0$, Then $\iota_{\Sigma}+\kappa_{\Sigma}$ is denoted by $\varrho_{\Sigma}$ for the sake of simplicity.

Remark 1

In the case $\Sigma=\mathbf{S}_{+}^{n-1}$, it follows that

• (I) $\iota_{\Sigma}=1$ and $\kappa_{\Sigma}=n-1$.

• (II) $h_{\Sigma}(\Xi)=\sqrt{\frac{2n}{w_{n}}}\cos\theta_{1}$, where $w_{n}$ is the surface area of $\mathbf{S}^{n-1}$.

It is easy to see that the set $\partial{\beth_{n}(\Sigma)}\cup\{ \infty\}$ is the Martin boundary of $\beth_{n}(\Sigma)$. For any $G\in\beth_{n}(\Sigma)$ and any $H\in\partial{\beth_{n}(\Sigma)} \cup\{\infty\}$, if the Martin kernel is denoted by $\mathcal{MK}(G,H)$, where a reference point is chosen in advance, then we see that (see [2], p.292)

$$\mathcal{MK}(G,\infty)=r^{\iota_{\Sigma}}h_{\Sigma}(\Xi)\quad \text{and}\quad \mathcal{MK}(G,O)=cr^{-\kappa_{\Sigma}}h_{\Sigma}( \Xi), $$

where $G=(r,\Xi)\in\beth_{n}(\Sigma)$ and c is a positive number.

We shall say that two positive real valued functions f and g are comparable and write $f\approx g$ if there exist two positive constants $c_{1}\leq c_{2}$ such that $c_{1}g\leq f\leq c_{2}g$.

Remark 2

Let $\Xi\in\Sigma$. Then $h_{\Sigma}(\Xi)$ and $\operatorname{dist}(\Xi,\partial{\Sigma})$ are comparable (see [3]).

Remark 3

Let $\varrho(G)=\operatorname{dist}(G,\partial {\beth_{n}(\Sigma)})$. Then $h_{\Sigma}(\Xi)$ and $\varrho(G)$ are comparable for any $(1,\Xi)\in\Sigma$ (see [4]).

Remark 4

Let $0\leq\alpha\leq n$. Then $h_{\Sigma}(\Xi)\leq c_{3}(\Sigma,n)\{h_{\Sigma}(\Xi)\}^{1- \alpha}$, where $c_{3}(\Sigma,n)$ is a constant depending on Σ and n (e.g. see [5], pp.126-128).

Definition 2

For any $G\in\beth_{n}(\Sigma)$ and any $H\in\beth_{n}(\Sigma)$. If the Green function in $\beth_{n}(\Sigma)$ is defined by $\mathcal{GF}_{\Sigma}(G,H)$, then:

• (I) The Poisson kernel can be defined by

  $$\mathcal{POI}_{\Sigma}(G,H)=\frac{\partial}{\partial n_{H}} \mathcal{GF}_{\Sigma}(G,H), $$

  where $\frac{\partial}{\partial n_{H}}$ denotes the differentiation at H along the inward normal into $\beth_{n}(\Sigma)$.

• (II) The Green potential on $\beth_{n}(\Sigma)$ can be defined by

  $$\mathcal{GF}_{\Sigma} \nu(G)= \int_{\beth_{n}(\Sigma)}\mathcal{GF} _{\Sigma}(G,H)\,d\nu(H), $$

  where $G\in\beth_{n}(\Sigma)$ and ν is a positive measure in $\beth_{n}(\Sigma)$.

Definition 3

For any $G\in\beth_{n}(\Sigma)$ and any $H\in\daleth_{n}(\Sigma)$. Let μ be a positive measure on $\daleth_{n}(\Sigma)$ and g be a continuous function on $\daleth_{n}(\Sigma)$. Then (see [6]):

• (I) The Poisson integral with μ can be defined by

  $$\mathcal{POI}_{\Sigma} \mu(G)= \int_{\daleth_{n}(\Sigma)} \mathcal{POI}_{\Sigma}(G,H)\,d\mu(H). $$

• (II) The Poisson integral with g can be defined by

  $$\mathcal{POI}_{\Sigma} [g](G)= \int_{\daleth_{n}(\Sigma)} \mathcal{POI}_{\Sigma}(G,H)g(H)\,d\sigma_{H}, $$

where $d\sigma_{H}$ is the surface area element on $\daleth_{n}( \Sigma)$.

Definition 4

Let μ be defined in Definition 3. Then the positive measure $\mu'$ is defined by

$$d\mu'=\textstyle\begin{cases} \frac{\partial h_{\Sigma}(\Omega)}{\partial n_{\Omega}} t^{- \kappa_{\Sigma}-1}\,d\mu& \mbox{on } \daleth_{n}(\Sigma; (1,+\infty)) , \\ 0 & \mbox{on } \mathbf{R}^{n}-\daleth_{n}(\Sigma; (1,+\infty)). \end{cases} $$

Definition 5

Let ν be any positive measure in $\beth_{n}(\Sigma)$ satisfying

$$ \mathcal{GF}_{\Sigma} \nu(G)\not\equiv+\infty $$

(1.1)

for any $G\in\beth_{n}(\Sigma)$. Then the positive measure $\nu'$ is defined by

$$d\nu'=\textstyle\begin{cases} h_{\Sigma}(\Omega) t^{-\kappa_{\Sigma}}\,d\nu& \mbox{on } \beth_{n}(\Sigma; (1,+\infty)) , \\ 0& \mbox{on } \mathbf{R}^{n}-\beth_{n}(\Sigma; (1,+\infty)). \end{cases} $$

Definition 6

Let ν be any positive measure in $\mathcal{T}_{n}$ such that (1.1) holds for any $G\in\beth _{n}(\Sigma)$. Then the positive measure $\nu_{1}$ is defined by

$$d\nu_{1}=\textstyle\begin{cases} h_{\mathbf{S}_{+}^{n-1}}(\Omega) t^{1-n}\,d\nu& \mbox{on } \mathcal{T}_{n}(1,+\infty) , \\ 0& \mbox{on } \mathbf{R}^{n}-\mathcal{T}_{n}(1,+\infty). \end{cases} $$

Definition 7

Let μ and ν be defined in Definitions 3 and 4, respectively. Then the positive measure ξ is defined by

$$d\xi=\textstyle\begin{cases} t^{-1-\kappa_{\Sigma}}\,d\xi' & \mbox{on } \overline{\beth_{n}(\Sigma; (1,+\infty))} , \\ 0& \mbox{on } \mathbf{R}^{n}-\overline{\beth_{n}(\Sigma; (1,+\infty))}, \end{cases} $$

where

$$d\xi'=\textstyle\begin{cases} \frac{\partial h_{\Sigma}(\Omega)}{\partial n_{\Omega}}\,d\mu(H) & \mbox{on } \in\daleth_{n}(\Sigma; (1,+\infty)) , \\ h_{\Sigma}(\Omega)t\,d\nu(H)& \mbox{on } \in\beth_{n}(\Sigma; (1,+\infty)). \end{cases} $$

Remark 5

Let $\Sigma=\mathbf{S}_{+}^{n-1}$. Then

$$\mathcal{GF}_{\mathbf{S}_{+}^{n-1}}(G,H)= \textstyle\begin{cases} \log \vert G-H^{\ast} \vert -\log \vert G-H\vert & \mbox{if } n=2, \\ \vert G-H\vert ^{2-n}-\vert G-H^{\ast} \vert ^{2-n} & \mbox{if } n\geq3, \end{cases} $$

where $G=(X,x_{n})$, $H^{\ast}=(Y,-y_{n})$, that is, $H^{\ast}$ is the mirror image of $H=(Y,y_{n})$ with respect to $\partial{\mathcal{T} _{n}}$. Hence, for the two points $G=(X,x_{n})\in\mathcal{T}_{n}$ and $H=(Y,y_{n})\in\partial{\mathcal{T}_{n}}$, we have

$$\begin{aligned} \mathcal{POI}_{\mathbf{S}_{+}^{n-1}}(G,H) =&\frac{\partial}{\partial n _{y}}\mathcal{GF}_{\mathbf{S}_{+}^{n-1}}(G,H) \\ =& \textstyle\begin{cases} 2x_{n}\vert G-H\vert ^{-2} & \mbox{if } n=2, \\ 2(n-2)x_{n}\vert G-H\vert ^{-n} & \mbox{if } n\geq3. \end{cases}\displaystyle \end{aligned}$$

Remark 6

Let $\Sigma=\mathbf{S}_{+}^{n-1}$. Then we define

$$d\varrho=\textstyle\begin{cases} \frac{d\varrho'}{\vert y\vert ^{n}} & \mbox{on } \overline{\mathcal{T}_{n}} , \\ 0& \mbox{on } \mathbf{R}^{n}-\overline{\mathcal{T}_{n}}, \end{cases} $$

where

$$d\varrho'(y)=\textstyle\begin{cases} d\mu& \mbox{on } \partial{\mathcal{T}_{n}} , \\ y_{n}\,d\nu& \mbox{on } \mathcal{T}_{n}. \end{cases} $$

Definition 8

Let λ be any positive measure on $\mathbf{R}^{n}$ having finite total mass. Then the maximal function $M(G;\lambda,\beta)$ is defined by

$$\mathfrak{M}(G;\lambda,\beta)=\sup_{ 0< \rho< \frac{r}{2}}\rho^{- \beta} \lambda\bigl(B(G,\rho)\bigr) $$

for any $G=(r,\Xi)\in\mathbf{R}^{n}-\{O\}$, where $\beta\geq0$. The exceptional set can be defined by

$$\mathbb{EX}(\epsilon; \lambda, \beta)=\bigl\{ G=(r,\Xi)\in\mathbf{R}^{n}- \{O\}; \mathfrak{M}(G;\lambda,\beta)r^{\beta}>\epsilon\bigr\} , $$

where ϵ is a sufficiently small positive number.

Remark 7

Let $\beta>0$ and $\lambda(\{P\})>0$ for any $P\neq O$. Then:

• (I) $\mathfrak{M}(G;\lambda,\beta)=+\infty$.

• (II) $\{G\in\mathbf{R}^{n}-\{O\}; \lambda(\{P\})>0\}\subset \mathbb{EX}(\epsilon; \lambda, \beta)$.

The boundary behavior of classical Green potential in $\mathcal{T} _{n}$ was proved by Huang in [7], Corollary and Remark 5.

Theorem A

Let g be a measurable function on $\partial{\mathcal{T}_{n}}$ satisfying

$$ \int_{\partial{\mathcal{T}_{n}}}y_{n}\bigl(1+\vert y\vert \bigr)^{-n}\,dy< \infty. $$

(1.2)

Then

$$ \mathcal{GF}_{\Sigma} \nu(x)=o\bigl(\vert x\vert \bigr) $$

(1.3)

for any $x\in\mathcal{T}_{n}-\mathbb{EX}(\epsilon;\nu_{1},n-1)$, where $\mathbb{EX}(\epsilon;\nu_{1},n-1)$ is a subset of $\mathcal{T}_{n}$ and has a covering $\{r_{k},R_{k}\}$ satisfying

$$ \sum_{k=0}^{\infty}\biggl( \frac{r_{k}}{R_{k}}\biggr)^{n-1}< \infty. $$

(1.4)

Results

Our first aim in this paper is also to consider boundary value problems for Green potential in a cone, which generalize Theorem A to the conical case. For similar results for Green-Sch potentials, we refer the reader to the paper by Li (see [1]).

The estimation of the Green potential at infinity is the following.

Theorem 1

If ν is a positive measure on $\beth_{n}(\Sigma)$ such that (1.1) holds for any $G\in\beth_{n}(\Sigma)$. Then

$$\mathcal{GF}_{\Sigma} \nu(G)=o\bigl(r^{\iota_{\Sigma}}\bigl\{ h_{\Sigma}( \Xi)\bigr\} ^{1-\alpha}\bigr) $$

for any $G=(r,\Xi)\in\beth_{n}(\Sigma)-\mathbb{EX}(\epsilon; \nu',n-\alpha)$ as $r \rightarrow\infty$, where $\mathbb{EX}( \epsilon;\nu',n-\alpha)$ is a subset of $\beth_{n}(\Sigma)$ and has a covering $\{r_{k},R_{k}\}$ satisfying

$$ \sum_{k=0}^{\infty}\biggl( \frac{r_{k}}{R_{k}}\biggr)^{n-\alpha}< \infty. $$

(2.1)

Corollary 1

Under the conditions of Theorem  1, $\mathcal{GF} _{\Sigma} \nu(G)=o(r^{\iota_{\Sigma}})$ for any $G=(r,\Xi)\in \beth_{n}(\Sigma)-\mathbb{EX}(\epsilon;\nu',n-1)$ as $r \rightarrow \infty$, where $\mathbb{EX}(\epsilon;\nu',n-1)$ is a subset of $\beth_{n}(\Sigma)$ and has a covering $\{r_{k},R_{k}\}$ satisfying (1.4).

Corollary 2

Under the conditions of Theorem  1, $\mathcal{GF} _{\Sigma} \nu(G)=o(r^{\iota_{\Sigma}}h_{\Sigma}(\Xi))$ for any $G=(r,\Xi)\in\beth_{n}(\Sigma)-\mathbb{EX}(\epsilon;\nu',n)$ as $r \rightarrow\infty$, where $\mathbb{EX}(\epsilon;\nu',n)$ is a subset of $\beth_{n}(\Sigma)$ and has a covering $\{r_{k},R_{k}\}$ satisfying

$$ \sum_{k=0}^{\infty}\biggl( \frac{r_{k}}{R_{k}}\biggr)^{n-1}< \infty. $$

(2.2)

Theorem B

see [8], Chapter 6, Theorem 6.2.1

Let $0< w(G)$ be a superharmonic function in $\mathcal{T}_{n}$. Then there exist a positive measure μ on $\partial\mathcal{T}_{n}$ and a positive measure ν on $\mathcal{T}_{n}$ such that $w(G)$ can be uniquely decomposed as

$$ w(x)=cx_{n}+\mathcal{POI}_{\mathbf{S}_{+}^{n-1}} \mu(x)+ \mathcal{GF} _{\mathbf{S}_{+}^{n-1}} \nu(x), $$

(2.3)

where $G\in\mathcal{T}_{n}$ and c is a nonnegative constant.

Theorem C

Let $0< w(G)$ be a superharmonic function in $\beth_{n}(\Sigma)$. Then there exist a positive measure μ on $\daleth_{n}(\Sigma)$ and a positive measure ν on $\beth_{n}( \Sigma)$ such that $w(G)$ can be uniquely decomposed as

$$ w(G)=c_{5}(w)\mathcal{MK}(G,\infty)+c_{6}(w) \mathcal{MK}(G,O)+ \mathcal{POI}_{\Sigma} \mu(G)+\mathcal{GF}_{\Sigma} \nu(G), $$

(2.4)

where $G\in\beth_{n}(\Sigma)$, $c_{5}(w)$ and $c_{6}(w)$ are two constants dependent on w satisfying

$$c_{5}(w)=\inf_{G\in \beth_{n}(\Sigma)}\frac{w(G)}{\mathcal{MK}(G, \infty)}\quad \textit{and}\quad c_{6}(w)=\inf_{G\in \beth_{n}(\Sigma)}\frac{w(G)}{\mathcal{MK}(G,O)}. $$

As an application of Theorem 1 and Lemma 3 in Section 2, we prove the following result.

Theorem 2

Let $0\leq\alpha< n$, ϵ be defined as in Theorem  1 and $w(G)$ ($\not\equiv+\infty$) ($G=(r,\Xi)\in\beth _{n}(\Sigma)$) be defined by (2.4). Then

$$w(G)=c_{5}(w)\mathcal{MK}(G,\infty)+c_{6}(w) \mathcal{MK}(G,O)+o\bigl(r ^{\iota_{\Sigma}}\bigl\{ h_{\Sigma}(\Xi)\bigr\} ^{1-\alpha}\bigr) $$

for any $G\in\beth_{n}(\Sigma)- \mathbb{EX}(\epsilon;\xi,n-\alpha)$ as $r \rightarrow\infty$, where $\mathbb{EX}(\epsilon;\xi,n- \alpha)$ is a subset of $\beth_{n}(\Sigma)$ and has a covering $\{r_{k},R_{k}\}$ satisfying (2.1).

Corollary 3

Under the conditions of Theorem  2,

$$w(G)=c_{5}(w)\mathcal{MK}(G,\infty)+c_{6}(w) \mathcal{MK}(G,O)+o\bigl(r ^{\iota_{\Sigma}}\bigr) $$

for any $G\in\beth_{n}(\Sigma)- \mathbb{EX}(\epsilon;\xi,n-1)$ as $r \rightarrow\infty$, where $\mathbb{EX}(\epsilon;\xi,n-1)$ is a subset of $\beth_{n}(\Sigma)$ and has a covering $\{r_{k},R_{k}\}$ satisfying (2.2).

Corollary 4

Under the conditions of Theorem  2,

$$w(G)=c_{5}(w)\mathcal{MK}(G,\infty)+c_{6}(w) \mathcal{MK}(G,O)+o\bigl(r ^{\iota_{\Sigma}}h_{\Sigma}(\Xi)\bigr) $$

for any $G\in\beth_{n}(\Sigma)- \mathbb{EX}(\epsilon;\xi,n)$ as $r \rightarrow\infty$, where $\mathbb{EX}(\epsilon;\xi,n)$ is a subset of $\beth_{n}(\Sigma)$ and has a covering $\{r_{k},R_{k}\}$ satisfying (1.4).

In $\mathcal{T}_{n}$, we have

Corollary 5

Let $w(x)$ ($\not\equiv+\infty$) ($x=(X,x_{n}) \in\mathcal{T}_{n}$) be defined by (2.3). Then

$$w(x)=cx_{n}+o\bigl(\vert x\vert \bigr) $$

for any $x\in\mathcal{T}_{n}- \mathbb{EX}(\epsilon;\varrho,n-1)$ as $\vert x\vert \rightarrow\infty$, where $\mathbb{EX}(\epsilon;\varrho,n-1)$ has a covering satisfying (2.2).

Corollary 6

Under the conditions of Corollary  5,

$$w(x)=cx_{n}+o(x_{n}) $$

for any $x\in\mathcal{T}_{n}- \mathbb{EX}(\epsilon;\varrho,n)$ as $\vert x\vert \rightarrow\infty$, where $\mathbb{EX}(\epsilon;\varrho,n)$ has a covering satisfying (1.4).

Lemmas

In order to prove our main results we need following lemmas.

Lemma 1

see [5], Lemma 2 and [9]

Let any $G=(r,\Xi)\in\beth_{n}(\Sigma)$ and any $H=(t,\Omega)\in\daleth_{n}(\Sigma)$, we have the following estimates:

$$ \mathcal{GF}_{\Sigma}(G,H)\leq M r^{-\kappa_{\Sigma }}t^{\iota_{ \Sigma}}h_{\Sigma}( \Xi)h_{\Sigma}(\Omega) $$

(3.1)

for $0<\frac{t}{r}\leq\frac{4}{5}$,

$$ \mathcal{GF}_{\Sigma}(G,H)\leq M r^{\iota_{\Sigma}}t^{-\kappa_{ \Sigma}}h_{\Sigma}( \Xi)h_{\Sigma}(\Omega) $$

(3.2)

for $0<\frac{r}{t}\leq\frac{4}{5}$ and

$$ \mathcal{GF}_{\Sigma}(G,H)\leq Mh_{\Sigma}(\Xi)t^{2-n}h_{\Sigma}( \Omega)+t^{-\kappa_{\Sigma}}h_{\Sigma}( \Omega)\Pi_{\Sigma}(G,H), $$

(3.3)

for $\frac{4r}{5}< t\leq\frac{5r}{4}$, where

$$\Pi_{\Sigma}(G,H)=\min\bigl\{ t^{\kappa_{\Sigma}} \vert G-H\vert ^{2-n}{h_{\Sigma}( \Omega)}^{-1}, Mrt^{\kappa_{\Sigma}+1} \vert G-H\vert ^{-n}h_{\Sigma}(\Xi)\bigr\} . $$

Lemma 2

see [10], Lemma 2

If λ is positive measure on $\mathbf{R}^{n}$ having finite total mass, then exceptional set $\mathbb{EX}(\epsilon; \lambda, \beta)$ has a covering $\{r_{k},R_{k}\}$ ($k=1,2,\ldots$) satisfying

$$\sum_{k=1}^{\infty}\biggl(\frac{r_{k}}{R_{k}} \biggr)^{\beta}< \infty. $$

The following result is due to Jiang et al. (see [10], Theorem 1), who are concerned with the boundary behaviors of Poisson integrals and their applications. For similar results in a half space, we refer the reader to the paper by Jiang and Huang (see [7]).

Lemma 3

Let $\mathcal{POI}_{\Sigma}\mu(G) \not\equiv+\infty$ for any $G=(r,\Xi)\in\beth_{n}(\Sigma)$, where μ is a positive measure on $\daleth_{n}(\Sigma)$. Then

$$ \mathcal{POI}_{\Sigma} \mu(G)=o\bigl(r^{\iota_{\Sigma}}\bigl\{ h_{\Sigma}( \Xi)\bigr\} ^{1-\alpha}\bigr) $$

(3.4)

for any $G\in\beth_{n}(\Sigma)-\mathbb{EX}(\epsilon; \mu',n- \alpha)$ as $r \rightarrow\infty$, where $\mathbb{EX}(\epsilon; \mu',n-\alpha)$ is a subset of $\beth_{n}(\Sigma)$ and has a covering $\{r_{k},R_{k}\}$ of satisfying (2.1).

Proof of Theorem 1

Let $G=(r,\Xi)$ be any point in $\beth_{n}(\Sigma; (L,+ \infty))-\mathbb{EX}(\epsilon; \nu', n-\alpha)$, where L is a sufficiently large number satisfying $r \geq\frac{5L}{4}$.

Put

$$\mathcal{GF}_{\Sigma}\nu(G)=\mathcal{GF}_{\Sigma}^{1}(G)+ \mathcal{GF}_{\Sigma}^{2}(G)+\mathcal{GF}_{\Sigma}^{3}(G), $$

where

$$\begin{aligned}& \mathcal{GF}_{\Sigma}^{1}(G)= \int_{\beth_{n}(\Sigma;(0,\frac{4}{5}r])} \mathcal{GF}_{\Sigma}(G,H)\,d\nu(H), \\& \mathcal{GF}_{\Sigma}^{2}(G)= \int_{\beth_{n}(\Sigma;(\frac{4}{5}r,\frac{5}{4}r))} \mathcal{GF} _{\Sigma}(G,H)\,d\nu(H), \\& \mathcal{GF}_{\Sigma}^{3}(G)= \int_{\beth_{n}(\Sigma;[\frac{5}{4}r,\infty))} \mathcal{GF}_{\Sigma }(G,H)\,d\nu(H). \end{aligned}$$

We have the following estimates:

$$\begin{aligned}& \begin{aligned}[b] \mathcal{GF}_{\Sigma}^{1}(G) &\leq Mr^{\iota_{\Sigma}}h_{\Sigma}( \Xi) \biggl(\frac{4}{5}r \biggr)^{-\varrho_{\Sigma}} \int_{\beth_{n}(\Sigma;(0,\frac{4}{5}r])}t^{\iota_{\Sigma}}h_{ \Sigma}(\Omega)\,d\nu(H) \\ &\leq M \epsilon r^{\iota_{\Sigma}}h_{\Sigma}(\Xi), \end{aligned} \end{aligned}$$

(4.1)

$$\begin{aligned}& \begin{aligned}[b] \mathcal{GF}_{\Sigma}^{3}(G) &\leq Mr^{\iota_{\Sigma}}h_{\Sigma}( \Xi) \int_{\beth_{n}(\Sigma;[\frac{5}{4}r,\infty))}t^{-\kappa_{ \Sigma}}h_{\Sigma}(\Omega)\,d\nu(H) \\ &\leq M \epsilon r^{\iota_{\Sigma}}h_{\Sigma}(\Xi), \end{aligned} \end{aligned}$$

(4.2)

from (3.1), (3.2), and [11], Lemma 1.

By (3.3), we have

$$\mathcal{GF}_{\Sigma}^{2}(G)\leq\mathcal{GF}_{\Sigma}^{21}(G)+ \mathcal{GF}_{\Sigma}^{22}(G), $$

where

$$\begin{aligned}& \mathcal{GF}_{\Sigma}^{21}(G)=Mh_{\Sigma}(\Xi) \int_{\beth_{n}(\Sigma;(\frac{4}{5}r,\frac{5}{4}r))}t^{2-n+ \kappa_{\Sigma}}\,d\nu'(H), \\& \mathcal{GF}_{\Sigma}^{22}(G)= \int_{\beth_{n}(\Sigma;(\frac{4}{5}r,\frac{5}{4}r))}\Pi_{\Sigma}(G,H)\,d\nu'(H). \end{aligned}$$

Then by [11], Lemma 1 we immediately get

$$\begin{aligned} \mathcal{GF}_{\Sigma}^{21}(G) \leq& \biggl( \frac{5}{4}\biggr)^{\iota_{\Sigma}}Mr ^{\iota_{\Sigma}}h_{\Sigma}(\Xi) \int_{\beth_{n}(\Sigma;(\frac{4}{5}r,\infty))}\,d\nu'(H) \\ \leq& M \epsilon r^{\iota_{\Sigma}}h_{\Sigma}(\Xi). \end{aligned}$$

(4.3)

In order to give the growth properties of $\mathcal{GF}_{\Sigma}^{22}(G)$. Take a sufficiently small positive number $k_{2}$ independent of G such that

$$ \Gamma(G)=\biggl\{ (t,\Omega)\in\beth_{n}\biggl(\Sigma; \biggl(\frac{4}{5}r, \frac{5}{4}r\biggr)\biggr); \bigl\vert (1, \Omega)-(1,\Xi)\bigr\vert < k_{2}\biggr\} \subset B\biggl(G, \frac{r}{2}\biggr). $$

(4.4)

The set $\beth_{n}(\Sigma;(\frac{4}{5}r,\frac{5}{4}r))$ can be split into two sets $\Gamma(G)$ and $\Gamma'(G)$, where $\Gamma'(G)=\beth _{n}(\Sigma;(\frac{4}{5}r,\frac{5}{4}r))-\Gamma(G)$. Write

$$\mathcal{GF}_{\Sigma}^{22}(G)= \mathcal{GF}_{\Sigma}^{221}(G)+ \mathcal{GF}_{\Sigma}^{222}(G), $$

where

$$\mathcal{GF}_{\Sigma}^{221}(G)= \int_{\Gamma(G)}\Pi_{\Sigma}(G,H)\,d\nu'(H),\qquad \mathcal{GF}_{\Sigma}^{222}(G)= \int_{\Gamma'(G)} \Pi_{\Sigma}(G,H)\,d\nu'(H). $$

For any $H\in\Gamma'(G)$ we have $\vert G-H\vert \geq k_{2}'r$, where $k_{2}'$ is a positive number. So [11], Lemma 1 gives

$$\begin{aligned} \mathcal{GF}_{\Sigma}^{222}(G) \leq& M \int_{\beth_{n}(\Sigma;(\frac{4}{5}r,\frac{5}{4}r))}rt^{\kappa_{ \Sigma}+1}h_{\Sigma}(\Xi)\vert G-H \vert ^{-n}\,d\nu'(H) \\ \leq& Mr^{\iota_{\Sigma}}h_{\Sigma}(\Xi) \int_{\beth_{n}(\Sigma;(\frac{4}{5}r,\infty))}\,d\nu'(H) \\ \leq& M \epsilon r^{\iota_{\Sigma}}h_{\Sigma}(\Xi). \end{aligned}$$

(4.5)

To estimate $\mathcal{GF}_{\Sigma}^{221}(G)$. Set

$$I_{l}(G)=\bigl\{ H\in\Gamma(G); 2^{l}\varrho(G)>\vert G-H \vert \geq2^{l-1}\varrho (G)\bigr\} , $$

where $l=0,\pm1,\pm2,\ldots$ and $\varrho(G)=\inf_{H\in\partial{ \beth_{n}(\Sigma)}}\vert G-H\vert$.

From Remark 7 it is easy to see that $\nu'(\{P\})=0$ for any $G=(r,\Xi)\notin\mathbb{EX}(\epsilon; \nu', n-\alpha)$. The function $\mathcal{GF}_{\Sigma}^{221}(G)$ can be divided into $\mathcal{GF}_{\Sigma}^{221}(G)=\mathcal{GF}_{\Sigma}^{2211}(G)+ \mathcal{GF}_{\Sigma}^{2212}(G)$, where

$$\begin{aligned}& \mathcal{GF}_{\Sigma}^{2211}(G)=\sum_{l=-\infty}^{-1} \int_{I_{l}(G)} \Pi_{\Sigma}(G,H)\,d\nu'(H), \\& \mathcal{GF}_{\Sigma}^{2212}(G)=\sum_{l=0}^{\infty} \int_{I_{l}(G)} \Pi_{\Sigma}(G,H)\,d\nu'(H). \end{aligned}$$

For any $H=(t,\Omega)\in I_{l}(p)$, we have $2^{-1}\varrho(G)\leq \varrho(H)\leq Mth_{\Sigma}(\Omega)$, because $\varrho(H)+\vert G-H\vert \geq\varrho(G)$. Then by Remark 3

$$\begin{aligned} \int_{I_{l}(G)}\Pi_{\Sigma}(G,H)\,d\nu'(H) \leq& \int_{I_{l}(G)}\frac{t ^{\kappa_{\Sigma}}}{ \vert G-H\vert ^{n-2}h_{\Sigma}(\Omega)}\,d\nu'(H) \\ \leq& M2^{(2-\alpha)i}r^{2-\alpha+\kappa_{\Sigma}}\bigl\{ h_{\Sigma}( \Xi)\bigr\} ^{1-\alpha}\frac{\nu'(B(G,2^{l}\varrho(G)))}{\{2^{l}\varrho (G)\}^{n-\alpha}} \\ \leq& M r^{\iota_{\Sigma}}\bigl\{ h_{\Sigma}(\Xi)\bigr\} ^{1-\alpha}r^{n- \alpha} \mathfrak{M}\bigl(G; \nu', n-\alpha\bigr) \end{aligned}$$

for $l=-1,-2,\ldots$ .

Moreover, we have

$$ \mathcal{GF}_{\Sigma}^{2211}(G)\leq M \epsilon r^{\iota_{\Sigma}}\bigl\{ h _{\Sigma}(\Xi)\bigr\} ^{1-\alpha} $$

(4.6)

for any $G=(r,\Xi)\notin\mathbb{EX}(\epsilon; \nu', n-\alpha)$.

Equation (4.4) shows that there exists an integer $l(G)>0$ such that $2^{l(G)}\varrho(G)\leq r<2^{l(G)+1}\varrho(G)$ and $I_{l}(G)= \varnothing$ for $l=l(G)+1,l(G)+2,\ldots$ . And Remark 3 shows that

$$\begin{aligned} \int_{I_{l}(G)}\Pi_{\Sigma}(G,H)\,d\nu'(H) \leq& Mrh_{\Sigma}( \Xi) \int_{I_{l}(G)}t^{\kappa_{\Sigma}+1}\vert G-H\vert ^{-n}\,d\nu'(H) \\ \leq& M2^{-i\alpha}r^{\iota_{\Sigma}}\bigl\{ h_{\Sigma}(\Xi)\bigr\} ^{1- \alpha}r^{n-\alpha}\nu'\bigl(I_{l}(G)\bigr) \bigl\{ 2^{l}\varrho(G)\bigr\} ^{\alpha-n} \end{aligned}$$

for $l=0,1,2,\ldots,l(G)$.

We have for any $G=(r,\Xi)\notin\mathbb{EX}(\epsilon; \nu', n- \alpha)$

$$\nu'\bigl(I_{l}(G)\bigr)\bigl\{ 2^{l} \varrho(G)\bigr\} ^{\alpha-n}\leq\nu'\bigl(B\bigl(G,2^{l} \varrho(G)\bigr)\bigr)\bigl\{ 2^{l}\varrho(G)\bigr\} ^{\alpha-n}\leq \mathfrak{M}\bigl(G; \nu', n-\alpha\bigr)< \epsilon r^{\alpha-n} $$

for $l=0,1,2,\ldots,l(G)-1$ and

$$\nu'\bigl(I_{l}(G)\bigr)\bigl\{ 2^{l} \varrho(G)\bigr\} ^{\alpha-n}\leq\nu'\bigl(\Gamma(G)\bigr) \biggl( \frac{r}{2}\biggr)^{\alpha-n}< \epsilon r^{\alpha-n}. $$

So

$$ \mathcal{GF}_{\Sigma}^{2212}(G)\leq M \epsilon r^{\iota_{\Sigma}}\bigl\{ h _{\Sigma}(\Xi)\bigr\} ^{1-\alpha}. $$

(4.7)

From (4.1), (4.2), (4.3), (4.5), (4.6), and (4.7) we obtain $\mathcal{GF}_{\Sigma}\nu(G)=o(r^{ \iota_{\Sigma}}\{h_{\Sigma}(\Xi)\}^{1-\alpha})$ for any $G=(r,\Xi)\in\beth_{n}(\Sigma; (L,+\infty))-\mathbb{EX}(\epsilon; \nu', n-\alpha)$ as $r\rightarrow\infty$, where L is sufficiently large number. Finally, Lemma 2 gives the conclusion of Theorem 1.