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.
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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
and if \(n\geq3\), then
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
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 \(\tau_{\Omega}\) 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
(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
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
where
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
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
In this paper, we consider the functions g satisfying
for \(0\leq p<\infty\) and \(\gamma\in\mathbf {R}\).
We define the positive measure λ on \(\mathbf{R}^{n}\) by
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
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
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
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
such that
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
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. \(0<\frac{r}{t}\leq\frac{4}{5}\));
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
Proof
Set
If \(P=(r,\Theta)\in E_{k}(\epsilon; \lambda, \beta)\), then there exists a positive number \(\rho(P)\) such that
\(E_{k}(\epsilon; \lambda, \beta)\) 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
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
Hence we obtain
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
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
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
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
where
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:
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,
By (2.3), we consider the inequality
where
We first have
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
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
which is similar to the estimate of \(PI_{41}(P)\).
We shall consider the case \(P=(r,\Theta)\in\Pi(b)\). Now put
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 \(2^{i(P)-1}\delta(P)\leq\frac{r}{2}<2^{i(P)}\delta(P)\).
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
for \(i=0,1,2,\ldots,i(P)\).
Since \(P=(r,\Theta)\notin E(\epsilon;\lambda, n-\alpha)\), we have
and
So
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.
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08 September 2020
This article has been retracted. Please see the retraction notice for more detail: https://doi.org/10.1186/s13660-015-0919-6.
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Acknowledgements
This work was completed while the third author was visiting the Department of Mathematics of the University of Delaware as a visiting professor, and he is grateful to the department for their support. The first author was supported by the Scientific and Technological Research Project of Henan Province (No. 152102310089). In the meanwhile, the authors wish to express their genuine thanks to the anonymous referees for careful reading and excellent comments on this manuscript.
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This article has been retracted. Please see the retraction notice for more detail:https://doi.org/10.1186/s13660-020-02486-7"
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Jiang, Z., Hou, L. & Peixoto-de-Büyükkurt, C. RETRACTED ARTICLE: Growth property at infinity of harmonic functions. J Inequal Appl 2015, 401 (2015). https://doi.org/10.1186/s13660-015-0919-6
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DOI: https://doi.org/10.1186/s13660-015-0919-6