Abstract
In this paper, we present new Poissontype 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.
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1 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 ndimensional Euclidean space and \(X=(x_{1},x_{2},\ldots,x_{n1})\). We introduce spherical coordinates for \(G=(r,\Xi)\) (\(\Xi=(\theta_{1},\theta _{2},\ldots ,\theta_{n1})\)) by \(x=r\),
where \(0\leq r<+\infty\), \(\frac{1}{2}\pi\leq\theta_{n1}<\frac{3}{2}\pi\) and \(0\leq\theta_{j}\leq\pi\) for \(1\leq j\leq n2\) (\(n\geq3\)).
We denote the unit sphere and the upper half unit sphere by \(\mathbf{ S}^{n1}\) and \(\mathbf{S}_{+}^{n1}\), respectively. Let \(\Sigma\subset \mathbf{ S}^{n1}\). 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 \(\mathbf{ R}_{+}\times\mathbf{S}_{+}^{n1}\) is called the upperhalf 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
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
where \(\Lambda_{n}\) is the Beltrami operator. Now we consider the boundary value problem
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 \(t^{2}+(n2)t\tau_{\Sigma}=0\), Then \(\iota _{\Sigma}+\kappa_{\Sigma}\) is denoted by \(\varrho_{\Sigma}\) for the sake of simplicity.
Remark 1
In the case \(\Sigma=\mathbf{S}_{+}^{n1}\), it follows that

(I)
\(\iota_{\Sigma}=1\) and \(\kappa_{\Sigma}=n1\).

(II)
\(h_{\Sigma}(\Xi)=\sqrt{\frac{2n}{ w_{n}}}\cos\theta_{1}\), where \(w_{n}\) is the surface area of \(\mathbf{S}^{n1}\).
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])
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.126128).
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 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:

(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
Definition 5
Let ν be any positive measure in \(\beth_{n}(\Sigma)\) satisfying
for any \(G\in\beth_{n}(\Sigma)\). Then the positive measure \(\nu'\) is defined by
Definition 6
Let μ and ν be defined in Definitions 3 and 4, respectively. Then the positive measure ξ is defined by
where
Remark 5
Let \(\Sigma=\mathbf{S}_{+}^{n1}\). Then
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
Remark 6
Let \(g(H)\) be a continuous function on \(\daleth_{n}(\Sigma)\). If \(d\mu =g\,d\sigma_{H}\), then we define
Remark 7
Let \(\Sigma=\mathbf{S}_{+}^{n1}\). Then we define
where
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
for any \(G=(r,\Xi)\in \mathbf{R}^{n}\{O\}\), where \(\beta\geq0\). The exceptional set can be defined by
where ϵ is a sufficiently small positive number.
Remark 8
Let \(\beta>0\) and \(\lambda(\{P\})>0\) for any \(P\neq O\). Then

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

(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 Poissontype 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 SchrödingerPoissontype inequalities for PoissonSchrödinger integrals and gave their related applications.
Theorem A
Let g be a measurable function on \(\partial{\mathcal{T}_{n}}\) satisfying
Then the harmonic function \(\mathcal{POI}_{\mathbf{S}_{+}^{n1}}[g](x)=\int_{\partial{\mathcal {T}_{n}}}\mathcal{POI}_{\mathbf{S}_{+}^{n1}}(x,y)g(y)\,dy\) satisfies
as \(x\rightarrow\infty\) in \(\mathcal{T}_{n}\).
2 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
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
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
Then the Poisson integral \(\mathcal{POI}_{\Sigma}[g](G)\) is harmonic in \(\beth_{n}(\Sigma)\) and
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 \(\Sigma=\mathbf{S}_{+}^{n1}\), 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 \(\Sigma=\mathbf{S}_{+}^{n1}\). We immediately have the following results from Theorem 1.
Corollary 2
If μ is a positive measure on \(\partial{\mathcal{T}_{n}}\) satisfying \(\mathcal{POI}_{\mathbf{S}_{+}^{n1}}\mu(x)\not\equiv+\infty\) for any \(x=(X,x_{n})\in\mathcal{T}_{n}\), then
for any \(x\in \mathcal{T}_{n}\mathbb{EX}(\epsilon;\mu',n1)\) as \(x \rightarrow \infty \), where \(\mathbb{EX}(\epsilon;\mu',n1)\) is a subset of \(\beth _{n}(\Sigma)\) and has a covering \(\{r_{k},R_{k}\}\) satisfying
Corollary 3
Let μ be defined as in Corollary 2. Then
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
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
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
where \(G\in\beth_{n}(\Sigma)\), \(c_{5}(w)\), and \(c_{6}(w)\) are two constants dependent of w satisfying
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
for any \(G\in\beth_{n}(\Sigma) \mathbb{EX}(\epsilon;\xi,n1)\) as \(r \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\xi,n1)\) 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 \(x\in\mathcal{T}_{n} \mathbb{EX}(\epsilon;\varrho,n1)\) as \(x \rightarrow\infty\), where \(\mathbb{EX}(\epsilon;\varrho,n1)\) is a subset of \(\beth_{n}(\Sigma )\) and has a covering satisfying (8).
3 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:
for \(0<\frac{t}{r}\leq\frac{4}{5}\),
for \(0<\frac{r}{t}\leq\frac{4}{5}\), and
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
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
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).
4 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
where
We have the following estimates:
from (12), (13), and [11], Lemma 4.
By (14), we write
where
We first have
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
for any \(G=(r,\Xi)\in\Lambda(k_{1})\), where
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
where \(H\in \daleth_{n}(\Sigma)\) and \(k_{1}'\) is a positive number. So
from [11], Lemma 4.
If \(G\in\Lambda(k_{1})\), we put
Since \(\daleth_{n}(\Sigma)\cap\{H\in\mathbf{R}^{n}: GH< \varrho(G)\}=\varnothing\), we have
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
for \(l=0,1,2,\ldots,l(G)\).
Since \(G=(r,\Xi)\notin\mathbb{EX}(\epsilon; \mu', n\alpha)\), we have
for \(l=0,1,2,\ldots,l(G)1\) and
So
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.
5 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
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)
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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31 August 2021
This article has been retracted. Please see the Retraction Notice for more detail: https://doi.org/10.1186/s13660021026869
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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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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/s13660021026869
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Luan, K., Vieira, J. RETRACTED ARTICLE: Poissontype inequalities for growth properties of positive superharmonic functions. J Inequal Appl 2017, 12 (2017). https://doi.org/10.1186/s1366001612787
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DOI: https://doi.org/10.1186/s1366001612787