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RETRACTED ARTICLE: Poisson-type inequalities for growth properties of positive superharmonic functions

This article was retracted on 31 August 2021

This article has been updated

Abstract

In this paper, we present new Poisson-type inequalities for Poisson integrals with continuous data on the boundary. The obtained inequalities are used to obtain growth properties at infinity of positive superharmonic functions in a smooth cone.

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 \(|x|=r\),

$$\left \{ \textstyle\begin{array}{l@{\quad}l} x_{n}=r\cos\theta_{1}, \qquad x_{1}=r(\prod_{j=1}^{n-1}\sin\theta_{j}),& n= 2, \\ x_{n-m+1}=r(\prod_{j=1}^{m-1}\sin\theta_{j})\cos\theta_{m}, & n\geq3, \end{array}\displaystyle \right . $$

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 be a subset of \(\beth _{n}(\Sigma )\). If there exists a sequence of 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 \(B_{k}\) (see [1]).

In spherical coordinate 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

  1. (I)

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

  2. (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])

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

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

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 [3]).

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 [4], 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:

  1. (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 \(\beth_{n}(\Sigma)\).

  2. (II)

    The Green potential in \(\beth_{n}(\Sigma)\) 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:

  1. (I)

    The Poisson integral with μ can be defined by

    $$\mathcal{POI}_{\Sigma} \mu(G)= \int_{\daleth_{n}(\Sigma)}\mathcal{POI}_{\Sigma}(G,H)\,d\mu(H). $$
  2. (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

$$d\mu'=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{\partial h_{\Sigma}(\Omega)}{\partial n_{\Omega}} t^{-\kappa _{\Sigma}-1}\,d\mu& \mbox{on } \daleth_{n}(\Sigma; (1,+\infty)) ,\\ 0 & \mbox{on } \mathbf{R}^{n}-\daleth_{n}(\Sigma; (1,+\infty)). \end{array}\displaystyle \right . $$

Definition 5

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

$$ \mathcal{GF}_{\Sigma} \nu(G)\not\equiv+\infty $$
(1)

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

$$d\nu'=\left \{ \textstyle\begin{array}{l@{\quad}l} h_{\Sigma}(\Omega) t^{-\kappa_{\Sigma}} \,d\nu& \mbox{on } \beth _{n}(\Sigma; (1,+\infty)) ,\\ 0& \mbox{on } \mathbf{R}^{n}-\beth_{n}(\Sigma; (1,+\infty)). \end{array}\displaystyle \right . $$

Definition 6

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

$$d\xi=\left \{ \textstyle\begin{array}{l@{\quad}l} t^{-1-\kappa_{\Sigma}}\, d\xi' & \mbox{on } \overline{\beth _{n}(\Sigma ; (1,+\infty))} ,\\ 0& \mbox{on } \mathbf{R}^{n}-\overline{\beth_{n}(\Sigma; (1,+\infty))}, \end{array}\displaystyle \right . $$

where

$$d\xi'=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{\partial h_{\Sigma}(\Omega)}{\partial n_{\Omega}}\,d\mu(H) & \mbox{on } \daleth_{n}(\Sigma; (1,+\infty)) ,\\ h_{\Sigma}(\Omega)t\,d\nu(H)& \mbox{on } \beth_{n}(\Sigma; (1,+\infty)). \end{array}\displaystyle \right . $$

Remark 5

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

$$\mathcal{GF}_{\mathbf{S}_{+}^{n-1}}(G,H)=\left \{ \textstyle\begin{array}{l@{\quad}l} \log|G-H^{\ast}|-\log|G-H| & \mbox{if } n=2, \\ |G-H|^{2-n}-|G-H^{\ast}|^{2-n} & \mbox{if } n\geq3, \end{array}\displaystyle \right . $$

where \(G=(X,x_{n})\), \(H^{\ast}=(Y,-y_{n})\), that is, \(H^{\ast}\) is the mirror image of \(H=(Y,y_{n})\) on \(\partial{\mathcal{T}_{n}}\). 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

$$\mathcal{POI}_{\mathbf{S}_{+}^{n-1}}(G,H)=\frac{\partial}{\partial n_{y}}\mathcal{GF}_{\mathbf{S}_{+}^{n-1}}(G,H)= \left \{ \textstyle\begin{array}{l@{\quad}l} 2x_{n}|G-H|^{-2} & \mbox{if } n=2, \\ 2(n-2)x_{n}|G-H|^{-n} & \mbox{if } n\geq3. \end{array}\displaystyle \right . $$

Remark 6

Let \(g(H)\) be a continuous function on \(\daleth_{n}(\Sigma)\). If \(d\mu =|g|\,d\sigma_{H}\), then we define

$$d\mu''=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{\partial h_{\Sigma}(\Omega)}{\partial n_{\Omega }}|g|t^{-1-\kappa _{\Sigma}}\,d\sigma_{H} & \mbox{on } \daleth_{n}(\Sigma; (1,+\infty)) ,\\ 0& \mbox{on } \mathbf{R}^{n}-\daleth_{n}(\Sigma; (1,+\infty)). \end{array}\displaystyle \right . $$

Remark 7

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

$$d\varrho=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{d\varrho'}{|y|^{n}} & \mbox{on } \overline{\mathcal {T}_{n}} ,\\ 0& \mbox{on } \mathbf{R}^{n}-\overline{\mathcal{T}_{n}}, \end{array}\displaystyle \right . $$

where

$$d\varrho'(y)=\left \{ \textstyle\begin{array}{l@{\quad}l} d\mu& \mbox{on } \partial{\mathcal{T}_{n}} ,\\ y_{n}d\nu& \mbox{on } \mathcal{T}_{n}. \end{array}\displaystyle \right . $$

Definition 7

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 8

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

  1. (I)

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

  2. (II)

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

Recently, Qiao and Wang (see [5], Corollary 2.1 with \(m=0\)) proved classical Poisson-type inequalities for Poisson integrals in a half space. Applications of them were also developed by Pang and Ychussie (see [6]) and Xue and Wang (see [7]). In particular, Huang (see [8]) further obtained Schrödinger-Poisson-type inequalities for Poisson-Schrödinger integrals and gave their related applications.

Theorem A

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

$$ \int_{\partial{\mathcal{T}_{n}}}\bigl|g(y)\bigr|\bigl(1+|y|\bigr)^{-n} \,dy< \infty. $$
(2)

Then the harmonic function \(\mathcal{POI}_{\mathbf{S}_{+}^{n-1}}[g](x)=\int_{\partial{\mathcal {T}_{n}}}\mathcal{POI}_{\mathbf{S}_{+}^{n-1}}(x,y)g(y)\,dy\) satisfies

$$ \mathcal{POI}_{\mathbf{S}_{+}^{n-1}}[g]=o\bigl(|x|\sec^{n-1} \theta_{1}\bigr) $$
(3)

as \(|x|\rightarrow\infty\) in \(\mathcal{T}_{n}\).

Results

Our first aim in this paper is to prove the following result, which is a generalization of Theorem A. For similar results with respect to Schrödinger operator, we refer the reader to the literature (see [5, 9]).

Theorem 1

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

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

$$ \sum_{k=0}^{\infty}\biggl( \frac{r_{k}}{R_{k}}\biggr)^{n-\alpha}< \infty. $$
(5)

Let \(d\mu=|g|\,d\sigma_{H}\) for any \(H=(t,\Omega)\in\daleth_{n}(\Sigma )\). Then we have the following result, which generalizes Theorem A to the conical case.

Corollary 1

If g is a measurable function on \(\daleth_{n}(\Sigma)\) satisfying

$$ \int_{1}^{\infty}\frac{\int_{\partial{\Sigma}}|g(H)|\,d_{\sigma _{\Omega}}}{t^{1+\iota_{\Sigma}}}\,dt< \infty. $$
(6)

Then the Poisson integral \(\mathcal{POI}_{\Sigma}[g](G)\) is harmonic in \(\beth_{n}(\Sigma)\) and

$$ \mathcal{POI}_{\Sigma}[g](G)=o\bigl(r^{\iota_{\Sigma}}\bigl\{ h_{\Sigma}(\Xi)\bigr\} ^{1-\alpha}\bigr) $$
(7)

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 \(\{r_{k},R_{k}\}\) satisfying (5).

Remark 9

If \(\Sigma=\mathbf{S}_{+}^{n-1}\), then it is easy to see that (6) is equivalent to (2) and (5) is a finite sum, then the set \(\mathbb{EX}(\epsilon; \mu'',0)\) is a bounded set and (7) reduces to (3) in the case \(\alpha=n\) from Remark 1.

Let \(\Sigma=\mathbf{S}_{+}^{n-1}\). We immediately have the following results from Theorem 1.

Corollary 2

If μ is a positive measure on \(\partial{\mathcal{T}_{n}}\) satisfying \(\mathcal{POI}_{\mathbf{S}_{+}^{n-1}}\mu(x)\not\equiv+\infty\) for any \(x=(X,x_{n})\in\mathcal{T}_{n}\), then

$$\mathcal{POI}_{\mathbf{S}_{+}^{n-1}} \mu(x)=0\bigl(|x|\bigr) $$

for any \(x\in \mathcal{T}_{n}-\mathbb{EX}(\epsilon;\mu',n-1)\) as \(|x| \rightarrow \infty \), where \(\mathbb{EX}(\epsilon;\mu',n-1)\) is a subset of \(\beth _{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) satisfying

$$ \sum_{k=0}^{\infty}\biggl( \frac{r_{k}}{R_{k}}\biggr)^{n-1}< \infty. $$
(8)

Corollary 3

Let μ be defined as in Corollary 2. Then

$$\mathcal{POI}_{\mathbf{S}_{+}^{n-1}} \mu(x)=0(x_{n}) $$

for any \(x\in \mathcal{T}_{n}-\mathbb{EX}(\epsilon;\mu',n)\) as \(|x| \rightarrow \infty \), where \(\mathbb{EX}(\epsilon;\mu',n)\) is a subset of \(\beth _{n}(\Sigma )\) and has a covering \(\{r_{k},R_{k}\}\) satisfying

$$ \sum_{k=0}^{\infty}\biggl( \frac{r_{k}}{R_{k}}\biggr)^{n}< \infty. $$
(9)

The following result is very well known. We quote it from [10].

Theorem B

see [10]

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 \(w(x)\) can be uniquely decomposed as

$$ w(x)=cx_{n}+\mathcal{POI}_{\mathbf{S}_{+}^{n-1}} \mu(x)+ \mathcal{GF}_{\mathbf{S}_{+}^{n-1}} \nu(x), $$
(10)

where \(x=(X,X_{n})\in\mathcal{T}_{n}\) and c is a nonnegative constant.

Theorem C

see [9], Theorem 2

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 in \(\beth_{n}(\Sigma)\) such that \(w(G)\) can be uniquely decomposed as

$$ w(G)=c_{5}(w)\mathcal{MK}(G,\infty)+c_{6}(w) \mathcal{MK}(G,O)+\mathcal {POI}_{\Sigma} \mu(G)+\mathcal{GF}_{\Sigma} \nu(G), $$
(11)

where \(G\in\beth_{n}(\Sigma)\), \(c_{5}(w)\), and \(c_{6}(w)\) are two constants dependent of 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 give the growth properties of positive superharmonic functions at infinity in a cone.

Theorem 2

Let \(w(G)\) (\(\not\equiv+\infty\)) (\(G=(r,\Xi)\in\beth_{n}(\Sigma)\)) be defined by (11). Then

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

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 (8).

Theorem 2 immediately gives the following corollary.

Corollary 4

Let \(w(x)\) (\(\not\equiv+\infty\)) (\(x=(X,x_{n})\in\mathcal{T}_{n}\)) be defined by (10). Then \(w(x)-cx_{n}=o(|x|)\) for any \(x\in\mathcal{T}_{n}- \mathbb{EX}(\epsilon;\varrho,n-1)\) as \(|x| \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\varrho,n-1)\) is a subset of \(\beth_{n}(\Sigma )\) and has a covering satisfying (8).

Lemmas

In order to prove our main results we need following lemmas. In this paper let M denote various constants independent of the variables in questions, which may be different from line to line.

Lemma 1

see [4], Lemma 2

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

$$ \mathcal{POI}_{\Sigma}(G,H)\leq M r^{-\kappa _{\Sigma}}t^{\iota_{\Sigma}-1}h_{\Sigma}( \Xi)\frac{\partial}{\partial n_{\Omega}}h_{\Sigma}(\Omega) $$
(12)

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

$$ \mathcal{POI}_{\Sigma}(G,H)\leq M r^{\iota _{\Sigma}}t^{-\kappa_{\Sigma}-1}h_{\Sigma}( \Xi)\frac{\partial }{\partial n_{\Omega}}h_{\Sigma}(\Omega) $$
(13)

for \(0<\frac{r}{t}\leq\frac{4}{5}\), and

$$ \mathcal{POI}_{\Sigma}(G,H)\leq Mh_{\Sigma}(\Xi )t^{1-n}\frac{\partial}{\partial n_{\Omega}}h_{\Sigma}(\Omega )+Mrh_{\Sigma}( \Xi)|G-H|^{-n}\frac{\partial}{\partial n_{\Omega}}h_{\Sigma}(\Omega) $$
(14)

for \(\frac{4r}{5}< t\leq\frac{5r}{4}\).

Lemma 2

see [5], Lemma 5

If \(\beta\geq0\) and 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 estimation of the Green potential at infinity is the following, which is due to [5].

Lemma 3

If ν is a positive measure on \(\beth_{n}(\Sigma)\) such that (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 (5).

Proof of Theorem 1

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

Put

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

where

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

We have the following estimates:

$$\begin{aligned} & \mathcal{POI}_{\Sigma}^{1}(G) \leq Mr^{\iota_{\Sigma}}h_{\Sigma}(\Xi) \biggl(\frac{4}{5}r \biggr)^{-\varrho_{\Sigma}} \int _{\daleth_{n}(\Sigma;(0,\frac{4}{5}r])}t^{\iota_{\Sigma}-1}\frac {\partial }{\partial n_{\Omega}}h_{\Sigma}( \Omega)\,d\mu(H) \\ &\hphantom{\mathcal{POI}_{\Sigma}^{1}(G)}\leq M \epsilon r^{\iota_{\Sigma}}h_{\Sigma}(\Xi), \end{aligned}$$
(15)
$$\begin{aligned} & \mathcal{POI}_{\Sigma}^{3}(G) \leq Mr^{\iota_{\Sigma}}h_{\Sigma}(\Xi) \int_{\daleth_{n}(\Sigma;[\frac {5}{4}r,\infty))}t^{-\kappa_{\Sigma}-1}\frac{\partial}{\partial n_{\Omega}}h_{\Sigma}( \Omega)\,d\mu(H) \\ &\hphantom{\mathcal{POI}_{\Sigma}^{3}(G)}\leq M \epsilon r^{\iota_{\Sigma}}h_{\Sigma}(\Xi), \end{aligned}$$
(16)

from (12), (13), and [11], Lemma 4.

By (14), we write

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

where

$$\begin{aligned} &\mathcal{POI}_{\Sigma}^{21}(G)=M \int_{\daleth_{n}(\Sigma;(\frac {4}{5}r,\frac{5}{4}r))}t^{\kappa_{\Sigma}+1}h_{\Sigma}(\Xi)t^{1-n}\,d \mu'(H),\\ &\mathcal{POI}_{\Sigma}^{22}(G)=M \int_{\daleth_{n}(\Sigma;(\frac {4}{5}r,\frac{5}{4}r))}t^{\kappa_{\Sigma}+1}rh_{\Sigma}(\Xi )|G-H|^{-n}\,d\mu'(H). \end{aligned} $$

We first have

$$\begin{aligned} \mathcal{POI}_{\Sigma}^{21}(G) \leq& Mr^{\iota_{\Sigma}}h_{\Sigma}(\Xi) \int_{\daleth_{n}(\Sigma;(\frac {4}{5}r,\infty))}d\mu'(H) \\ \leq& M \epsilon r^{\iota_{\Sigma}}h_{\Sigma}(\Xi) \end{aligned}$$
(17)

from [11], Lemma 4.

Next, we shall estimate \(\mathcal{POI}_{\Sigma}^{22}(G)\). We can find a number \(k_{1}\) satisfying \(k_{1}\geq0\) and

$$\daleth_{n}\biggl(\Sigma;\biggl(\frac{4}{5}r, \frac{5}{4}r\biggr)\biggr)\subset B\biggl(G,\frac{r}{2}\biggr) $$

for any \(G=(r,\Xi)\in\Lambda(k_{1})\), where

$$\Lambda(k_{1})=\Bigl\{ G=(r,\Xi)\in\beth_{n}(\Sigma); \inf _{z\in\partial \Sigma }\bigl|(1,\Xi)-(1,z)\bigr|< k_{1}, 0< r< \infty\Bigr\} . $$

Then the set \(\beth_{n}(\Sigma)\) can be split into two sets \(\Lambda (k_{1})\) and \(\beth_{n}(\Sigma)-\Lambda(k_{1})\).

Let \(G=(r,\Xi)\in\beth_{n}(\Sigma)-\Lambda(k_{1})\). Then

$$|G-H|\geq k_{1}'r, $$

where \(H\in \daleth_{n}(\Sigma)\) and \(k_{1}'\) is a positive number. So

$$\begin{aligned} \mathcal{POI}_{\Sigma}^{22}(G) \leq& Mr^{\iota_{\Sigma}}h_{\Sigma}(\Xi) \int_{\daleth_{n}(\Sigma;(\frac {4}{5}r,\infty))}d\mu'(H) \\ \leq& M \epsilon r^{\iota_{\Sigma}}h_{\Sigma}(\Xi) \end{aligned}$$
(18)

from [11], Lemma 4.

If \(G\in\Lambda(k_{1})\), we put

$$F_{l}(G)=\biggl\{ H\in\daleth_{n}\biggl(\Sigma;\biggl( \frac{4}{5}r,\frac{5}{4}r\biggr)\biggr); 2^{l-1}\varrho(G) \leq|G-H|< 2^{l}\varrho(G)\biggr\} . $$

Since \(\daleth_{n}(\Sigma)\cap\{H\in\mathbf{R}^{n}: |G-H|< \varrho(G)\}=\varnothing\), we have

$$\mathcal{POI}_{\Sigma}^{22}(G)=M\sum _{i=1}^{l(G)} \int _{F_{l}(G)}t^{\kappa _{\Sigma}+1}rh_{\Sigma}( \Xi)|G-H|^{-n}\,d\mu'(H), $$

where \(l(G)\) is a positive integer satisfying \(2^{l(G)-1}\varrho(G)\leq\frac{r}{2}<2^{l(G)}\varrho(G)\).

By Remark 3 we have \(rh_{\Sigma}(\Xi)\leq M\varrho(G)\) (\(G=(r,\Xi)\in\beth_{n}(\Sigma)\)), and hence

$$\begin{aligned} \int_{F_{l}(G)}\frac{t^{\kappa_{\Sigma}+1}rh_{\Sigma}(\Xi )}{|G-H|^{n}}\,d\mu'(H) \leq& Mr^{\kappa_{\Sigma}-\alpha+2}\bigl\{ h_{\Sigma}(\Xi)\bigr\} ^{1-\alpha}\mu '\bigl(F_{l}(G)\bigr)\bigl\{ 2^{l}\varrho(G) \bigr\} ^{\alpha-n} \end{aligned}$$

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

Since \(G=(r,\Xi)\notin\mathbb{EX}(\epsilon; \mu', n-\alpha)\), we have

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

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

$$\mu'\bigl(F_{l(G)}(G)\bigr)\bigl\{ 2^{l}\varrho(G) \bigr\} ^{\alpha-n}\leq\mu'\biggl(B\biggl(G,\frac {r}{2} \biggr)\biggr) \biggl(\frac{r}{2}\biggr)^{\alpha-n}\leq\epsilon r^{\alpha-n}. $$

So

$$ \mathcal{POI}_{\Sigma}^{22}(G)\leq M \epsilon r^{\iota_{\Sigma}}\bigl\{ h_{\Sigma}(\Xi)\bigr\} ^{1-\alpha}. $$
(19)

From (15), (16), (17), (18), (19), and Remark 4, we obtain \(\mathcal{POI}_{\Sigma}\mu(G)=o(r^{\iota_{\Sigma}}\{h_{\Sigma}(\Xi)\} ^{1-\alpha})\) for any \(G=(r,\Xi)\in\beth_{n}(\Sigma; (L,+\infty))-\mathbb{EX}(\epsilon; \mu', n-\alpha)\) as \(r\rightarrow\infty\), where L is a sufficiently large real number. With Lemma 3 we have the conclusion of Theorem 1.

Proof of Corollary 1

Let \(G=(r,\Xi) \) be a fixed point in \(\beth_{n}(\Sigma)\). Then there exists a number R satisfying \(\max\{\frac{5r}{4},1\}< R\). There exists a positive constant \(M'\) such that

$$ \mathcal{POI}_{\Sigma}(G,H)\leq M' r^{\iota_{\Sigma}}t^{-\kappa _{\Sigma}-1}h_{\Sigma}(\Xi) $$
(20)

from Remark 2 and (13), where \(H=(t,\Omega)\in\daleth _{n}(\Sigma )\) satisfying \(0<\frac{r}{t}\leq \frac{4}{5}\).

Let \(M=M'c_{n}^{-1}r^{\iota_{\Sigma}}h_{\Sigma}(\Xi)\). Then we have from (6) and (20)

$$\begin{aligned} \int_{\daleth_{n}(\Sigma;(R,+\infty))}\bigl|g(H)\bigr|\mathcal{POI}_{\Sigma }(G,H)\,d \sigma_{H} \leq& M \int_{R}^{\infty}t^{-\iota_{\Sigma}-1}\biggl( \int_{\partial{\Sigma }}\bigl|g(t,\Omega )\bigr|\,d_{\sigma_{\Omega}}\biggr)\,dt< \infty. \end{aligned}$$

For any \(G\in\beth_{n}(\Sigma)\), it is easy to see that \(\mathcal {POI}_{\Sigma}[g](G)\) is finite, which means that \(\mathcal{POI}_{\Sigma}[g](G)\) is a harmonic function of \(G\in \beth_{n}(\Sigma)\). Meanwhile, Theorem 1 gives (7). The proof of Corollary 1 is completed.

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Acknowledgements

The project is partially supported by the Applied Technology Research and the Development Foundation of Heilongjiang Province (Grant No. GC13A308).The authors would like to thank the referees and the editor for their careful reading and some useful comments on improving the presentation of this paper.

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Correspondence to John Vieira.

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JV completed the main study. KL pointed out some mistakes and verified the calculation. Both authors read and approved the final manuscript.

This article has been retracted. Please see the retraction notice for more detail: https://doi.org/10.1186/s13660-021-02686-9

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Luan, K., Vieira, J. RETRACTED ARTICLE: Poisson-type inequalities for growth properties of positive superharmonic functions. J Inequal Appl 2017, 12 (2017). https://doi.org/10.1186/s13660-016-1278-7

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Keywords

  • Poisson-type inequality
  • continuous data
  • growth property