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*.