RETRACTED ARTICLE: Growth property at infinity of harmonic functions

Abstract

This paper gives the growth property of certain harmonic functions at infinity in an n-dimensional cone, which generalize the results obtained by Huang and Qiao (Abstr. Appl. Anal. 2012:203096, 2012), Xu et al. (Bound. Value Probl. 2013:262, 2013), Yang and Ren (Proc. Indian Acad. Sci. Math. Sci. 124(2): 175-178, 2014) and Zhao and Yamada (J. Inequal. Appl. 2014:497, 2014) to the conical case.

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 $\mathbf{R}^{n}$ ($n\geq2$) 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 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 and the closure of a set S in $\mathbf{R}^{n}$ are denoted by ∂S and $\overline{\mathbf{S}}$, 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 shall say that a set $E\subset C_{n}(\Omega)$ has a covering $\{r_{k}, R_{k}\}$ if there exists a sequence of balls $\{B_{k}\}$ with centers in $C_{n}(\Omega)$ such that $E\subset\bigcup_{k=1}^{\infty} B_{k}$, where $r_{k}$ is the radius of $B_{k}$ and $R_{k}$ is the distance from the origin to the center of $B_{k}$. We shall also write $h_{1}\approx h_{2}$ for two positive functions $h_{1}$ and $h_{2}$ if and only if there exists a positive constant a such that $a^{-1}h_{1}\leq h_{2}\leq ah_{1}$.

The unit sphere and the upper half unit sphere 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 $\Xi\times\Omega$. 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 $C_{n}(\Omega)$, we denote the set $\mathbf{R}_{+}\times\Omega$ in $\mathbf{R}^{n}$ with the domain Ω on $\mathbf{S}^{n-1}$ ($n\geq2$). We call it a cone. Then $T_{n}$ is a special cone obtained by putting $\Omega=\mathbf{S}_{+}^{n-1}$. We denote the sets $I\times\Omega$ and $I\times\partial{\Omega}$ with an interval on R by $C_{n}(\Omega;I)$ and $S_{n}(\Omega;I)$. By $S_{n}(\Omega)$ we denote $S_{n}(\Omega; (0,+\infty))$, which is $\partial{C_{n}(\Omega)}-\{O\}$.

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

$$x_{1}=r\Biggl(\prod_{j=1}^{n-1} \sin\theta_{j}\Biggr) \quad(n\geq2),\qquad x_{n}=r\cos \theta_{1}, $$

and if $n\geq3$, then

$$x_{n-k+1}=r\Biggl(\prod_{j=1}^{k-1} \sin\theta_{j}\Biggr)\cos\theta_{k}\quad (2\leq k\leq n-1), $$

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

Let Ω be a domain on $\mathbf{S}^{n-1}$ ($n\geq2$) with smooth boundary. Consider the Dirichlet problem

$$\begin{aligned}& (\Lambda_{n}+\tau)f=0 \quad\mbox{on }\Omega,\\& f=0 \quad\mbox{on }\partial{\Omega}, \end{aligned}$$

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 $\tau_{\Omega}$ and the normalized positive eigenfunction corresponding to $\tau_{\Omega}$ by $f_{\Omega}(\Theta)$, $\int_{\Omega}\{f_{\Omega}(\Theta)\}^{2}\,d\sigma_{\Theta}=1$, where $d\sigma_{\Theta}$ is the surface area on $S^{n-1}$. We denote the solutions of the equation $t^{2}+(n-2)t-\tau_{\Omega}=0$ by $\alpha_{\Omega}$, $-\beta_{\Omega}$ ($\alpha_{\Omega}$, $\beta_{\Omega}>0$) and write $\delta_{\Omega}$ for $\alpha_{\Omega}+\beta_{\Omega}$. If $\Omega=\mathbf{S}_{+}^{n-1}$, then $\alpha_{\Omega}=1$, $\beta_{\Omega}=n-1$ and $f_{\Omega}(\Theta)=(2n s_{n}^{-1})^{1/2}\cos\theta_{1}$, where $s_{n}$ is the surface area $2\pi^{n/2}\{\Gamma(n/2)\}^{-1}$ of $\mathbf{S}^{n-1}$.

To simplify our consideration in the following, we shall assume that 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 (e.g. see [5], pp.88-89, for the definition of $C^{2,\alpha}$-domain). Then there exist two positive constants $c_{1}$ and $c_{2}$ such that

$$ c_{1}\operatorname{dist}(\Theta,\partial{\Omega})\leq f_{\Omega}(\Theta)\leq c_{2}\operatorname{dist}(\Theta ,\partial{\Omega}) \quad(\Theta\in\Omega). $$

(1.1)

(By modifying Miranda’s method [6], pp.7-8, we can prove this equality.)

Let $\delta(P)=\operatorname{dist}(P,\partial{C_{n}(\Omega)})$, we have

$$ f_{\Omega}(\Theta)\approx\delta(P), $$

(1.2)

for any $P=(1,\Theta)\in\Omega$ (see [7]).

We denote the Green function of $C_{n}(\Omega)$ by $G_{C_{n}(\Omega)}(P,Q)$ ($P\in C_{n}(\Omega)$, $Q\in C_{n}(\Omega)$). The Poisson integral $PI_{C_{n}(\Omega)}[g](P)$ with respect to $C_{n}(\Omega)$ is defined by

$$PI_{C_{n}(\Omega)} [g](P)=\frac{1}{c_{n}} \int_{S_{n}(\Omega)}\frac{\partial}{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)g(Q)\,d \sigma_{Q}, $$

where

$$c_{n}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 2\pi, & n=2, \\ (n-2)s_{n},& n\geq3, \end{array}\displaystyle \right . $$

g is a measurable function on $S_{n}(\Omega)$, $d\sigma_{Q}$ is the surface area element on $S_{n}(\Omega)$ and $\frac{\partial}{\partial n_{Q}}$ denotes the differentiation at Q along the inward normal into $C_{n}(\Omega)$.

Remark 1

(see [2])

Let $\Omega=S_{+}^{n-1}$. Then

$$G_{T_{n}}(P,Q)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \log|P-Q^{\ast}|-\log|P-Q|, & n=2, \\ |P-Q|^{2-n}-|P-Q^{\ast}|^{2-n},& n\geq3, \end{array}\displaystyle \right . $$

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

$$PI_{T_{n}}(P,Q)=\frac{\partial}{\partial n_{Q}}G_{T_{n}}(P,Q)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 2|P-Q|^{-2}x_{n}, & n=2, \\ 2(n-2)|P-Q|^{-n}x_{n},& n\geq3. \end{array}\displaystyle \right . $$

In this paper, we consider the functions g satisfying

$$ \int_{S_{n}(\Omega)}\frac{|g(Q)|^{p}}{1+t^{\gamma}}\,d\sigma_{Q}< \infty $$

(1.3)

for $0\leq p<\infty$ and $\gamma\in\mathbf {R}$.

We define the positive measure λ on $\mathbf{R}^{n}$ by

$$d\lambda(Q)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} |g(Q)|^{p}t^{-\gamma}\,d\sigma_{Q}, & Q=(t,\Phi)\in S_{n}(\Omega; (1,+\infty)) ,\\ 0,& Q\in\mathbf{R}^{n}-S_{n}(\Omega; (1,+\infty)), \end{array}\displaystyle \right . $$

where p and γ are defined as above. If g is a measurable function on $\partial{C_{n}(\Omega)}$ satisfying (1.3), we remark that the total mass of λ is finite.

Let $\epsilon>0$ and $\beta\geq0$. For each $P=(r,\Theta)\in\mathbf{R}^{n}-\{O\}$, the maximal function is defined by

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

The set $\{P=(r,\Theta)\in\mathbf{R}^{n}-\{O\}; M(P;\lambda,\beta)r^{\beta}>\epsilon\}$ is denoted by $E(\epsilon; \lambda, \beta)$.

As in $T_{n}$, Huang et al. (see [1–3]) have proved the following result. For a similar result in the half-plane, we refer the reader to the paper by Zhao and Yamada (see [4]).

Theorem A

Letgbe a measurable function on $\partial{T_{n}}$satisfying

$$ \int_{\partial{T_{n}}}\frac{|g(Q)|}{1+|Q|^{n}}\,dQ< \infty. $$

(1.4)

Then the harmonic function $PI_{T_{n}}[g](P)=\int_{\partial{T_{n}}}PI_{T_{n}}(P,Q)g(Q)\,dQ$satisfies $PI_{T_{n}}[g]= o(r\sec^{n-1}\theta_{1})$as $r\rightarrow\infty$in $T_{n}$, where $PI_{T_{n}}(P,Q)$is the general Poisson kernel for then-dimensional half space; see Remark  1.

Our aim in this paper is the study of the growth property of $PI_{C_{n}(\Omega)}[g](P)$ in a cone.

Theorem 1

Let $0\leq\alpha\leq n$, $0\leq p<\infty$, $\gamma>(-\alpha_{\Omega}-n+2)p+n-1$and

$$\begin{aligned} \alpha_{\Omega}>\frac{\gamma-n+1}{p}\quad\textit{in the case }p>1,\\ \alpha_{\Omega}\geq\gamma-n+1\quad\textit{in the case }p=1. \end{aligned} $$

Ifgis a measurable function on $\partial{C_{n}(\Omega)}$satisfying (1.3), then $PI_{C_{n}(\Omega)}[g](P)$is a harmonic function of $P\in C_{n}(\Omega)$and there exists a covering $\{r_{k},R_{k}\}$of $E(\epsilon;\lambda,n-\alpha)$ ($\subset C_{n}(\Omega)$) satisfying

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

(1.5)

such that

$$ \lim_{r \rightarrow\infty, P\in C_{n}(\Omega)-E(\epsilon; \lambda,n-\alpha)} r^{\frac{n-\gamma-1}{p}}\bigl\{ f_{\Omega}(\Theta) \bigr\} ^{np-1-\frac{n-\alpha}{p}} PI_{C_{n}(\Omega)}[g](P)=0. $$

(1.6)

Remark 2

In the case $\Omega=S_{+}^{n-1}$, $p=1$, and $\gamma=\alpha=n$, (1.3) is equivalent to (1.4) and (1.5) is a finite sum, then the set $E(\epsilon;\lambda,0)$ is a bounded set and (1.6) holds in $T_{n}$. This is just the result of Qiao-Huang.

Remark 3

In the case $p=1$, $\gamma=n$, and $\alpha=1$, Theorem 1 generalizes Xu-Yang [2], Theorem 1, to the conical case.

2 Lemmas

Throughout this paper, let M denote various constants independent of the variables in question, which may be different from line to line.

Lemma 1

$$\begin{aligned}& \frac{\partial}{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q) \leq M r^{-\beta_{\Omega}}t^{\alpha_{\Omega}-1}f_{\Omega}( \Theta) \end{aligned}$$

(2.1)

$$\begin{aligned}& \biggl(\textit{resp. }\frac{\partial}{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)\leq M r^{\alpha _{\Omega}}t^{-\beta_{\Omega}-1}f_{\Omega}( \Theta)\biggr) \end{aligned}$$

(2.2)

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

$$ \frac{\partial}{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)\leq M\frac{f_{\Omega}(\Theta)}{t^{n-1}}+M \frac{rf_{\Omega}(\Theta)}{|P-Q|^{n}}, $$

(2.3)

for any $P=(r,\Theta)\in C_{n}(\Omega)$and any $Q=(t,\Phi)\in S_{n}(\Omega; (\frac{4}{5}r,\frac{5}{4}r))$.

Proof

These results immediately follow from [8], Lemma 4 and Remark, and (1.1). □

Lemma 2

Let $\epsilon>0$, $\beta\geq0$andλbe any positive measure on $\mathbf{R}^{n}$ ($n\geq2$) having finite total mass. Then $E(\epsilon; \lambda, \beta)$has a covering $\{r_{k},R_{k}\}$ ($k=1,2,\ldots$) satisfying

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

Proof

Set

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

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

$$\biggl(\frac{\rho(P)}{r}\biggr)^{\beta}\leq \frac{\lambda(B(P,\rho(P)))}{\epsilon}. $$

$E_{k}(\epsilon; \lambda, \beta)$ can be covered by the union of a family of balls $\{B(P_{k,i},\rho_{k,i}):P_{k,i}\in E_{k}(\epsilon; \lambda, \beta)\}$ ($\rho_{k,i}=\rho(P_{k,i})$). By the Vitali lemma (see [9]), there exists $\Lambda_{k} \subset E_{k}(\epsilon; \lambda, \beta)$, which is at most countable, such that $\{B(P_{k,i},\rho_{k,i}):P_{k,i}\in\Lambda_{k} \}$ are disjoint and $E_{k}(\epsilon; \lambda, \beta) \subset \bigcup_{P_{k,i}\in\Lambda_{k}} B(P_{k,i}, 5\rho_{k,i})$.

Therefore

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

On the other hand, note that $\bigcup_{P_{k,i}\in\Lambda_{k}} B(P_{k,i},\rho_{k,i}) \subset\{P=(r,\Theta):2^{k-1}\leq r<2^{k+2}\}$, so that

$$\sum_{P_{k,i} \in\Lambda_{k}}\biggl(\frac{5\rho_{k,i}}{|P_{k,i}|} \biggr)^{\beta} \leq 5^{\beta}\sum_{P_{k,i}\in\Lambda_{k}} \frac{\lambda(B(P_{k,i},\rho _{k,i}))}{\epsilon} \leq\frac{5^{\beta}}{\epsilon} \lambda\bigl(C_{n}\bigl( \Omega;\bigl[2^{k-1},2^{k+2}\bigr)\bigr)\bigr). $$

Hence we obtain

$$\sum_{k=1}^{\infty}\sum _{P_{k,i} \in\Lambda_{k}}\biggl(\frac{\rho_{k,i}}{|P_{k,i}|}\biggr)^{\beta} \leq \sum _{k=1}^{\infty}\frac{ \lambda(C_{n}(\Omega ;[2^{k-1},2^{k+2})))}{\epsilon} \leq \frac{3\lambda(\mathbf{R}^{n})}{\epsilon}. $$

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

$$\sum_{k,i}\biggl(\frac{\rho_{k,i}}{|P_{k,i}|} \biggr)^{\beta}\leq\frac{3\lambda(\mathbf{R}^{n})}{\epsilon}+6^{\beta}< +\infty, $$

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

We only prove the case $p>0$ and $p\neq1$, because the case $0\leq p\leq1$ can be proved similarly.

For any fixed $P=(r,\Theta)\in C_{n}(\Omega)$, take a number satisfying $R>\max(1,\frac{5}{4}r)$. If $\alpha_{\Omega}>\frac{\gamma-n+1}{p}$ and $\frac{1}{p}+\frac{1}{q}=1$, then $\{-\beta_{\Omega}-1+\frac{\gamma}{p}\}q+n-1<0$.

By (1.3), (2.2), and Hölder’s inequality, we have

$$\begin{aligned} &\frac{1}{c_{n}} \int_{S_{n}(\Omega;(R,\infty))}\biggl|\frac{\partial }{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)\biggr|\bigl|g(Q)\bigr|\,d \sigma_{Q} \\ &\quad\leq M' \int_{S_{n}(\Omega;(R,\infty))}t^{-\beta_{\Omega}-1}\bigl|g(Q)\bigr|\,d\sigma _{Q}\\ &\quad\leq M' \biggl( \int_{S_{n}(\Omega;(R,\infty))}\bigl|g(Q)\bigr|^{p}t^{-\gamma}\,d\sigma _{Q} \biggr)^{\frac{1}{p}} \biggl( \int_{S_{n}(\Omega;(\frac{5}{4}r,\infty))}t^{(-\beta_{\Omega}+\frac {\gamma}{p}-1)q}\,d\sigma_{Q} \biggr)^{\frac{1}{q}}\\ &\quad< \infty, \end{aligned}$$

where $M'={c_{n}}^{-1}Mr^{\alpha_{\Omega}}$. Thus $PI_{C_{n}(\Omega)}[g](P)$ is finite for any $P\in C_{n}(\Omega)$. Since $\frac{\partial}{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)$ is a harmonic function of $P\in C_{n}(\Omega)$ for any $Q\in S_{n}(\Omega)$, $PI_{C_{n}(\Omega)}[g](P)$ is also a harmonic function of $P\in C_{n}(\Omega)$.

For any $\epsilon>0$, there exists $R_{\epsilon}>1$ such that

$$\int_{S_{n}(\Omega;(R_{\epsilon},\infty))}\frac{|g(Q)|^{p}}{1+t^{\gamma }}\,d\sigma_{Q}< \epsilon. $$

Take any point $P=(r,\Theta)\in C_{n}(\Omega; (R_{\epsilon},+\infty))-E(\epsilon;\lambda, n-\alpha)$ such that $r>\frac{5}{4}R_{\epsilon}$, and write

$$PI\bigl(C_{n}(\Omega),m;g\bigr)\leq PI_{1}(P)+PI_{2}(P)+PI_{3}(P)+PI_{4}(P)+PI_{5}(P), $$

where

$$\begin{aligned}& PI_{1}(P)=\frac{1}{c_{n}} \int_{S_{n}(\Omega;(0,1])}\biggl|\frac{\partial }{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)\biggr|\bigl|g(Q)\bigr|\,d \sigma_{Q}, \\& PI_{2}(P)=\frac{1}{c_{n}} \int_{S_{n}(\Omega;(1,R_{\epsilon}])}\biggl|\frac{\partial }{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)\biggr|\bigl|g(Q)\bigr|\,d \sigma_{Q}, \\& PI_{3}(P)=\frac{1}{c_{n}} \int_{S_{n}(\Omega;(R_{\epsilon},\frac{4}{5}r])}\biggl|\frac {\partial }{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)\biggr|\bigl|g(Q)\bigr|\,d \sigma_{Q}, \\& PI_{4}(P)=\frac{1}{c_{n}} \int_{S_{n}(\Omega;(\frac{4}{5}r,\frac {5}{4}r))}\biggl|\frac{\partial }{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)\biggr|\bigl|g(Q)\bigr|\,d \sigma_{Q},\\& PI_{5}(P)=\frac{1}{c_{n}} \int_{S_{n}(\Omega;[\frac{5}{4}r,\infty])}\biggl|\frac {\partial }{\partial n_{Q}}G_{C_{n}(\Omega)}(P,Q)\biggr|\bigl|g(Q)\bigr|\,d \sigma_{Q}. \end{aligned}$$

If $\gamma>(-\alpha_{\Omega}-n+2)p+n-1$, then $\{\alpha_{\Omega}-1+\frac{\gamma}{p}\}q+n-1>0$. By (2.1) and Hölder’s inequality we have the following growth estimates:

$$\begin{aligned} & PI_{2}(P) \leq Mr^{-\beta_{\Omega}}f_{\Omega}( \Theta) \int_{S_{n}(\Omega;(1,R_{\epsilon}])}t^{\alpha_{\Omega}-1}\bigl|g(Q)\bigr|\,d\sigma_{Q} \\ &\hphantom{PI_{2}(P)}\leq Mr^{-\beta_{\Omega}}f_{\Omega}(\Theta) \biggl( \int_{S_{n}(\Omega;(1,R_{\epsilon}])}\bigl|g(Q)\bigr|^{p}t^{-\gamma}\,d \sigma_{Q} \biggr)^{\frac{1}{p}} \biggl( \int _{S_{n}(\Omega;(1,R_{\epsilon}])}t^{(\alpha_{\Omega}-1+\frac{\gamma }{p})q}\,d\sigma_{Q} \biggr)^{\frac{1}{q}} \\ &\hphantom{PI_{2}(P)} \leq M r^{-\beta_{\Omega}}R_{\epsilon}^{\alpha_{\Omega}+n-2+\frac{\gamma -n+1}{p}}f_{\Omega}( \Theta), \end{aligned}$$

(3.1)

$$\begin{aligned} &PI_{1}(P)\leq Mr^{-\beta_{\Omega}}f_{\Omega}( \Theta), \end{aligned}$$

(3.2)

$$\begin{aligned} &PI_{3}(P)\leq M\epsilon r^{\frac{\gamma-n+1}{p}}f_{\Omega}( \Theta). \end{aligned}$$

(3.3)

If $\alpha_{\Omega}>\frac{\gamma-n+1}{p}$, then $\{-\beta_{\Omega}-1+\frac{\gamma}{p}\}q+n-1<0$. We obtain (2.2) and Hölder’s inequality,

$$\begin{aligned} PI_{5}(P) \leq& Mr^{\alpha_{\Omega}}f_{\Omega}( \Theta) \int_{S_{n}(\Omega;[\frac{5}{4}r,\infty ))}t^{-\beta_{\Omega}-1}\bigl|g(Q)\bigr|\,d\sigma_{Q} \\ \leq& Mr^{\alpha_{\Omega}}f_{\Omega}(\Theta) \biggl( \int_{S_{n}(\Omega;[\frac {5}{4}r,\infty))}\bigl|g(Q)\bigr|^{p}t^{-\gamma}\,d \sigma_{Q} \biggr)^{\frac{1}{p}} \biggl( \int_{S_{n}(\Omega;[\frac{5}{4}r,\infty))}t^{(-\beta_{\Omega}-1+\frac {\gamma}{p})q}\,d\sigma_{Q} \biggr)^{\frac{1}{q}} \\ \leq& M\epsilon r^{\frac{\gamma-n+1}{p}}f_{\Omega}(\Theta). \end{aligned}$$

(3.4)

By (2.3), we consider the inequality

$$PI_{4}(P)\leq PI_{41}(P)+PI_{42}(P), $$

where

$$\begin{aligned}& PI_{41}(P)=Mf_{\Omega}(\Theta) \int_{S_{n}(\Omega;(\frac{4}{5}r,\frac {5}{4}r))}t^{1-n}\bigl|g(Q)\bigr|\,d\sigma_{Q}, \\& PI_{42}(P)=Mrf_{\Omega}(\Theta) \int_{S_{n}(\Omega;(\frac{4}{5}r,\frac {5}{4}r))}\frac{|g(Q)|}{|P-Q|^{n}}\,d\sigma_{Q}. \end{aligned}$$

We first have

$$\begin{aligned} PI_{41}(P) \leq& Mf_{\Omega}(\Theta) \int_{S_{n}(\Omega;(\frac{4}{5}r,\frac{5}{4}r))}t^{\alpha _{\Omega}-\beta_{\Omega}-1}\bigl|g(Q)\bigr|\,d\sigma_{Q} \\ \leq& Mr^{\alpha_{\Omega}}f_{\Omega}(\Theta) \int_{S_{n}(\Omega;(\frac{4}{5}r,\infty ))}t^{-\beta_{\Omega}-1}\bigl|g(Q)\bigr|\,d\sigma_{Q} \\ \leq& M\epsilon r^{\frac{\gamma-n+1}{p}} f_{\Omega}(\Theta), \end{aligned}$$

(3.5)

which is similar to the estimate of $PI_{5}(P)$.

Next, we shall estimate $PI_{42}(P)$. Take a sufficiently small positive number b such that $S_{n}(\Omega;(\frac{4}{5}r,\frac{5}{4}r))\subset B(P,\frac{1}{2}r)$ for any $P=(r,\Theta)\in\Pi(b)$, where

$$\Pi(b)=\Bigl\{ P=(r,\Theta)\in C_{n}(\Omega); \inf_{z\in\partial\Omega }\bigl|(1, \Theta)-(1,z)\bigr|< b, 0< r< \infty\Bigr\} $$

and divide $C_{n}(\Omega)$ into two sets $\Pi(b)$ and $C_{n}(\Omega)-\Pi(b)$.

If $P=(r,\Theta)\in C_{n}(\Omega)-\Pi(b)$, then there exists a positive $b'$ such that $|P-Q|\geq{b}'r$ for any $Q\in S_{n}(\Omega)$, and hence

$$\begin{aligned} PI_{42}(P) \leq&Mf_{\Omega}(\Theta) \int_{S_{n}(\Omega;(\frac{4}{5}r,\frac{5}{4}r))}t^{1-n} \bigl|g(Q)\bigr|\,d\sigma_{Q} \\ \leq& M\epsilon r^{\frac{\gamma-n+1}{p}} f_{\Omega}(\Theta), \end{aligned}$$

(3.6)

which is similar to the estimate of $PI_{41}(P)$.

We shall consider the case $P=(r,\Theta)\in\Pi(b)$. Now put

$$H_{i}(P)=\biggl\{ Q\in S_{n}\biggl(\Omega;\biggl( \frac{4}{5}r,\frac{5}{4}r\biggr)\biggr); 2^{i-1}\delta(P) \leq|P-Q|< 2^{i}\delta(P)\biggr\} . $$

Since $S_{n}(\Omega)\cap\{Q\in\mathbf{R}^{n}: |P-Q|< \delta(P)\}=\varnothing$, we have

$$PI_{42}(P)=M\sum_{i=1}^{i(P)} \int_{H_{i}(P)}rf_{\Omega}(\Theta)\frac {|g(Q)|}{|P-Q|^{n}}\,d \sigma_{Q}, $$

where $i(P)$ is a positive integer satisfying $2^{i(P)-1}\delta(P)\leq\frac{r}{2}<2^{i(P)}\delta(P)$.

If $\alpha_{\Omega}>\frac{\gamma-\alpha+1}{p}$, then $\{-\beta_{\Omega}-1+\frac{n-\alpha+\gamma}{p}\}q+n-1<0$. By (1.2), we have $rf_{\Omega}(\Theta)\leq M\delta(P)$ ($P=(r,\Theta)\in C_{n}(\Omega)$). By Hölder’s inequality we obtain

$$\begin{aligned} & \int_{H_{i}(P)}rf_{\Omega}(\Theta)\frac{|g(Q)|}{|P-Q|^{n}}\,d \sigma_{Q} \\ &\quad\leq2^{(1-i)n}f_{\Omega}(\Theta){\delta(P)}^{\frac{\alpha-n}{p}} \int _{H_{i}(P)}r{\delta(P)}^{\frac{n-\alpha}{p}-n}\bigl|g(Q)\bigr|\,d \sigma_{Q} \\ &\quad\leq M\bigl\{ f_{\Omega}(\Theta)\bigr\} ^{1-n+\frac{n-\alpha}{p}}{ \delta(P)}^{\frac {\alpha-n}{p}} \int_{H_{i}(P)}r^{1-n+\frac{n-\alpha}{p}}\bigl|g(Q)\bigr|\,d\sigma_{Q} \\ &\quad\leq Mr^{\alpha_{\Omega}}\bigl\{ f_{\Omega}(\Theta)\bigr\} ^{1-n+\frac{n-\alpha }{p}}{\delta(P)}^{\frac{\alpha-n}{p}} \int_{H_{i}(P)}t^{-\beta_{\Omega}-1+\frac{n-\alpha}{p}}\bigl|g(Q)\bigr|\,d\sigma_{Q} \\ &\quad\leq Mr^{\alpha_{\Omega}}\bigl\{ f_{\Omega}(\Theta)\bigr\} ^{1-n+\frac{n-\alpha }{p}}{\delta(P)}^{\frac{\alpha-n}{p}} \biggl( \int_{H_{i}(P)}\bigl|g(Q)\bigr|^{p}t^{-\gamma}\,d \sigma_{Q} \biggr)^{\frac{1}{p}} \\ &\qquad{}\times \biggl( \int_{S_{n}(\Omega;(\frac{4}{5}r,\infty))}t^{\{-\beta_{\Omega}-1+\frac{n-\alpha+\gamma}{p}\}q}\,d\sigma_{Q} \biggr)^{\frac{1}{q}} \\ &\quad\leq M\epsilon r^{\frac{1-\alpha+\gamma}{p}}\bigl\{ f_{\Omega}(\Theta)\bigr\} ^{1-n+\frac{n-\alpha}{p}} \biggl(\frac{ \lambda(H_{i}(P))}{\{2^{i}\delta(P)\}^{n-\alpha}} \biggr)^{\frac{1}{p}} \end{aligned}$$

for $i=0,1,2,\ldots,i(P)$.

Since $P=(r,\Theta)\notin E(\epsilon;\lambda, n-\alpha)$, we have

$$\frac{\lambda(H_{i}(P))}{\{2^{i}\delta(P)\}^{n-\alpha}}\leq\frac{\lambda (B(P,2^{i}\delta(P)))}{\{2^{i}\delta(P)\}^{n-\alpha}}\leq M(P;\lambda, n-\alpha)\leq\epsilon r^{\alpha-n}\quad \bigl(i=0,1,2,\ldots,i(P)-1\bigr) $$

and

$$\frac{\lambda(H_{i(P)}(P))}{\{2^{i}\delta(P)\}^{n-\alpha}}\leq\frac {\lambda(B(P,\frac{r}{2}))}{(\frac{r}{2})^{n-\alpha}}\leq\epsilon r^{\alpha-n}. $$

So

$$ PI_{42}(P)\leq M \epsilon r^{\frac{\gamma-n+1}{p}}\bigl\{ f_{\Omega}( \Theta)\bigr\} ^{1-n+\frac{n-\alpha }{p}}. $$

(3.7)

Combining (3.1)-(3.7), we finally obtain $PI_{C_{n}(\Omega)}[g](P)=o(r^{\frac{\gamma-n+1}{p}}\{f_{\Omega}(\Theta)\} ^{1-n+\frac{n-\alpha}{p}})$ as $r\rightarrow\infty$, where $P=(r,\Theta)\in C_{n}(\Omega; (R_{\epsilon},+\infty))-E(\epsilon;\lambda, n-\alpha)$. Thus we complete the proof of Theorem 1 by Lemma 2.