Abstract
Our aim in this paper is to obtain Matsaev type inequalities about harmonic functions on smooth cones, which generalize the results obtained by Xu, Yang and Zhao in a half space.
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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 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 \(x_{n}=r\cos\theta_{1}\).
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).
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 \(\varphi(\Theta)=(2n w_{n}^{-1})^{1/2}\cos\theta_{1}\).
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
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 \(u(z)\).
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 \(u(P)\).
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 \(u(P)\).
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 \(u(P)\).
Remark
From the corollary, we know that conditions (1.1) and (1.2) may be replaced with the weaker conditions
and
respectively.
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
Lemma 2
for any \(P=(r,\Theta)\in C_{n}(\Omega)\)and any \(Q=(t,\Phi)\in S_{n}(\Omega)\)satisfying \(0<\frac{t}{r}\leq\frac{4}{5}\),
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
where \(P=(r,\Theta)\in C_{n}(\Omega)\)and \(Q=(R,\Phi)\in S_{n}(\Omega;R)\).
3 Proof of Theorem 1
Lemma 1 applied to \(u=u^{+}-u^{-}\) gives
It immediately follows from (1.3) that
and
Notice that
Hence from (3.1), (3.2), (3.3), and (3.4) we have
and
Equation (3.6) gives
Thus
By the Riesz decomposition theorem (see [7]), for any \(P=(r,\Theta)\in C_{n}(\Omega;(0,R))\) we have
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
Then from (2.1) and (3.7) we have
and
By (2.2), we consider the inequality
where
and
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 \(C_{n}(\Omega)-\Pi(k)\).
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 \(I_{31}(P)\).
We shall consider the case \(P=(r,\Theta)\in\Pi(k)\). Now put
where \(\delta(P)=\inf_{Q\in\partial{C_{n}(\Omega)}}|P-Q|\).
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 \(2^{i(P)-1}\delta(P)\leq\frac{r}{2}<2^{i(P)}\delta(P)\).
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 \(i=0,1,2,\ldots,i(P)\).
So
From (3.12), (3.13), (3.14), and (3.15) we see that
On the other hand, we have from (2.3) and (3.5) that
We thus obtain (3.10), (3.11), (3.16), and (3.17) that
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
Similar to the estimate of \(I_{3}(P)\) in Case 1 we have
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
which is the conclusion of Theorem 1.
Change history
10 March 2020
The Editors-in-Chief have retracted this article [1] because it significantly overlaps with a previously published article [2]. In addition, the identity of the corresponding author could not be verified: Roskilde University have confirmed that Beatriz Ychussie has not been affiliated with their institution. The authors have not responded to correspondence regarding this retraction.
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Acknowledgements
This work was partially supported by NSF Grant DMS-0913205.
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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.
The Editors-in-Chief have retracted this article because it significantly overlaps with a previously published article (Lei Qiao & Guoshuang Pan 2016). In addition, the identity of the corresponding author could not be verified: Roskilde University have confirmed that Beatriz Ychussie has not been affiliated with their institution. The authors have not responded to correspondence regarding this retraction.
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Pang, S., Ychussie, B. RETRACTED ARTICLE: Matsaev type inequalities on smooth cones. J Inequal Appl 2015, 108 (2015). https://doi.org/10.1186/s13660-015-0621-8
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DOI: https://doi.org/10.1186/s13660-015-0621-8