1 Introduction

Let \(\mathbf{R}^{n}\) denote the n-dimensional Euclidean space, where \(n\geq3\). We denote two points L and N in \(\mathbf{R}^{n}\) by \(L=(x',x_{n})\) and \(N=(y',y_{n})\), respectively, where \(x'=(x_{1},x_{2},\ldots,x_{n-1}) \), \(y'=(y_{1},y_{2},\ldots,y_{n-1}) \), \(x_{n} \in\mathbf{R}\) and \(y_{n} \in\mathbf{R}\). The Euclidean distance of them is denoted by \(\vert L-N\vert \). Let E be a subset of \(\mathbf{R}^{n}\), we denote the boundary and closure of it by ∂E and E̅, respectively.

The set

$$\bigl\{ L=\bigl(x',x_{n}\bigr)\in\mathbf{R}^{n}; x_{n}>0\bigr\} , $$

is denoted by \(\mathcal{T}_{n}\), which is called the upper half-space. Let F be a subset of \(\mathbf{R}_{+}\cup\{0\}\). Then two sets

$$\bigl\{ L=\bigl(x',x_{n}\bigr)\in\mathcal{T}_{n}; \vert L\vert \in F\bigr\} \quad \mbox{and}\quad \bigl\{ N= \bigl(y',0\bigr)\in\partial\mathcal{T}_{n}; \vert N \vert \in F\bigr\} $$

are denoted by \(\mathcal{T}_{n}E\) and \(\partial\mathcal{T}_{n}E\), respectively.

Let \(B_{n}(r)\) denote the open ball with center at the origin and radius r, where \(r>0\). By \(S_{n}(r)\) we denote \(\mathcal{T}_{n}\cap\partial B_{n}(r)\). When g is a function defined by \(\sigma_{n}(r)=\mathcal{T}_{n}\cap B_{n}(r)\), the mean of g is defined by

$$\mathrm{M}(g) (r)=\frac{2s_{n}}{r^{n-1}} \int_{\sigma_{n}(r)}g(L)\,d\sigma_{L}, $$

where \(s_{n}\) is the surface area of \(B_{n}(1)\) and \(d\sigma_{L}\) is the surface element on \(B_{n}(r)\) at \(L\in\sigma_{n}(r)\).

Let \(h(L)\) be a function on \(\mathcal{T}_{n}\). In this paper we denote \(h^{+}=\max\{h,0\}\), \(h^{-}=-\min\{h,0\}\) and \([c]\) is the integer part of c, where \(c\in\mathbf {R}\). Let \(\partial/\partial n\) denote differentiation along the inward normal into \(\mathcal{T}_{n}\). We use the Lebesgue measure \(dL=dx'\,dx_{n}\), where \(dx'=dx_{1}\cdots\,dx_{n-1}\).

Let f be a continuous function on \(\partial\mathcal{T}_{n}\). If h is a harmonic function on \(\mathcal{T}_{n}\) and

$$\lim_{L\rightarrow N\in\partial\mathcal{T}_{n}, L\in\mathcal {T}_{n}(\Omega)}\frac{\partial h(L)}{\partial x_{n}}=f(N), $$

then we say that h is a solution of the Neumann problem on \(\mathcal {T}_{n}\) with respect to f.

The uniqueness and the existence of solutions of the Neumann problem on \(\mathcal{T}_{n}\) with a continuous function on \(\partial\mathcal{T}_{n}\) were given by Su (see [1, 2]).

Theorem A

(see [3], Theorem 1)

Let \(f(N) \) (\(N=(y',0)\)) be a function continuous on \(\partial\mathcal{T}_{n}\) such that

$$ \int_{\partial\mathcal{T}_{n}}\bigl\vert f\bigl(y'\bigr) \bigl(\bigr\vert 1+\bigl\vert y'\bigr\vert \bigr)^{2-n} \,dy'< +\infty. $$
(1.1)

Then the Neumann integral

$$\mathbb{H}_{0,n}[f](L)=-\rho_{n} \int_{\partial\mathcal{T}_{n}}f(N)\vert L-N\vert ^{2-n}\,dN $$

is a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f satisfying

$$\mathrm{M}\bigl(\mathbb{H}_{0,n}[f]\bigr) (r)=O(1) $$

as \(r\rightarrow+\infty\), where \(\rho_{n}=2\{(n-2)s_{n}\}^{-1}\).

Theorem B

(see [3], Theorem 3)

Let k be a positive integer, f be a continuous function on \(\partial \mathcal{T}_{n}\) such that (1.1) holds and \(h(L)\) be a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f satisfying

$$\mathrm{M}\bigl(h^{+}\bigr) (r)=o\bigl(r^{k}\bigr) $$

as \(r\rightarrow+\infty\). Then

$$h(L)=\mathbb{H}_{0,n}(f) (L)+ \left \{ \textstyle\begin{array}{@{}l@{\quad}l} d & \textit{when } k=1, \\ \Pi(x')+\sum_{j=1}^{[\frac{k}{2} ]}\frac{(-1)^{j}}{(2j)!}x_{n}^{2j}\Delta^{j} \Pi(x') &\textit{when } k\geq2, \end{array}\displaystyle \right . $$

for any \(L=(x',x_{n})\), where d is a constant, \(\Pi(x')\) is a polynomial of degree less than k on \(\partial\mathcal{T}_{n}\) and

$$\Delta^{j}= \biggl(\frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\partial ^{2}}{\partial x_{2}^{2}}+\cdots+ \frac{\partial^{2}}{\partial x_{n-1}^{2}} \biggr) \quad (j=1,2\ldots). $$

Recently, Ren and Yang (see [4]) extended Theorems A and B by defining generalised Neumann integrals with continuous functions under less restricted conditions than (1.1). Meanwhile, they also proved that for any continuous function f on \(\partial\mathcal{T}_{n}\) there exists a solution of Neumann problem on \(\mathcal{T}_{n}\). To state them, we need some preliminaries.

Let L and N be two points on \(\mathcal{T}_{n}\) and \(\partial\mathcal {T}_{n}\), respectively. By \(\langle L,N\rangle\) we denote the usual inner product in \(\mathbf{R}^{n}\). We denote

$$\vert L-N\vert ^{2-n}=\sum_{k=0}^{\infty}d_{k,n} \vert N\vert ^{-k-n+2}\vert L\vert ^{k}G_{k,n}(t), $$

where \(\vert N\vert >\vert L\vert \),

$$t=\vert L\vert ^{-1}\vert N\vert ^{-1}\langle L,N \rangle, \quad d_{k,n}= \begin{pmatrix} k+n-3\\ k \end{pmatrix} $$

and \(G_{k,n}\) is the n-dimensional Legendre polynomial of degree k.

As in [2], we shall use the following generalised Dirichlet kernel. For a non-negative integer l, two points \(L\in\mathcal{T}_{n}\) and \(N\in\partial\mathcal{T}_{n}\), we put

$$ \mathbb{V}_{l,n}(L,N)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} -\rho_{n+1}\sum_{k=0}^{l-1}d_{k,n}\vert N\vert ^{-n-k+2}\vert L\vert ^{k}G_{k,n}(t) & \mbox{when } \vert N\vert \geq1 \mbox{ and } l\geq1, \\ 0 & \mbox{when } \vert N\vert < 1 \mbox{ and } l\geq1,\\ 0 & \mbox{when } l=0. \end{array}\displaystyle \right . $$
(1.2)

The generalised Neumann kernel \(\mathbb{K}_{l,n}(L,N)\) on \(\mathcal {T}_{n}\) is defined by (see [2])

$$\mathbb{K}_{l,n}(L,N)=\mathbb{K}_{0,n}(L,N)- \mathbb{V}_{l,n}(L,N), $$

where \(L\in\mathcal{T}_{n}\), \(N\in\partial\mathcal{T}_{n}\) and

$$\mathbb{K}_{0,n}(L,N)=-\alpha_{n}\vert L-N\vert ^{2-n}. $$

As for similar generalised Dirichlet kernel in a half plane and smooth cone, we refer the reader to the papers by Yang and Ren (see [5]), Zhao and Yamada (see [6]) and Su (see [1]).

Let \(f(N)\) be a continuous function on \(\partial\mathcal{T}_{n}\). Then the generalised Neumann integral on \(\mathcal{T}_{n}\) can be defined by

$$\mathbb{H}_{l,n}[f](L)= \int_{\partial\mathcal{T}_{n}}f(N)\mathbb{K}_{l,n}(L,N)\,dN. $$

Ren and Yang proved the following results.

Theorem C

(see [4], Corollary 1)

Let \(1< p< \infty\), \(n+\beta-2>-(n-1)(p-1)\) and

$$1-\frac{1-\beta}{p}< m< 2-\frac{1-\beta}{p}. $$

Let \(f(N) \) (\(N=(y',0)\)) be a continuous function on \(\partial\mathcal {T}_{n}\) such that

$$ \int_{\partial{H}}\bigl\vert f\bigl(y'\bigr)\bigr\vert ^{p}\bigl(1+\bigl\vert y'\bigr\vert \bigr)^{2-\beta-n}\,dy'< \infty. $$
(1.3)

Then the generalised Neumann integral \(\mathbb{H}_{l,n}[f](L)\) is a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f satisfying

$$\mathrm{M}\bigl(\bigl\vert \mathbb{H}_{l,n}[f]\bigr\vert \bigr) (r)=O\bigl(\vert x\vert ^{1+\frac{\beta-1}{p}}\sec ^{n-2}\theta\bigr) $$

as \(r\rightarrow+\infty\).

Theorem D

(see [4], Theorem 3)

Let \(1\leq p< \infty\), \(\beta>1-p\), l be a positive integer and

$$\begin{aligned}& 1-\frac{1-\beta}{p}< m< 2-\frac{1-\beta}{p} \quad\textit{when } p>1, \\& \beta\leq m< \beta+1 \quad\textit{when } p=1. \end{aligned}$$

Let \(f(N)\) be a continuous function on \(\partial\mathcal{T}_{n}\) satisfying (1.3). If \(h(L)\) is a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f such that

$$\lim_{r \rightarrow\infty, L=(r,\Theta)\in H} h^{+}(L)=o\bigl(r^{l+[1+\frac{\beta-1}{p}]}\bigr), $$

then

$$h(L)=N_{m}[f](L)+\Pi\bigl(x'\bigr)+\sum _{j=1}^{ [\frac{l+[1+\frac{\beta -1}{p}]}{2} ]}\frac{(-1)^{j}}{(2j)!}x_{n}^{2j} \Delta^{j} \Pi\bigl(x'\bigr) $$

for any \(L=(x',x_{n})\), where d is a constant, \(\Pi(x')\) is a polynomial of degree less than \(l+[1+\frac{\beta-1}{p}]\) on \(\partial \mathcal{T}_{n}\).

From Theorems A, B, C and D, it is easy to see that the continuous boundary function f grows slowly on \(\partial\mathcal{T}_{n}\). It is natural to ask what will happen if f is replaced by a fast-growing continuous function on \(\partial\mathcal{T}_{n}\). In this paper, we shall solve this problem and explicitly give a new solution of the Neumann problem on \(\partial\mathcal{T}_{n}\).

Define

$$\varepsilon_{0}=\limsup_{r\rightarrow\infty}\tau^{-1}(r)r \tau'(r)\log r< 1, $$

where \(\tau(r)\) is a nondecreasing and continuously differentiable function satisfying \(\tau(r)\geq1\) for any \(r\in\mathbf{R}^{+}\cup\{0\}\).

From these we see that there is a sufficiently large positive number r such that for any \(t>r\)

$$ \tau(e) (\ln t)^{\epsilon_{0}+\epsilon}>\tau(t), $$
(1.4)

where ϵ is a sufficiently small positive number satisfying \(\epsilon_{0}+\epsilon<1\).

Let \(\mathfrak{A}_{\varpi}\) be the set of continuous functions \(f(N)\) (\(N=(y',0)\)) on \(\partial\mathcal{T}_{n}\) satisfying

$$ \int_{\partial\mathcal{T}_{n}}\bigl\vert f\bigl(y'\bigr)\bigr\vert \bigl(1+\bigl\vert y'\bigr\vert \bigr)^{3-n-\varpi-\tau (\vert y'\vert )} \,dy'< +\infty, $$
(1.5)

where ϖ is a real number such that \(\varpi>2\).

2 Results

Now we state our results.

Theorem 1

If \(f\in\mathfrak{A}_{\varpi}\), then generalised Neumann integral \(\mathbb{H}_{[\tau(\vert y'\vert )+\varpi],n}[f](L)\) is a solution of the Neumann problem on \(\mathcal{T}_{n}\) with respect to f.

Then we shall prove that if the negative part of a harmonic function satisfies a fast-growing condition, then its positive part satisfies the similar condition. That is to say, the condition of Theorem 1 may be replaced by a weaker integral condition. To state this result, we also need some notations.

Let \(\mathfrak{B}_{\varpi}\) be the set of continuous functions \(f(N)\) (\(N=(y',y_{n})\)) on \(\mathcal{T}_{n}\) satisfying

$$ \int_{\mathcal{T}_{n}}\bigl\vert f(N)\bigr\vert \bigl(1+\vert N\vert \bigr)^{1-n-\varpi-\tau(\vert N\vert )}y_{n}\,dN < +\infty. $$
(2.1)

By \(\mathfrak{C}_{\varpi}\) we denote the set of all continuous functions \(h(N)\) on \(\overline{\mathcal{T}_{n}}\), harmonic on \(\mathcal {T}_{n}\) with \(h^{-}(N)\in\mathfrak{B}_{\varpi}\) and \(h^{-}(y')\in\mathfrak {A}_{\varpi}\).

Theorem 2

The conclusion of Theorem  1 remains valid if its condition is replaced by \(h\in\mathfrak{C}_{\varpi}\).

Theorem 3

If \(h\in\mathfrak{C}_{\varpi}\), then there exists a harmonic function \(\Lambda(L)\) with normal derivative vanishes on \(\partial\mathcal{T}_{n}\) such that

$$h(L)=\Lambda(L)+\mathbb{H}_{[\tau(\vert y'\vert )+\varpi],n}[h](L), $$

where \(L\in\overline{\mathcal{T}}_{n}\).

3 Lemmas

Lemma 1

Let \(L\in\mathcal{T}_{n}\) and \(N\in\partial\mathcal{T}_{n}\) such that \(\vert N\vert \geq\max\{1,2\vert L\vert \}\). Then (see [7])

$$\bigl\vert \mathbb{K}_{l,n}(L,N)\bigr\vert \leq M \vert N\vert ^{-l-n+2}\vert L\vert ^{l}, $$

where M is a positive constant.

Lemma 2

Let \(\mathbb{W}(L,N)\) (\(N\in\partial \mathcal{T}_{n}\)) be a locally integrable function for any fixed point \(L\in\mathcal{T}_{n}\), \(g(N)\) be a upper semicontinuous and locally integrable function on \(\partial\mathcal{T}_{n}\). Set

$$\mathbb{K}(L,N)=\mathbb{K}_{0,n}(L,N)-\mathbb{W}(L,N) $$

for any \(N\in\partial\mathcal{T}_{n}\) and \(L\in\mathcal{T}_{n}\).

Suppose that the following two conditions hold:

  1. (I)

    There are a positive number R and a neighborhood \(B(N^{*})\) of \(N^{*}\) (\(\in\partial\mathcal{T}_{n}\)) satisfying

    $$\int_{\partial\mathcal{T}_{n}[R,+\infty)\cup\partial\mathcal {T}_{n}(-\infty,-R]}\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial}{\partial x_{n}}\mathbb {K}(L,N)\biggr\vert \,dN< \epsilon, $$

    where \(\epsilon> 0\).

  2. (II)

    There exists a positive number R satisfying

    $$\limsup_{L\rightarrow N^{*},L\in\mathcal{T}_{n}} \int_{\partial\mathcal {T}_{n}(-R,R)}\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial}{\partial x_{n}}\mathbb{W}(L,N)\biggr\vert \,dN=0 $$

    for any \(N^{*}\in\partial\mathcal{T}_{n}\).

Then

$$ \limsup_{L\rightarrow N^{*}\in\partial\mathcal{T}_{n},L\in\mathcal {T}_{n}} \int_{\partial\mathcal{T}_{n}}g(N)\frac{\partial}{\partial x_{n}}\mathbb{W}(L,N)\,dN\leq g \bigl(N^{*}\bigr). $$
(3.1)

Proof

Let \(N^{*}\) be any point of \(\partial \mathcal{T}_{n}\) and ϵ be any positive number. There exists a positive number \(R^{*}\) satisfying

$$ \int_{\partial\mathcal{T}_{n}[R^{*},+\infty)\cup\partial\mathcal {T}_{n}(-\infty,-R^{*}]}\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial}{\partial x_{n}}\mathbb {K}(L,N)\biggr\vert \,dN\leq\frac{\epsilon}{2} $$
(3.2)

for any \(L=(x',x_{n})\in\mathcal{T}_{n}\cap B(N^{*})\) from (I).

Let ϕ be a continuous function on \(\partial\mathcal{T}_{n}\) such that \(0\leq\phi\leq1\) and

$$\phi=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1 & \mbox{if } \partial\mathcal{T}_{n}[-R^{*},R^{*}], \\ 0 &\mbox{if } \partial\mathcal{T}_{n}(-\infty,-2R^{*})\cup\partial \mathcal{T}_{n}(2R^{*},+\infty). \end{array}\displaystyle \right . $$

Let \(\mathbb{K}_{0,n}^{j}(L,N)\) be the Neumann function of \(\mathcal{T}_{n}(-j,j)\), where j is a positive integer. Since

$$\Gamma_{j}(L,N)=\mathbb{K}_{0,n}(L,N)-\mathbb{K}_{0,n}^{j}(L,N) $$

on \(\mathcal{T}_{n}(-j,j)\) converges monotonically to 0 as \(j\rightarrow\infty\), we can find an integer \(j^{*}\) satisfying \(j^{*}>2R^{*}\) such that

$$ \int_{\partial\mathcal{T}_{n}(-2R^{*},2R^{*})}\bigl\vert \phi(N)g(N)\bigr\vert \biggl\vert \frac{\partial }{\partial x_{n}}\Gamma_{j^{*}}(L,N)\biggr\vert \,d\sigma< \frac{\epsilon}{4} $$
(3.3)

for any \(L=(x',x_{n})\in B(N^{*})\cap\mathcal{T}_{n}\).

Then we have from (3.2) and (3.3) that

$$\begin{aligned} \int_{\partial\mathcal{T}_{n}}g(N)\frac{\partial}{\partial x_{n}}\mathbb {K}(L,N)\,dN \leq{}& \int_{\partial\mathcal{T}_{n}(-2R^{*},2R^{*})}g(N)\frac{\partial \mathbb{K}_{0,n}^{j^{*}}(L,N) }{\partial x_{n}}\phi(N)\,dN \\ &{}+ \int_{\partial\mathcal{T}_{n}(-2R^{*},2R^{*})}\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial \Gamma_{j^{*}}(L,N)}{\partial x_{n}}\biggr\vert \bigl\vert \phi(N)\bigr\vert \,dN \\ &{}+ \int_{\partial\mathcal{T}_{n}(-2R^{*},2R^{*})}\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial \mathbb{W}(L,N)}{\partial x_{n}}\biggr\vert \,dN \\ &{}+ 2 \int_{\partial\mathcal{T}_{n}[R^{*},+\infty)\cup\partial\mathcal {T}_{n}(-\infty,-R^{*}] }\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial\mathbb{K}(L,N)}{\partial x_{n}}\biggr\vert \,dN \\ \leq{}& \int_{S_{n}(\Gamma;(-2R^{*},2R^{*}))}g(N)\frac{\partial \mathbb{K}_{0,n}^{j^{*}}(L,N) }{\partial x_{n}}\phi(N)\,dN \\ &{}+ \int_{\partial\mathcal{T}_{n}(-2R^{*},2R^{*})}\bigl\vert g(N)\bigr\vert \biggl\vert \frac{\partial \mathbb{W}(L,N)}{\partial x_{n}}\biggr\vert \,dN+\frac{5}{4}\epsilon \end{aligned}$$
(3.4)

for any \(L=(x',x_{n})\in\mathcal{T}_{n}\cap B(N^{*})\).

Consider an upper semicontinuous function

$$\psi(N)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \phi(N)g(N) & \mbox{if } \partial\mathcal {T}_{n}[-2R^{*},2R^{*}], \\ 0&\mbox{if } \partial\mathcal{T}_{n}[-j^{*},j^{*}]-\partial\mathcal {T}_{n}[-2R^{*},2R^{*}] \end{array}\displaystyle \right . $$

on \(\partial\mathcal{T}_{n}(-j^{*},j^{*})\) and denote the Perron-Wiener-Brelot solution of the Neumann problem on \(\mathcal{T}_{n}(-j^{*},j^{*})\) by \(\mathbb{H}_{\psi}(L;\mathcal {T}_{n}(-j^{*},j^{*}))\). We know that

$$ \int_{\partial\mathcal{T}_{n}(-2R^{*},2R^{*})}g(N)\frac{\partial \mathbb{K}_{0,n}^{j^{*}}(L,N) }{\partial x_{n}}\phi(N)\,dN= \mathbb{H}_{\psi}\bigl(L;\mathcal{T}_{n}\bigl(-j^{*},j^{*}\bigr) \bigr). $$

We also have

$$\limsup_{L\rightarrow N^{*}, L\in\mathcal{T}_{n}}\mathbb{H}_{\psi }\bigl(L; \mathcal{T}_{n}\bigl(-j^{*},j^{*}\bigr)\bigr)\leq\limsup _{N\in\partial T_{n}, N\rightarrow N^{*}}\psi(N)=g\bigl(N^{*}\bigr). $$

Hence we obtain

$$\limsup_{L\rightarrow N^{*},L\in\mathcal{T}_{n}} \int_{\partial\mathcal {T}_{n}(-2R^{*},2R^{*})}g(N)\frac{\partial \mathbb{K}_{0,n}^{j^{*}}(L,N) }{\partial x_{n}}\phi(N)\,dN\leq g\bigl(N^{*} \bigr), $$

which together with (II) and (3.4) gives (3.1). □

Lemma 3

Let \(r>1\) and \(h(N)\) (\(N=(y',y_{n})\)) be a function harmonic on \(\mathcal {T}_{n}\). Then

$$\int_{S_{n}(r)}r^{-1-n}h(N)ny_{n}\,dN+ \int_{\partial\mathcal{T}_{n}(1,r)}h\bigl(y'\bigr) \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr) \,dy'=d_{1}+d_{2}r^{-n}, $$

where

$$d_{1} = \int_{S_{n}(1)}y_{n} \biggl((n-1)h(N)+\frac{\partial h(N)}{\partial n} \biggr)\,dN $$

and

$$d_{2}= \int_{S_{n}(1)}y_{n} \biggl(h(N)-\frac{\partial h(N)}{\partial n} \biggr)\,dN. $$

4 Proof of Theorem 1

We have from (1.4)

$$ M_{1}(r)\geq(2r)^{\tau(k+1)+\varpi+1 }k^{\frac{2-\varpi}{2}} $$
(4.1)

for any \(k>k_{r}=[2r]+1\), where \(M_{1}(r)\) is a positive constant dependent only on r.

We have for any \(L\in\mathcal{T}_{n}\) and \(\vert L\vert \leq R\)

$$\begin{aligned} & \sum_{k=k_{r}}^{\infty}\int_{\partial\mathcal {T}_{n}[k,k+1)}\bigl\vert f\bigl(y'\bigr)\bigr\vert \bigl(2\vert L\vert \bigr)^{[\tau(\vert y'\vert )+\varpi] } \bigl\vert y'\bigr\vert ^{2-n-[\tau(\vert y'\vert )+\varpi]}\,dy' \\ &\quad\leq \sum_{k=k_{r}}^{\infty}k^{\frac{2-\varpi}{2}}(2r)^{1+\varpi+\tau(k+1)} \int_{\partial\mathcal{T}_{n}[k,k+1)}2\bigl\vert f\bigl(y'\bigr)\bigr\vert \bigl(1+\bigl\vert y'\bigr\vert \bigr)^{1-n-\frac{\varpi -2}{2}-\tau(\vert y'\vert )} \,dy' \\ &\quad\leq 2M_{1}(r) \int_{\partial\mathcal{T}_{n}[k_{r},+\infty )}\bigl\vert f\bigl(y'\bigr)\bigr\vert \bigl(1+\bigl\vert y'\bigr\vert \bigr)^{1-n-\frac{\varpi-2}{2}-\tau(\vert y'\vert )} \,dy' \\ &\quad< +\infty \end{aligned}$$
(4.2)

from Lemma 1 and (1.5). So \(\mathbb{H}_{[\tau(\vert y'\vert )+\varpi ],n}(L)\) is absolutely convergent.

Next we shall prove that

$$\lim_{L\rightarrow N',L=(x',x_{n})\in\mathcal{T}_{n}}\frac{\partial\mathbb {H}_{[\tau(\vert y'\vert )+\varpi],n}(L)}{\partial x_{n}}=h\bigl(N'\bigr) $$

for any \(N'=(y',0)\in\partial\mathcal{T}_{n}\). By applying Lemma 2 to \(-g(y')\) and \(g(y')\) by setting

$$\mathbb{W}(L,N)=\mathbb{V}_{[\tau(\vert y'\vert )+\varpi],n}(L,N), $$

then we shall see that (I) and (II) hold. Take any \(N'=(y',0)\in\partial\mathcal{T}_{n}\) and any \(\epsilon>0\). There exists a number R (\({>}\max\{2(\delta+y'),1\}\)) satisfying

$$\int_{\partial\mathcal{T}_{n}[R,+\infty)\cup\partial\mathcal {T}_{n}(-\infty,-R]}\bigl\vert f(N)\bigr\vert \biggl\vert \frac{\partial}{\partial x_{n}}\mathbb{K}_{[\tau (\vert y'\vert )+\varpi],n}(L,N)\biggr\vert \,dN< \epsilon $$

for any \(L\in\mathcal{T}_{n} \cap U(N',\delta)\) from (1.5) and (4.2), which is (I) in Lemma 2. To see (II), we only need to observe from (1.2) that for any \(N'\in\partial\mathcal{T}_{n}\)

$$\limsup_{L=(x',x_{n})\rightarrow N^{*},L\in\mathcal{T}_{n}}\frac{\partial }{\partial x_{n}}\mathbb{V}_{[\tau(\vert y'\vert )+\varpi],n}(L,N)=0. $$

So Theorem 1 is proved.

5 Proof of Theorem 2

Lemma 2 gives

$$\begin{aligned} & P_{-}(r)+ \int_{\partial\mathcal{T}_{n}(1,r)}h^{-}\bigl(y'\bigr) \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr) \,dy' \\ &\quad= P_{+}(r)+ \int_{\partial\mathcal{T}_{n}(1,r)}h^{+}\bigl(y'\bigr) \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr) \,dy'- d_{1}-d_{2}r^{-n}, \end{aligned}$$

where

$$P_{\pm}(r)= \int_{\sigma_{n}(r)}nh^{\pm}(y)r^{-n-1}y_{n} \,dN. $$

Since \(h\in\mathfrak{C}_{\varpi}\), we obtain by (2.1)

$$\begin{aligned} \int_{1}^{+\infty}P_{-}(r)r^{2-\varpi-\tau(r)}\,dr = n \int_{\mathcal{T}_{n}(1,+\infty)}h^{-}(N)y_{n}\vert N\vert ^{1-\varpi-n-\tau (\vert N\vert )}\,dN < +\infty. \end{aligned}$$
(5.1)

We have by (1.5)

$$\begin{aligned} & \int_{1}^{+\infty}r^{2-\varpi-\tau(r)} \biggl( \int_{\partial\mathcal {T}_{n}(1,r)}h^{-}\bigl(y'\bigr) \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr) \,dy' \biggr)\,dr \\ &\quad = \int_{\partial\mathcal{T}_{n}(1,+\infty)}h^{-}\bigl(y'\bigr) \biggl( \int _{\vert y'\vert }^{\infty}r^{2-\varpi-\tau(r)} \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr)\,dr \biggr) \,dy' \\ &\quad\leq \frac{n}{n+1} \int_{\partial\mathcal{T}_{n}(1,+\infty)} h^{-}\bigl(y'\bigr)\bigl\vert y'\bigr\vert ^{3-\varpi-n-\tau(\vert y'\vert )}\,dy' \\ &\quad< +\infty. \end{aligned}$$
(5.2)

From (5.1), (5.2) and Lemma 2, we see that

$$\begin{aligned} & \int_{1}^{+\infty}r^{\frac{2-\varpi}{2}-\tau(r)} \biggl( \int_{\partial \mathcal{T}_{n}(1,r)}h^{+}\bigl(y'\bigr) \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr) \,dy' \biggr)\,dr \\ &\quad = \int_{\partial\mathcal{T}_{n}[1,+\infty)}h^{+}\bigl(y'\bigr) \biggl( \int _{\vert y'\vert }^{\infty}r^{\frac{2-\varpi}{2}-\tau(r)} \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr)\,dr \biggr) \,dy' \\ &\quad\leq \int_{1}^{+\infty}P_{-}(r)r^{\frac{2-\varpi}{2}-\tau(r)}\,dr- \int_{1}^{+\infty}r^{\frac{2-\varpi}{2}-\tau(r)} \bigl( d_{1}+d_{2}r^{-n} \bigr)\,dr \\ & \qquad{} + \int_{1}^{+\infty}r^{\frac{2-\varpi}{2}-\tau(r)} \biggl( \int_{\partial \mathcal{T}_{n}(1,r)}h^{-}\bigl(y'\bigr) \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr) \,dy' \biggr)\,dr \\ &\quad< +\infty. \end{aligned}$$
(5.3)

Set

$$\mathbb{Q}(\varpi)=\lim_{\vert y'\vert \rightarrow\infty} \int_{\vert y'\vert }^{\infty}r^{\frac{2-\varpi}{2}-\tau(r)} \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr)\,dr \bigl\vert y'\bigr\vert ^{-3+\varpi +n+\tau(\vert y'\vert )}. $$

It is easy to see that

$$\mathbb{Q}(\varpi)=+\infty, $$

from (1.4), which shows that

$$M_{2}\bigl\vert y'\bigr\vert ^{3-\varpi-n-\tau(\vert y'\vert )}\leq \int_{\vert y'\vert }^{\infty}r^{\frac {2-\varpi}{2}-\tau(r)} \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr)\,dr $$

for any \(\vert y'\vert \geq1\), where \(M_{2}\) is a positive constant.

It follows that

$$\begin{aligned} & M_{2} \int_{\partial\mathcal{T}_{n}[1,+\infty)} h^{+}\bigl(y'\bigr)\bigl\vert y'\bigr\vert ^{3-\varpi-n-\tau(\vert y'\vert )}\,dx' \\ &\quad\leq \int_{\partial\mathcal{T}_{n}[1,+\infty)}h^{+}\bigl(y'\bigr) \int _{\vert y'\vert }^{\infty}r^{\frac{2-\varpi}{2}-\tau(r)} \bigl(\bigl\vert y'\bigr\vert ^{-n}-r^{-n} \bigr)\,dr \,dy' \\ &\quad< +\infty \end{aligned}$$

from (5.3).

Then Theorem 2 is proved from \(\vert h\vert =h^{+}+h^{-}\).

6 Proof of Theorem 3

Put \(h'(L)= h(L)-\mathbb{H}_{[\tau(\vert y'\vert )+\varpi],n}(L)\). Then it is easy to see that \(h'(L)\) is harmonic on \(\mathcal{T}_{n}\) with normal derivative vanishes on \(\partial\mathcal{T}_{n} \) and \(h'(L)\) can be continuously extended to \(\overline{\mathcal{T}_{n}}\). By applying the Schwarz reflection principle [8], p.68, to \(h'(L)\), it follows that there is a function harmonic on \(\mathcal{T}_{n}\) satisfying \(h(L^{*})=-h'(L)=-(h(L)-\mathbb{H}_{[\tau(\vert y'\vert )+\varpi],n}(L))\) for \(L\in \overline{T}_{n}\), where ∗ denotes reflection in \(\partial\mathcal{T}_{n}\) just as \(L^{*}=(x', -x_{n})\). Thus \(h(L)=\Lambda(L)+\mathbb{H}_{[\tau(|y'|)+\varpi],n}(L)\) for all \(L \in \overline{\mathcal{T}}_{n} \), where \(\Lambda(L)\) is a harmonic function on \(\mathcal{T}_{n}\) with normal derivative which vanishes continuously on \(\partial\mathcal{T}_{n}\). Theorem 3 is proved.