Let Γ be the subset of the upper half unit sphere. The set \({\mathbf {R}}_{+}\times\Gamma\) in \({\mathbf {R}}^{n}\) is called a cone. We denote it by \(\mathfrak{C}_{n}(\Gamma)\), where \(\Gamma\subset{\mathbf {S}}_{1}\). The sets \(I\times\Gamma\) and \(I\times \partial{\Gamma}\) with an interval on R are denoted by \(\mathfrak{C}_{n}(\Gamma;I)\) and \(\mathfrak{S}_{n}(\Gamma;I)\), respectively. We denote \(\mathfrak {C}_{n}(\Gamma)\cap S_{R}\) and \(\mathfrak{S}_{n}(\Gamma; (0,+\infty))\) by \(\mathfrak {S}_{n}(\Gamma; R)\) and \(\mathfrak{S}_{n}(\Gamma)\), respectively.
Furthermore, we denote by dσ (resp. \(dS_{R}\)) the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(\partial{\mathfrak{C}_{n}(\Gamma)}\) (resp. \(S_{R}\)) and by dw the elements of the Euclidean volume in \({\mathbf {R}}^{n}\).
It is well known (see, e.g., [2], p.41) that
$$\begin{aligned} &{\Delta^{*}\varphi(\Theta)+\lambda\varphi(\Theta)=0\quad \textrm{in } \Gamma,} \\ \\ &{\varphi(\Theta)=0\quad\textrm{on } \partial{\Gamma},} \end{aligned}$$
(1.1)
where \(\Delta^{*}\) is the Laplace-Beltrami operator. We denote the least positive eigenvalue of this boundary value problem (1.1) by λ and the normalized positive eigenfunction corresponding to λ by \(\varphi(\Theta)\), \(\int_{\Gamma}\varphi^{2}(\Theta)\,dS_{1}=1\).
We remark that the function \(r^{\aleph^{\pm}}\varphi(\Theta)\) is harmonic in \(\mathfrak{C}_{n}(\Gamma)\), belongs to the class \(C^{2}(\mathfrak{C}_{n}(\Gamma )\backslash\{O\})\) and vanishes on \(\mathfrak{S}_{n}(\Gamma)\), where
$$2\aleph^{\pm}=-n+2\pm\sqrt{(n-2)^{2}+4\lambda}. $$
For simplicity we shall write χ instead of \(\aleph^{+}-\aleph^{-}\).
For simplicity we shall assume that the boundary of the domain Γ is twice continuously differentiable, \(\varphi\in C^{2}(\overline{\Gamma})\) and \(\frac{\partial\varphi}{\partial n}>0\) on ∂Γ. Then (see [3], p.7-8)
$$ \operatorname{dist}(\Theta,\partial{\Gamma})\approx\varphi(\Theta), $$
(1.2)
where \(\Theta\in\Gamma\).
Let \(\delta(P)=\operatorname{dist}(P,\partial{\mathfrak{C}_{n}(\Gamma)})\), we have
$$ \varphi(\Theta)\approx\delta(P), $$
(1.3)
for any \(P=(1,\Theta)\in\Gamma\) (see [4]).
Let \(u(r,\Theta)\) be a function on \(\mathfrak{C}_{n}(\Gamma)\). For any given \(r\in{\mathbf {R}}_{+}\), the integral
$$\int_{\Gamma}u(r,\Theta)\varphi(\Theta)\,d S_{1}, $$
is denoted by \(\mathcal{N}_{u}(r)\), when it exists. The finite or infinite limits
$$\lim_{r\rightarrow\infty}r^{-\aleph^{+}}\mathcal{N}_{u}(r)\quad\textrm{and}\quad \lim_{r\rightarrow0}r^{-\aleph^{-}}\mathcal{N}_{u}(r) $$
are denoted by \(\mathscr{U}_{u}\) and \(\mathscr{V}_{u}\), respectively, when they exist.
Remark 1
A function \(g(t)\) on \((0,\infty)\) is \(\mathbb{A}_{d_{1},d_{2}}\)-convex if and only if \(g(t)t^{d_{2}}\) is a convex function of \(t^{d}\)
\((d=d_{1}+d_{2})\) on \((0,\infty)\), or, equivalently, if and only if \(g(t)t^{-d_{1}}\) is a convex function of \(t^{-d}\) on \((0,\infty)\).
Remark 2
\(\mathcal{N}_{u}(r)\) is a \(\mathbb{A}_{\aleph^{+},\gamma-1}\)-convex on \((0,\infty)\), where u is a subharmonic function on \(\mathfrak{C}_{n}(\Gamma)\) such that
$$ \limsup_{P\in\mathfrak{C}_{n}(\Gamma),P\rightarrow Q \in \partial{\mathfrak{C}_{n}(\Gamma)}}u(P)\leq c, $$
(1.4)
where c is a nonnegative number (see [5]).
The function
$$\mathbb{P}_{\mathfrak{C}_{n}(\Gamma)}(P,Q)=\frac{\partial\mathbb {G}_{\mathfrak{C}_{n}(\Gamma)}(P,Q)}{\partial n_{Q}} $$
is called the ordinary Poisson kernel, where \(\mathbb{G}_{\mathfrak{C}_{n}(\Gamma)}\) is the Green function.
The Poisson integral of g relative to \(\mathfrak{C}_{n}(\Gamma)\) is defined by
$$\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)} [g](P)=\frac{1}{c_{n}} \int_{\mathfrak{S}_{n}(\Gamma)}\mathbb {P}_{\mathfrak{C}_{n}(\Gamma)}(P,Q)g(Q)\,d\sigma, $$
where g is a continuous function on \(\partial{\mathfrak{C}_{n}(\Gamma)}\) and \(\frac{\partial}{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into \(\mathfrak{C}_{n}(\Gamma)\).
We set functions f satisfying
$$ \int_{\mathfrak{S}_{n}(\Gamma)}\frac{ \vert f(t,\Phi) \vert ^{p}}{1+t^{\gamma }}\,d\sigma< \infty, $$
(1.5)
where \(-1< p<+\infty\) and
$$\frac{-\aleph^{+}-n+2}{p}< \gamma< \frac{-\aleph^{+}-n+2}{p}+n-1. $$
Let \(-1< p<+\infty\). we denote \(\mathcal{A}_{\Gamma}\) the class of all measurable functions \(g(t,\Phi)\)
\((Q=(t,\Phi)=(Y, y_{n})\in \mathfrak{C}_{n}(\Gamma))\) satisfying the following inequality:
$$ \int_{\mathfrak{C}_{n}(\Gamma)}\frac{ \vert g(t,\Phi) \vert ^{p-1}\varphi }{1+t^{\gamma-3}}\,dw< \infty $$
and the class \(\mathcal{B}_{\Gamma}\), consists of all measurable functions \(h(t,\Phi)\)
\(((t,\Phi)=(Y, y_{n})\in\mathfrak{S}_{n}(\Gamma))\) satisfying
$$ \int_{\mathfrak{S}_{n}(\Gamma)}\frac{ \vert h(t,\Phi) \vert ^{q}}{1+t^{\gamma}}\frac{\partial\varphi}{\partial n}\,d\sigma < \infty, $$
where \(q>0\).
We will also consider the class of all continuous functions \(u(t,\Phi)\)
\(((t,\Phi)\in\overline{\mathfrak{C}_{n}(\Gamma)})\) harmonic in \(\mathfrak{C}_{n}(\Gamma)\) with \(u^{+}(t,\Phi)\in \mathcal{A}_{\Gamma}\)
\(((t,\Phi)\in\mathfrak{C}_{n}(\Gamma))\) and \(u^{+}(t,\Phi)\in\mathcal{B}_{\Gamma}\)
\(((t,\Phi)\in\mathfrak {S}_{n}(\Gamma))\) is denoted by \(\mathcal{C}_{\Gamma}\) (see [6]).
In 2015, Jiang, Hou and Peixoto-de-Büyükkurt (see [7]) obtained the following result.
Theorem A
Let
g
be a measurable function on
\(\partial{T_{n}}\)
such that
$$\int_{\partial{T_{n}}}\bigl(1+ \vert Q \vert \bigr)^{2-n} \bigl\vert g(Q) \bigr\vert \,dQ< \infty. $$
Then the harmonic function
\(\mathbb{PI}_{T_{n}}[g]\)
satisfies
\(\mathbb{PI}_{T_{n}}[g](P)=o(r^{2}\sec^{n-3}\theta_{1})\)
as
\(r\rightarrow\infty\)
in
\(T_{n}\).
Recently, Wang, Huang and N. Yamini (see [8]) generalized Theorem
A
to the conical case.
Theorem B
Let
g
be a continuous function on
\(\partial {\mathfrak{C}_{n}(\Gamma)}\)
satisfying (1.5) with
\(p=q=1\)
and
\(\gamma=\aleph^{+}+1-\aleph^{-}\). Then
$$\mathscr{U}_{\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[g]}=\mathscr {U}_{\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[ \vert g \vert ]}=0. $$
The remainder of the paper is organized as follows: in Section
2, we shall give our main theorem; in Section
3, some necessary lemmas are given; in Section
4, we shall prove the main result.