RETRACTED ARTICLE: Levin’s type boundary behaviors for functions harmonic and admitting certain lower bounds

Abstract

In this paper, we prove Levin’s type boundary behaviors for functions harmonic and admitting certain lower bounds, which extend Pan, Qiao and Deng’s inequalities for analytic functions in a half-space.

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 S̅, 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 $x_{n}=r\cos\theta_{1}$.

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 ${\partial}/{\partial n_{Q}}$ 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

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

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

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

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by $\varphi(\Theta)$,

$$\int_{\Omega}\varphi^{2}(\Theta)\, dS_{1}=1. $$

In order to ensure the existence of λ and 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 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).

We note that each function $r^{\aleph^{\pm}}\varphi(\Theta)$ is harmonic in $C_{n}(\Omega)$, belongs to the class $C^{2}(C_{n}(\Omega )\backslash\{O\})$ and vanishes on $S_{n}(\Omega)$, where

$$2\aleph^{\pm}=-n+2\pm\sqrt{(n-2)^{2}+4\lambda}. $$

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 $\varphi(\Theta)=(2n w_{n}^{-1})^{1/2}\cos\theta_{1}$.

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

$$\mathcal {PI}_{\Omega}(P,Q)=\frac{1}{c_{n}}\frac{\partial}{\partial n_{Q}}G_{\Omega}(P,Q), $$

where $Q\in S_{n}(\Omega)$, $c_{n}= 2\pi$ if $n=2$ and $c_{n}= (n-2)w_{n}$ if $n\geq3$.

The estimate we deal with has a long history which can be traced back to Levin’s type boundary behaviors for functions harmonic from below (see, for example, Levin [1], 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

$$u(z)\leq A_{1}R^{\rho}, \quad z\in T_{2}, R>1, \rho>1 $$

and

$$\bigl\vert u(z)\bigr\vert \leq A_{1}, \quad R\leq1, z\in{ \overline{T}}_{2}. $$

Then

$$u(z)\geq-A_{1}A_{2}\bigl(1+R^{\rho}\bigr) \sin^{-1}\alpha, $$

where $z=Re^{i\alpha}\in{T}_{2}$and $A_{2}$is a constant independent of $A_{1}$, R, αand the function $u(z)$.

Recently, Pan et al. [2] considered Theorem A in the n-dimensional 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

$$ u(P)\leq A_{3}R^{\rho}, \quad P\in T_{n}, R>1, \rho>n-1 $$

(1.1)

and

$$ \bigl\vert u(P)\bigr\vert \leq A_{3}, \quad R\leq1, P\in \overline{T}_{n}, $$

(1.2)

then

$$u(P)\geq-A_{3}A_{4}\bigl(1+R^{\rho}\bigr) \cos^{1-n}\theta_{1}, $$

where $P\in T_{n}$and $A_{4}$is a constant independent of $A_{3}$, R, $\theta_{1}$and the function $u(P)$.

Now we have the following.

Theorem 1

LetKbe a constant, $u(P)$ ($P=(R,\Theta)$) be harmonic on $C_{n}(\Omega)$and continuous on ̅. If

$$ u(P)\leq KR^{\rho(R)}, \quad P=(R,\Theta)\in C_{n}\bigl( \Omega;(1,\infty)\bigr), \rho(R)>\aleph^{+} $$

(1.3)

and

$$ u(P)\geq-K, \quad R\leq1, P=(R,\Theta) \in \overline{C_{n}(\Omega)}, $$

(1.4)

then

$$u(P)\geq-KM\bigl(1+\rho(R)R^{\rho(R)}\bigr)\varphi^{1-n}\theta, $$

where $P\in C_{n}(\Omega)$, $\rho(R)$is nondecreasing in $[1,+\infty)$andMis a constant independent ofK, R, $\varphi(\theta)$and the function $u(P)$.

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 ̅. If

$$u(P)\leq KR^{\rho}, \quad P=(R,\Theta)\in C_{n}\bigl( \Omega;(1,\infty)\bigr), \rho>\aleph^{+} $$

and

$$u(P)\geq-K, \quad R\leq1, P=(R,\Theta) \in \overline{C_{n}(\Omega)}, $$

then

$$u(P)\geq-KM\bigl(1+R^{\rho}\bigr)\varphi^{1-n}\theta, $$

where $P\in C_{n}(\Omega)$, Mis a constant independent ofK, R, $\varphi(\theta)$and the function $u(P)$.

Remark

(see [2])

From corollary, we know that conditions (1.1) and (1.2) may be replaced with weaker conditions

$$u(P)\leq A_{3}R^{\rho}, \quad P\in T_{n}, R>1, \rho>1 $$

and

$$u(P)\geq-A_{3}, \quad R\leq1, P\in\overline{T}_{n}, $$

respectively.

2 Lemma

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

Lemma 1

(see [3–5])

$$ \mathcal{PI}_{\Omega}(P,Q)\leq M r^{\aleph^{-}}t^{\aleph^{+}-1}\varphi( \Theta)\frac{\partial\varphi( \Phi)}{\partial n_{\Phi}} $$

(2.1)

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

$$ \mathcal{PI}_{\Omega}(P,Q)\leq M\frac{\varphi(\Theta)}{t^{n-1}}\frac{\partial\varphi( \Phi)}{\partial n_{\Phi}}+M \frac{r\varphi(\Theta)}{|P-Q|^{n}}\frac{\partial\varphi( \Phi)}{\partial n_{\Phi}} $$

(2.2)

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

Let $G_{\Omega,R}(P,Q)$be the Green function of $C_{n}(\Omega,(0,R))$. Then

$$ \frac{\partial G_{\Omega,R}(P,Q)}{\partial R}\leq M r^{\aleph^{+}}R^{\aleph^{-}-1}\varphi(\Theta)\varphi( \Phi), $$

(2.3)

where $P=(r,\Theta)\in C_{n}(\Omega)$and $Q=(R,\Phi)\in S_{n}(\Omega;R)$.

3 Proof of theorem

Applied Carleman’s formula (see [6–8]) 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

$$ \chi\int_{S_{n}(\Omega;R)}\frac{u^{+}\varphi}{R^{1-\aleph^{-}}}\, d S_{R} \leq MKR^{\rho(R)-\aleph^{+}} $$

(3.2)

and

$$\begin{aligned}& \int_{S_{n}(\Omega;(1,R))}u^{+} \biggl(\frac{1}{t^{-\aleph^{-}}}- \frac {t^{\aleph^{+}}}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, d\sigma_{Q} \\& \quad \leq MK\int_{1}^{R} \biggl(r^{\rho(r)-\aleph^{+}-1}- \frac{r^{\rho(r)-\aleph^{-}-1}}{R^{\chi }} \biggr) \frac{\partial\varphi}{\partial n} \, dr \\& \quad \leq MKR^{\rho(R)-\aleph^{+}}. \end{aligned}$$

(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

$$ \chi\int_{S_{n}(\Omega;R)}\frac{u^{-}\varphi}{R^{1-\aleph^{-}}} \, d S_{R} \leq MKR^{\rho(R)-\aleph^{+}} $$

(3.5)

and

$$ \int_{S_{n}(\Omega;(1,R))}u^{-} \biggl(\frac{1}{t^{-\aleph^{-}}}- \frac {t^{\aleph^{+}}}{R^{\chi}} \biggr) \frac{\partial\varphi}{\partial n} \, d\sigma_{Q} \leq MKR^{\rho(R)-\aleph^{+}}. $$

(3.6)

And (3.6) gives

$$\begin{aligned}& \int_{S_{n}(\Omega;(1,R))}u^{-}t^{\aleph^{-}}\frac{\partial \varphi}{\partial n} \, d\sigma_{Q} \\& \quad \leq MK \frac{(\rho(R)+1)^{\chi}}{(\rho(R)+1)^{\chi}-(\rho(R))^{\chi}} \biggl(\frac{\rho(R)+1}{\rho(R)}R \biggr)^{\rho(\frac{\rho(R)+1}{\rho (R)}R)-\aleph^{+}}. \end{aligned}$$

Thus

$$ \int_{S_{n}(\Omega;(1,R))}u^{-}t^{\aleph^{-}}\frac{\partial\varphi }{\partial n} \, d\sigma_{Q} \leq MK\rho(R)R^{\rho(R)-\aleph^{+}}. $$

(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

$$ -u(P)=\sum_{i=1}^{4} I_{i}(P) $$

(3.9)

from (3.8), where

$$\begin{aligned}& I_{1}(P)=\int_{S_{n}(\Omega;(0,1])}\mathcal {PI}_{\Omega}(P,Q)-u(Q) \, d\sigma_{Q}, \\& I_{2}(P)=\int_{S_{n}(\Omega;(1,\frac{4}{5}r])}\mathcal {PI}_{\Omega}(P,Q)-u(Q) \, d\sigma_{Q}, \\& I_{3}(P)=\int_{S_{n}(\Omega;(\frac{4}{5}r,R))}\mathcal {PI}_{\Omega}(P,Q)-u(Q) \, d\sigma_{Q}, \\& I_{4}(P)=\int_{S_{n}(\Omega;R)}\mathcal {PI}_{\Omega}(P,Q)-u(Q) \, d\sigma_{Q}. \end{aligned}$$

Then from (2.1) and (3.7) we have

$$ I_{1}(P)\leq MK\varphi(\Theta) $$

(3.10)

and

$$ I_{2}(P) \leq MK\rho(R)R^{\rho(R)}\varphi(\Theta). $$

(3.11)

By (2.2), we consider the inequality

$$ I_{3}(P)\leq I_{31}(P)+I_{32}(P), $$

(3.12)

where

$$I_{31}(P)=M\int_{S_{n}(\Omega;(\frac{4}{5}r,R))}\frac{-u(Q) \varphi(\Theta)}{t^{n-1}} \frac{\partial\varphi( \Phi)}{\partial n_{\Phi}}\, d\sigma_{Q} $$

and

$$I_{32}(P)=Mr\varphi(\Theta)\int_{S_{n}(\Omega;(\frac{4}{5}r,R))} \frac{-u(Q) r\varphi(\Theta)}{|P-Q|^{n}} \frac{\partial\varphi( \Phi)}{\partial n_{\Phi}}\, d\sigma_{Q}. $$

We first have

$$ I_{31}(P) \leq MK\rho(R)R^{\rho(R)}\varphi(\Theta) $$

(3.13)

from (3.7). Next, we shall estimate $I_{32}(P)$. Take a sufficiently small positive number k such that

$$S_{n}\biggl(\Omega;\biggl(\frac{4}{5}r,R\biggr)\biggr)\subset B \biggl(P,\frac{1}{2}r\biggr) $$

for any $P=(r,\Theta)\in\Pi(k)$, where

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

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

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

$$ I_{32}(P) \leq MK\rho(R)R^{\rho(R)}\varphi(\Theta), $$

(3.14)

which is similar to the estimate of $I_{31}(P)$.

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

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

where

$$\delta(P)=\inf_{Q\in \partial{C_{n}(\Omega)}}|P-Q|. $$

Since

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

we have

$$I_{32}(P)=M\sum_{i=1}^{i(P)}\int _{H_{i}(P)}\frac{-u(Q)r\varphi(\Theta)}{|P-Q|^{n}}\frac {\partial \varphi( \Phi)}{\partial n_{\Phi}}\, d \sigma_{Q}, $$

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

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

$$\int_{H_{i}(P)}\frac{-u(Q)r\varphi(\Theta)}{|P-Q|^{n}}\frac{\partial \varphi( \Phi)}{\partial n_{\Phi}}d \sigma_{Q}\leq MK\rho(R)R^{\rho(R)}\varphi^{1-n}(\Theta) $$

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

So

$$ I_{32}(P)\leq MK\rho(R)R^{\rho(R)}\varphi^{1-n}( \Theta). $$

(3.15)

From (3.12), (3.13), (3.14) and (3.15) we see that

$$ I_{3}(P)\leq MK\rho(R)R^{\rho(R)}\varphi^{1-n}( \Theta). $$

(3.16)

On the other hand, we have from (2.3) and (3.5) that

$$ I_{4}(P) \leq MKR^{\rho(R)}\varphi(\Theta). $$

(3.17)

We thus obtain from (3.10), (3.11), (3.16) and (3.17) that

$$ -u(P)\leq MK\bigl(1+\rho(R)R^{\rho(R)}\bigr)\varphi^{1-n}(\Theta). $$

(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 that $-u(P)= I_{1}(P)+I_{5}(P)+I_{4}(P)$, where $I_{1}(P)$ and $I_{4}(P)$ are defined in Case 1 and

$$I_{5}(P)=\int_{S_{n}(\Omega;(1,R))}\mathcal {PI}_{\Omega}(P,Q)-u(Q) \, d\sigma_{Q}. $$

Similar to the estimate of $I_{3}(P)$ in Case 1 we have

$$ I_{5}(P)\leq MK\rho(R)R^{\rho(R)}\varphi^{1-n}( \Theta), $$

(3.19)

which together with (3.10) and (3.17) gives (3.18).

Case 3. $P=(r,\Theta)\in C_{n}(\Omega;(0,\frac{4}{5}])$.

It is evident from (1.4) that we have $-u\leq K$, which also gives (3.18).

From (3.18) we finally have

$$u(P)\geq -KM\bigl(1+\rho(R)R^{\rho(R)}\bigr)\varphi^{1-n}\theta, $$

which is the conclusion of Theorem 1.