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 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 \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \({\mathbf{R}}^{n}\) which are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).
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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 \(\Xi\times\Omega\). 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 \({T}_{n}\).
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 \({\mathbf{R}}^{n}\). \(S_{r}=\partial{B(O,r)}\). 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. 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 \(S_{n}(\Omega; r)\) we denote \(C_{n}(\Omega)\cap S_{r}\). By \(S_{n}(\Omega)\) we denote \(S_{n}(\Omega; (0,+\infty))\) which is \(\partial{C_{n}(\Omega)}-\{O\}\).
We use the standard notations \(u^{+}=\max\{u,0\}\) and \(u^{-}=-\min\{u,0\}\). Further, we denote by \(w_{n}\) the surface area \(2\pi^{n/2}\{\Gamma(n/2)\}^{-1}\) of \({\mathbf{S}}^{n-1}\), by \({\partial}/{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into \(C_{n}(\Omega)\), by \(dS_{r}\) the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(S_{r}\) and by dw the elements of the Euclidean volume in \({\mathbf{R}}^{n}\).
Let Ω be a domain on \({\mathbf{S}}^{n-1}\) with smooth boundary. Consider the Dirichlet problem
where \(\Lambda_{n}\) is the spherical part of the Laplace operator \(\Delta_{n}\),
We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by \(\varphi(\Theta)\), \(\int_{\Omega}\varphi^{2}(\Theta)\, dS_{1}=1\). 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 (e.g. see [1], pp.88-89, for the definition of \(C^{2,\alpha}\)-domain). Then \(\varphi\in C^{2}(\overline{\Omega})\) and \({\partial\varphi}/{\partial n}>0\) on ∂Ω (here and below, \({\partial}/{\partial n}\) denotes differentiation along the interior normal).
$$\begin{aligned}& (\Lambda_{n}+\lambda)\varphi=0\quad \text{on } \Omega, \\& \varphi=0 \quad \text{on } \partial{\Omega}, \end{aligned}$$
$$\Delta_{n}=\frac{n-1}{r}\frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial r^{2}}+ \frac{\Lambda_{n}}{r^{2}}. $$
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We note that each function
is harmonic in \(C_{n}(\Omega)\), belongs to the class \(C^{2}(C_{n}(\Omega )\backslash\{O\})\) and vanishes on \(S_{n}(\Omega)\), where
In the sequel, for the sake of brevity, we shall write χ instead of \(\aleph^{+}-\aleph^{-}\). If \(\Omega={\mathbf{S}}^{n-1}_{+}\), then \(\aleph^{+}=1\), \(\aleph^{-}=1-n\), and \(\varphi(\Theta)=(2n w_{n}^{-1})^{1/2}\cos\theta_{1}\).
$$r^{\aleph^{\pm}}\varphi(\Theta) $$
$$2\aleph^{\pm}=-n+2\pm\sqrt{(n-2)^{2}+4\lambda}. $$
Let \(G_{\Omega}(P,Q)\) (\(P=(r,\Theta)\), \(Q=(t,\Phi)\in C_{n}(\Omega)\)) be the Green function of \(C_{n}(\Omega)\). Then the ordinary Poisson kernel relative to \(C_{n}(\Omega)\) is defined by
where \(Q\in S_{n}(\Omega)\) and
$$\mathcal{PI}_{\Omega}(P,Q)=\frac{1}{c_{n}}\frac{\partial}{\partial n_{Q}}G_{\Omega}(P,Q), $$
$$c_{n}=\left \{ \begin{array}{l@{\quad}l} 2\pi & \mbox{if } n=2, \\ (n-2)w_{n} & \mbox{if } n\geq3. \end{array} \right . $$
The estimate we deal with has a long history which can be traced back to Matsaev’s estimate of harmonic functions from below (see, for example, Levin [2], p.209).
Theorem A
Let
\(A_{1}\)be a constant, \(u(z)\) (\(|z|=R\)) be harmonic on
\(T_{2}\)and continuous on
\({\partial T}_{2}\). Suppose that
and
Then
where
\(z=Re^{i\alpha}\in{T}_{2}\)and
\(A_{2}\)is a constant independent of
\(A_{1}\), R, α, and the function
\(u(z)\).
$$u(z)\leq A_{1}R^{\rho}, \quad z\in T_{2}, R>1, \rho>1 $$
$$\bigl\vert u(z)\bigr\vert \leq A_{1}, \quad R\leq1, z\in{ \overline{T}}_{2}. $$
$$u(z)\geq-A_{1}A_{2}\bigl(1+R^{\rho}\bigr) \sin^{-1}\alpha, $$
Recently, Xu et al. [3‐5] considered Theorem A in the n-dimensional (\(n\geq2\)) case and obtained the following result.
Theorem B
Let
\(A_{3}\)be a constant, \(u(P)\) (\(|P|=R\)) be harmonic on
\(T_{n}\)and continuous on
\(\overline{T}_{n}\). If
and
then
where
\(P\in T_{n}\)and
\(A_{4}\)is a constant independent of
\(A_{3}\), R, \(\theta_{1}\), and the function
\(u(P)\).
$$ u(P)\leq A_{3}R^{\rho}, \quad P\in T_{n}, R>1, \rho>n-1 $$
(1.1)
$$ \bigl\vert u(P)\bigr\vert \leq A_{3}, \quad R\leq1, P\in \overline{T}_{n}, $$
(1.2)
$$u(P)\geq-A_{3}A_{4}\bigl(1+R^{\rho}\bigr) \cos^{1-n}\theta_{1}, $$
Now we have the following.
Theorem 1
LetKbe a constant, \(u(P)\) (\(P=(R,\Theta)\)) be harmonic on
\(C_{n}(\Omega)\)and continuous on
\(\overline{C_{n}(\Omega)}\). If
and
then
where
\(P\in C_{n}(\Omega)\), N (≥1) is a sufficiently large number, \(\rho(R)\)is nondecreasing in
\([1,+\infty)\)andMis a constant independent ofK, R, \(\varphi(\theta)\), and the function
\(u(P)\).
$$ u(P)\leq KR^{\rho(R)}, \quad P=(R,\Theta)\in C_{n}\bigl( \Omega;(1,\infty)\bigr), \rho(R)>\aleph^{+} $$
(1.3)
$$ u(P)\geq-K, \quad R\leq1, P=(R,\Theta) \in \overline{C_{n}(\Omega)}, $$
(1.4)
$$u(P)\geq -KM \biggl(1+\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)} \biggr) \varphi ^{1-n}\theta, $$
By taking \(\rho(R)\equiv\rho\), we obtain the following corollary, which generalizes Theorem B to the conical case.
Corollary
LetKbe a constant, \(u(P)\) (\(P=(R,\Theta)\)) be harmonic on
\(C_{n}(\Omega)\)and continuous on
\(\overline{C_{n}(\Omega)}\). If
and
then
where
\(P\in C_{n}(\Omega)\), Mis a constant independent ofK, R, \(\varphi(\theta)\), and the function
\(u(P)\).
$$u(P)\leq KR^{\rho},\quad P=(R,\Theta)\in C_{n}\bigl(\Omega;(1, \infty)\bigr), \rho>\aleph^{+} $$
$$u(P)\geq-K, \quad R\leq1, P=(R,\Theta) \in \overline{C_{n}(\Omega)}, $$
$$u(P)\geq-KM\bigl(1+R^{\rho}\bigr)\varphi^{1-n}\theta, $$
2 Lemmas
Throughout this paper, let M denote various constants independent of the variables in question, which may be different from line to line.
Carleman’s formula (see [6]) connects the modulus and the zeros of a function analytic in a complex plane (see, for example, [7], p.224). I Miyamoto and H Yoshida generalized it to subharmonic functions in an n-dimensional cone (see [8, 9]).
Lemma 1
If
\(R>1\)and
\(u(t,\Phi)\)is a subharmonic function on a domain containing
\(C_{n}(\Omega;(1,R))\), then
where
$$\begin{aligned}& \int_{C_{n}(\Omega;(1,R))} \biggl(\frac{1}{t^{-\aleph^{-}}}-\frac {t^{\aleph^{+}}}{R^{\chi}} \biggr) \varphi\Delta u \, dw \\& \quad =\chi\int_{S_{n}(\Omega;R)}\frac{u\varphi}{R^{1-\aleph^{-}}} \, d S_{R} + \int_{S_{n}(\Omega;(1,R))}u \biggl(\frac{1}{t^{-\aleph^{-}}}-\frac {t^{\aleph^{+}}}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, d\sigma_{Q}+d_{1}+ \frac{d_{2}}{R^{\chi}}, \end{aligned}$$
$$d_{1}=\int_{S_{n}(\Omega;1)}\aleph^{-}u\varphi- \varphi\frac{\partial u}{\partial n} \, dS_{1} \quad \textit{and} \quad d_{2}=\int_{S_{n}(\Omega;1)}\varphi \frac{\partial u}{\partial n}- \aleph^{+}u\varphi \, dS_{1}. $$
Lemma 2
$$ \mathcal{PI}_{\Omega}(P,Q)\leq M r^{\aleph^{-}}t^{\aleph^{+}-1}\varphi( \Theta)\frac{\partial \varphi( \Phi)}{\partial n_{\Phi}} $$
(2.1)
$$ \mathcal{PI}_{\Omega}(P,Q)\leq M\frac{\varphi(\Theta)}{t^{n-1}}\frac{\partial\varphi( \Phi)}{\partial n_{\Phi}}+M \frac{r\varphi(\Theta)}{|P-Q|^{n}}\frac{\partial\varphi( \Phi)}{\partial n_{\Phi}} $$
(2.2)
Let
\(G_{\Omega,R}(P,Q)\)be the Green function of
\(C_{n}(\Omega,(0,R))\). Then
where
\(P=(r,\Theta)\in C_{n}(\Omega)\)and
\(Q=(R,\Phi)\in S_{n}(\Omega;R)\).
$$ \frac{\partial G_{\Omega,R}(P,Q)}{\partial R}\leq M r^{\aleph^{+}}R^{\aleph^{-}-1}\varphi(\Theta)\varphi( \Phi), $$
(2.3)
3 Proof of Theorem 1
Lemma 1 applied to \(u=u^{+}-u^{-}\) gives
$$\begin{aligned}& \chi\int_{S_{n}(\Omega;R)}\frac{u^{+}\varphi}{R^{1-\aleph^{-}}}\, d S_{R}+\int _{S_{n}(\Omega;(1,R))}u^{+} \biggl(\frac{1}{t^{-\aleph ^{-}}}- \frac{t^{\aleph^{+}}}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, d\sigma_{Q}+d_{1}+ \frac{d_{2}}{R^{\chi}} \\& \quad =\chi\int_{S_{n}(\Omega;R)}\frac{u^{-}\varphi}{R^{1-\aleph^{-}}} \, d S_{R}+\int _{S_{n}(\Omega;(1,R))}u^{-} \biggl(\frac{1}{t^{-\aleph ^{-}}}- \frac{t^{\aleph^{+}}}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, d\sigma_{Q}. \end{aligned}$$
(3.1)
It immediately follows from (1.3) that
and
$$ \chi\int_{S_{n}(\Omega;R)}\frac{u^{+}\varphi}{R^{1-\aleph^{-}}}\, d S_{R} \leq MKR^{\rho(R)-\aleph^{+}} $$
(3.2)
$$\begin{aligned}& \int_{S_{n}(\Omega;(1,R))}u^{+} \biggl( \frac{1}{t^{-\aleph^{-}}}-\frac {t^{\aleph^{+}}}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, d \sigma_{Q} \\& \quad \leq \int_{S_{n}(\Omega;(1,R))}Kt^{\rho(t)+\aleph^{+}} \biggl( \frac {1}{t^{\chi}}-\frac{1}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, d \sigma_{Q} \\& \quad \leq MK\int_{1}^{R} \biggl(r^{\rho(r)-\aleph^{+}-1}- \frac{r^{\rho(r)-\aleph ^{-}-1}}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, dr \\& \quad \leq MK\int_{1}^{R} r^{\rho(R)-\aleph^{+}-1} \, dr \\& \quad \leq \frac{MK}{\rho(R)-\aleph^{+}}R^{\rho(R)-\aleph^{+}} \\& \quad \leq MKR^{\rho(R)-\aleph^{+}}. \end{aligned}$$
(3.3)
Notice that
$$ d_{1}+\frac{d_{2}}{R^{\chi}} \leq MKR^{\rho(R)-\aleph^{+}}. $$
(3.4)
Hence from (3.1), (3.2), (3.3), and (3.4) we have
and
$$ \chi\int_{S_{n}(\Omega;R)}\frac{u^{-}\varphi}{R^{1-\aleph^{-}}} \, d S_{R} \leq MKR^{\rho(R)-\aleph^{+}} $$
(3.5)
$$ \int_{S_{n}(\Omega;(1,R))}u^{-} \biggl(\frac{1}{t^{-\aleph^{-}}}- \frac {t^{\aleph^{+}}}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n}\, d\sigma_{Q} \leq MKR^{\rho(R)-\aleph^{+}}. $$
(3.6)
Equation (3.6) gives
$$\begin{aligned}& \int_{S_{n}(\Omega;(1,R))}u^{-}t^{\aleph^{-}}\frac{\partial\varphi }{\partial n} \, d\sigma_{Q} \\& \quad \leq \frac{(N+1)^{\chi}}{(N+1)^{\chi}-N^{\chi}}\int_{S_{n}(\Omega ;(1,\frac{N+1}{N}R))}u^{-} \biggl(\frac{1}{t^{-\aleph^{-}}}-\frac {t^{\aleph^{+}}}{(\frac{N+1}{N}R)^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, d \sigma_{Q} \\& \quad \leq \frac{(N+1)^{\chi}}{(N+1)^{\chi}-N^{\chi}}MK\biggl(\frac {N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)-\aleph^{+}} \\& \quad \leq MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)-\aleph^{+}}. \end{aligned}$$
Thus
$$ \int_{S_{n}(\Omega;(1,R))}u^{-}t^{\aleph^{-}}\frac{\partial\varphi }{\partial n}\, d\sigma_{Q} \leq MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)-\aleph^{+}}. $$
(3.7)
By the Riesz decomposition theorem (see [7]), for any \(P=(r,\Theta)\in C_{n}(\Omega;(0,R))\) we have
$$\begin{aligned} -u(P) =&\int_{S_{n}(\Omega;(0,R))}\mathcal{PI}_{\Omega }(P,Q)-u(Q)\, d \sigma_{Q} \\ &{}+\int_{S_{n}(\Omega;R)}\frac{\partial G_{\Omega ,R}(P,Q)}{\partial R}-u(Q)\, dS_{R}. \end{aligned}$$
(3.8)
Now we distinguish three cases.
Case 1. \(P=(r,\Theta)\in C_{n}(\Omega;(\frac {5}{4},\infty))\) and \(R=\frac{5}{4}r\).
Since \(-u(x)\leq u^{-}(x)\), we obtain
from (3.8), where
$$ -u(P)=\sum_{i=1}^{4} I_{i}(P) $$
(3.9)
$$\begin{aligned}& I_{1}(P)=\int_{S_{n}(\Omega;(0,1])}\mathcal{PI}_{\Omega }(P,Q)-u(Q) \, d\sigma_{Q}, \\& I_{2}(P)=\int_{S_{n}(\Omega;(1,\frac {4}{5}r])} \mathcal{PI}_{\Omega}(P,Q)-u(Q)\, d\sigma_{Q}, \\& I_{3}(P)=\int_{S_{n}(\Omega;(\frac{4}{5}r,R))}\mathcal{PI}_{\Omega }(P,Q)-u(Q) \, d\sigma_{Q} \quad \text{and} \\& I_{4}(P)=\int _{S_{n}(\Omega ;R)}\mathcal{PI}_{\Omega}(P,Q)-u(Q)\, d \sigma_{Q}. \end{aligned}$$
Then from (2.1) and (3.7) we have
and
$$ I_{1}(P)\leq MK\varphi(\Theta) $$
(3.10)
$$\begin{aligned} I_{2}(P) \leq& r^{\aleph^{-}}\varphi(\Theta) \biggl( \frac{4}{5}r\biggr)^{\chi-1}\int_{S_{n}(\Omega;(1,\frac{4}{5}r])}-u(Q)t^{\aleph^{-}} \frac{\partial \varphi( \Phi)}{\partial n_{\Phi}}\, d\sigma_{Q} \\ \leq& MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)}\varphi(\Theta). \end{aligned}$$
(3.11)
By (2.2), we consider the inequality
where
and
$$ I_{3}(P)\leq I_{31}(P)+I_{32}(P), $$
(3.12)
$$I_{31}(P)=M\int_{S_{n}(\Omega;(\frac{4}{5}r,R))}\frac{-u(Q) \varphi (\Theta)}{t^{n-1}} \frac{\partial\varphi( \Phi)}{\partial n_{\Phi}}\, d\sigma_{Q} $$
$$I_{32}(P)=Mr\varphi(\Theta)\int_{S_{n}(\Omega;(\frac {4}{5}r,R))} \frac{-u(Q) r\varphi(\Theta)}{|P-Q|^{n}}\frac{\partial \varphi( \Phi)}{\partial n_{\Phi}}\, d\sigma_{Q}. $$
We first have
from (3.7). Next, we shall estimate \(I_{32}(P)\). Take a sufficiently small positive number k such that \(S_{n}(\Omega;(\frac{4}{5}r,R))\subset B(P,\frac{1}{2}r)\) for any \(P=(r,\Theta)\in\Pi(k)\), where
and divide \(C_{n}(\Omega)\) into two sets \(\Pi(k)\) and \(C_{n}(\Omega)-\Pi(k)\).
$$\begin{aligned} I_{31}(P) \leq& M\varphi(\Theta)r^{1-n-\aleph^{-}}\int _{S_{n}(\Omega;(\frac {4}{5}r,R))}-u(Q)t^{\aleph^{-}}\frac{\partial \varphi( \Phi)}{\partial n_{\Phi}} \, d \sigma_{Q} \\ \leq& MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)}\varphi(\Theta) \end{aligned}$$
(3.13)
$$\Pi(k)=\Bigl\{ P=(r,\Theta)\in C_{n}(\Omega); \inf_{(1,z)\in\partial \Omega} \bigl\vert (1,\Theta)-(1,z)\bigr\vert < k, 0<r<\infty\Bigr\} , $$
If \(P=(r,\Theta)\in C_{n}(\Omega)-\Pi(k)\), then there exists a positive \(k'\) such that \(|P-Q|\geq{k}'r\) for any \(Q\in S_{n}(\Omega)\), and hence
which is similar to the estimate of \(I_{31}(P)\).
$$\begin{aligned} I_{32}(P) \leq&M \int_{S_{n}(\Omega;(\frac{4}{5}r,R))} \frac{-u(Q) \varphi(\Theta)}{t^{n-1}}\frac{\partial\varphi( \Phi)}{\partial n_{\Phi}}\, d\sigma_{Q} \\ \leq& MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)}\varphi(\Theta), \end{aligned}$$
(3.14)
We shall consider the case \(P=(r,\Theta)\in\Pi(k)\). Now put
where \(\delta(P)=\inf_{Q\in\partial{C_{n}(\Omega)}}|P-Q|\).
$$H_{i}(P)=\biggl\{ Q\in S_{n}\biggl(\Omega;\biggl( \frac{4}{5}r,R\biggr)\biggr); 2^{i-1}\delta(P) \leq|P-Q|< 2^{i}\delta(P)\biggr\} , $$
Since \(S_{n}(\Omega)\cap\{Q\in{\mathbf{R}}^{n}: |P-Q|< \delta(P)\}=\varnothing\), we have
where \(i(P)\) is a positive integer satisfying \(2^{i(P)-1}\delta(P)\leq\frac{r}{2}<2^{i(P)}\delta(P)\).
$$I_{32}(P)=M\sum_{i=1}^{i(P)}\int _{H_{i}(P)}\frac{-u(Q)r\varphi(\Theta )}{|P-Q|^{n}}\frac{\partial\varphi( \Phi)}{\partial n_{\Phi}}\, d \sigma_{Q}, $$
Since \(r\varphi(\Theta)\leq M\delta(P)\) (\(P=(r,\Theta)\in C_{n}(\Omega)\)), similar to the estimate of \(I_{31}(P)\) we obtain
for \(i=0,1,2,\ldots,i(P)\).
$$\begin{aligned}& \int_{H_{i}(P)}\frac{-u(Q)r\varphi(\Theta)}{|P-Q|^{n}}\frac {\partial \varphi( \Phi)}{\partial n_{\Phi}}\, d \sigma_{Q} \\& \quad \leq \int_{H_{i}(P)}r\varphi(\Theta)\frac{-u(Q)}{(2^{i-1}\delta (P))^{n}} \frac{\partial \varphi( \Phi)}{\partial n_{\Phi}}\, d\sigma_{Q} \\& \quad \leq M2^{(1-i)n}\varphi^{1-n}(\Theta) \int _{H_{i}(P)}t^{1-n}-u(Q)\frac{\partial\varphi( \Phi)}{\partial n_{\Phi}}\, d \sigma_{Q} \\& \quad \leq MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)}\varphi ^{1-n}(\Theta) \end{aligned}$$
So
$$ I_{32}(P)\leq MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)}\varphi ^{1-n}(\Theta). $$
(3.15)
From (3.12), (3.13), (3.14), and (3.15) we see that
$$ I_{3}(P)\leq MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)}\varphi ^{1-n}(\Theta). $$
(3.16)
On the other hand, we have from (2.3) and (3.5) that
$$\begin{aligned} I_{4}(P) \leq&M r^{\aleph^{+}}\varphi(\Theta)\int _{S_{n}(\Omega;R)}\frac {-u(Q)\varphi}{R^{1-\aleph^{-}}}\, d S_{R} \\ \leq& MKR^{\rho(R)}\varphi(\Theta). \end{aligned}$$
(3.17)
We thus obtain (3.10), (3.11), (3.16), and (3.17) that
$$ -u(P)\leq MK \biggl(1+\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)} \biggr) \varphi^{1-n}(\Theta). $$
(3.18)
Case 2. \(P=(r,\Theta)\in C_{n}(\Omega;(\frac{4}{5},\frac{5}{4}])\) and \(R=\frac{5}{4}r\).
Equation (3.8) gives
where \(I_{1}(P)\) and \(I_{4}(P)\) are defined in Case 1 and
$$-u(P)= I_{1}(P)+I_{5}(P)+I_{4}(P), $$
$$I_{5}(P)=\int_{S_{n}(\Omega;(1,R))}\mathcal {PI}_{\Omega}(P,Q)-u(Q) \, d\sigma_{Q}. $$
Similar to the estimate of \(I_{3}(P)\) in Case 1 we have
which together with (3.10) and (3.17) gives (3.18).
$$I_{5}(P)\leq MK\biggl(\frac{N+1}{N}R\biggr)^{\rho(\frac{N+1}{N}R)}\varphi ^{1-n}(\Theta), $$
Case 3. \(P=(r,\Theta)\in C_{n}(\Omega;(0,\frac{4}{5}])\).
Acknowledgements
This work was partially supported by NSF Grant DMS-0913205.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by the corresponding author BY. SP and BY prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.