Growth estimates for modified Neumann integrals in a half-space

Abstract

Our aim in this paper is to deal with the growth properties for modified Neumann integrals in a half-space of ${\mathbf{R}}^n$. As an application, the solutions of Neumann problems in it for a slowly growing continuous function are also given.

1 Introduction and main results

Let R and ${\mathbf{R}}_{+}$ be the sets of all real numbers and of all positive real numbers, respectively. Let ${\mathbf{R}}^n$ ($n\ge 3$) denote the n-dimensional Euclidean space with points $x=(x^{\prime },x_n)$, where $x^{\prime }=(x_1,x_2,\dots ,x_{n-1})\in {\mathbf{R}}^{n-1}$ and $x_n\in \mathbf{R}$. The boundary and closure of an open set Ω of ${\mathbf{R}}^n$ are denoted by ∂ Ω and $\overline{\mathrm{\Omega }}$, respectively. For $x\in {\mathbf{R}}^n$ and $r>0$, let $B_n(x,r)$ denote the open ball with center at x and radius r in ${\mathbf{R}}^n$.

The upper half-space is the set $H=\{(x^{\prime },x_n)\in {\mathbf{R}}^n:x_n>0\}$, whose boundary is ∂H. For a set F, $F\subset {\mathbf{R}}_{+}\cup \{0\}$, we denote $\{x\in H;|x|\in F\}$ and $\{x\in \partial H;|x|\in F\}$ by HF and $\partial HF$, respectively. We identify ${\mathbf{R}}^n$ with ${\mathbf{R}}^{n-1}\times \mathbf{R}$ and ${\mathbf{R}}^{n-1}$ with ${\mathbf{R}}^{n-1}\times \{0\}$, writing typical points $x,y\in {\mathbf{R}}^n$ as $x=(x^{\prime },x_n)$, $y=(y^{\prime },y_n)$, where $y^{\prime }=(y_1,y_2,\dots ,y_{n-1})\in {\mathbf{R}}^{n-1}$. Let θ be the angle between x and ${\hat{e}}_n$, i.e., $x_n=|x|cos\theta$ and $0\le \theta <\pi /2$, where ${\hat{e}}_n$ is the i th unit coordinate vector and ${\hat{e}}_n$ is normal to ∂H.

We shall say that a set $E\subset H$ has a covering $\{r_j,R_j\}$ if there exists a sequence of balls $\{B_j\}$ with centers in H such that $E\subset {\bigcup }_{j=0}^{\mathrm{\infty }}{B}_j$, where $r_j$ is the radius of $B_j$ and $R_j$ is the distance between the origin and the center of $B_j$.

For positive functions $g_1$ and $g_2$, we say that $g_1\lesssim g_2$ if $g_1\le M{g}_2$ for some positive constant M. Throughout this paper, let M denote various constants independent of the variables in question. Further, we use the standard notations, $[d]$ is the integer part of d and $d=[d]+\{d\}$, where d is a positive real number.

Given a continuous function f in ∂H, we say that h is a solution of the Neumann problem in H with f, if h is a harmonic function in H and

$$\underset{x\in H,x\to y^{\prime }}{lim}\frac{\partial }{\partial x_n}h(x)=f\left(y^{\prime }\right)$$

for every point $y^{\prime }\in \partial H$.

For $x\in {\mathbf{R}}^n$ and $y^{\prime }\in {\mathbf{R}}^{n-1}$, consider the kernel function

$$K_n(x,y^{\prime })=-\frac{{\beta }_n}{|x-y^{\prime }{|}^{n-2}},$$

where ${\beta }_n=2/(n-2){\sigma }_n$ and ${\sigma }_n$ is the surface area of the n-dimensional unit sphere. It has the expression

$$K_n(x,y^{\prime })=\sum _{k=0}^{\mathrm{\infty }}\frac{|x{|}^k}{|y{|}^{n+k-2}}{C}_k^{\frac{n-2}{2}}\left(\frac{x\cdot y^{\prime }}{|x||y^{\prime }|}\right),$$

where $C_k^{\frac{n}{2}}(t)$ is the ultraspherical (Gegenbauer) polynomials [1]. The series converges for $|y^{\prime }|>|x|$, and each term in it is a harmonic function of x.

The Neumann integral is defined by

$$N[f](x)={\int }_{\partial H}{K}_n(x,y^{\prime })f\left(y^{\prime }\right)d{y}^{\prime },$$

where f is a continuous function on ∂H, ${\alpha }_n=2/n{\sigma }_n$ and ${\sigma }_n={\pi }^{\frac{n}{2}}/\mathrm{\Gamma }(1+\frac{n}{2})$ is the volume of the unit n-ball.

The Neumann integral $N[f](x)$ is a solution of the Neumann problem on H with f if (see [[2], Theorem 1 and Remarks])

$${\int }_{\partial H}\frac{f(y^{\prime })}{{(1+|y^{\prime }|)}^{n-2}}d{y}^{\prime }<\mathrm{\infty }.$$

In this paper, we consider functions f satisfying

$${\int }_{\partial H}\frac{|f(y^{\prime }){|}^p}{{(1+|y^{\prime }|)}^{n+\alpha -2}}d{y}^{\prime }<\mathrm{\infty }$$

(1.1)

for $1\le p<\mathrm{\infty }$ and $\alpha \in \mathbf{R}$.

For this p and α, we define the positive measure μ on ${\mathbf{R}}^n$ by

$$d\mu \left(y^{\prime }\right)=\{\begin{array}{ll}|f(y^{\prime }){|}^p|y^{\prime }{|}^{-n-\alpha +2}d{y}^{\prime },& y^{\prime }\in \partial H(1,+\mathrm{\infty }),\\ 0,& Q\in {\mathbf{R}}^n-\partial H(1,+\mathrm{\infty }).\end{array}$$

If f is a measurable function on ∂H satisfying (1.1), we remark that the total mass of μ is finite.

Let $ϵ>0$ and $\delta \ge 0$. For each $x\in {\mathbf{R}}^n$, the maximal function $M(x;\mu ,\delta )$ is defined by

$$M(x;\mu ,\delta )=\underset{0<\rho <\frac{|x|}{2}}{sup}\frac{\mu (B_n(x,r))}{{\rho }^{\delta }}.$$

The set $\{x\in {\mathbf{R}}^n;M(x;\mu ,\delta )>ϵ\}$ is denoted by $E(ϵ;\mu ,\delta )$.

To obtain the Neumann solution for the boundary data f, as in [3–6], we use the following modified kernel function defined by

$$L_{n,m}(x,y^{\prime })=\{\begin{array}{ll}-{\beta }_n{\sum }_{k=0}^{m-1}\frac{|x{|}^k}{|y{|}^{n+k-2}}{C}_k^{\frac{n-2}{2}}(\frac{x\cdot y^{\prime }}{|x||y^{\prime }|}),& |y^{\prime }|\ge 1m\ge 1,\\ 0,& |y^{\prime }|<1m\ge 1,\\ 0,& m=0\end{array}$$

for a non-negative integer m.

For $x\in {\mathbf{R}}^n$ and $y^{\prime }\in {\mathbf{R}}^{n-1}$, the generalized Neumann kernel is defined by

$$K_{n,m}(x,y^{\prime })=K_n(x,y^{\prime })-L_{n,m}(x,y^{\prime })(m\ge 0).$$

Since $|x{|}^k{C}_k^{\frac{n-2}{2}}(\frac{x\cdot y^{\prime }}{|x||y^{\prime }|})$ ($k\ge 0$) is harmonic in H (see [4]), $K_{n,m}(\cdot ,y^{\prime })$ is also harmonic in H for any fixed $y^{\prime }\in \partial H$. Also, $K_{n,m}(x,y^{\prime })$ will be of order $|y^{\prime }{|}^{-(n+m-2)}$ as $y^{\prime }\to \mathrm{\infty }$ (see [[7], Theorem D]).

Put

$$N_m[f](x)={\int }_{\partial H}{K}_{n,m}(x,y^{\prime })f\left(y^{\prime }\right)d{y}^{\prime },$$

where f is a continuous function on ∂H. Here, note that $N_0[f](x)$ is nothing but the Neumann integral $N[f](x)$.

The following result is due to Siegel and Talvila (see [[5], Corollary 2.1]). For similar results with respect to the Schrödinger operator in a half-space, we refer readers to papers by Su (see [8]).

Theorem A If f is a continuous function on ∂H satisfying (1.1) with $p=1$ and $\alpha =m$, then

$$\underset{|x|\to \mathrm{\infty },x\in H}{lim}{N}_m[f](x)=o(|x{|}^m{sec}^{n-2}\theta ).$$

(1.2)

The next result deals with a type of uniqueness of solutions for the Neumann problem on H (see [[9], Theorem 3]).

Theorem B Let l be a positive integer and m be a non-negative integer. If f is a continuous function on ∂H satisfying

$${\int }_{\partial H}\frac{|f(y^{\prime })|}{{(1+|y^{\prime }|)}^{n+m-2}}d{y}^{\prime }<\mathrm{\infty },$$

and h is a solution of the Neumann problem on H with f such that

$$\underset{|x|\to \mathrm{\infty },x\in H}{lim}{h}^{+}(x)=o(|x{|}^{l+m}),$$

then

$$h(x)=N_m[f](x)+\mathrm{\Pi }\left(x^{\prime }\right)+\sum _{j=1}^{[\frac{l+m}{2}]}\frac{{(-1)}^j}{(2j)!}{x}_n^{2j}{\mathrm{\Delta }}^j\mathrm{\Pi }\left(x^{\prime }\right)$$

for any $x=(x^{\prime },x_n)\in H$, where $h^{+}(x)$ is the positive part of h,

$${\mathrm{\Delta }}^j=(\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\cdots +\frac{{\partial }^2}{\partial x_{n-1}^2})(j=1,2\dots )$$

and $\mathrm{\Pi }(x^{\prime })$ is a polynomial of $x^{\prime }\in {\mathbf{R}}^{n-1}$ of degree less than $l+m$.

Our first aim is to be concerned with the growth property of $N_m[f]$ at infinity and establish the following theorem.

Theorem 1 Let $1\le p<\mathrm{\infty }$, $0\le \beta \le (n-2)p$, $n+\alpha -2>-(n-1)(p-1)$ and

$$\begin{array}{c}1-\frac{1-\alpha }{p}<m<2-\frac{1-\alpha }{p}\mathit{\text{if }}p>1,\hfill \\ \alpha \le m<\alpha +1\mathit{\text{if }}p=1.\hfill \end{array}$$

If f is a measurable function on ∂ satisfying (1.1), then there exists a covering $\{r_j,R_j\}$ of $E(ϵ;\mu ,(n-2)p-\beta )$ (⊂H) satisfying

$$\sum _{j=0}^{\mathrm{\infty }}{\left(\frac{r_j}{R_j}\right)}^{(n-2)p-\beta }<\mathrm{\infty }$$

(1.3)

such that

$$\underset{|x|\to \mathrm{\infty },x\in H-E(ϵ;\mu ,(n-2)p-\beta )}{lim}{N}_m[f](x)=o(|x{|}^{1+\frac{\alpha -1}{p}}{sec}^{\frac{\beta }{p}}\theta ).$$

(1.4)

Corollary 1 Let $1<p<\mathrm{\infty }$, $n+\alpha -2>-(n-1)(p-1)$ and

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

If f is a measurable function on ∂H satisfying (1.1), then

$$\underset{|x|\to \mathrm{\infty },x\in H}{lim}{N}_m[f](x)=o(|x{|}^{1+\frac{\alpha -1}{p}}{sec}^{n-2}\theta ).$$

(1.5)

As an application of Theorem 1, we now show the solution of the Neumann problem with continuous data on H.

Theorem 2 Let p, β, α and m be defined as in Theorem  1. If f is a continuous function on ∂H satisfying (1.1), then the function $N_m[f]$ is a solution of the Neumann problem on H with f and (1.4) holds, where the exceptional set $E(ϵ;\mu ,(n-2)p-\beta )$ (⊂H) has a covering $\{r_j,R_j\}$ satisfying (1.3).

Remark In the case $p=1$, $\alpha =m$ and $\beta =n-2$, then (1.3) is a finite sum and the set $E(ϵ;\mu ,0)$ is a bounded set. So (1.4) holds in H. That is to say, (1.2) holds. This is just the result of Theorem A.

Corollary 2 Let $1\le p<\mathrm{\infty }$, $n+\alpha -2>-(n-1)(p-1)$ and

$$\begin{array}{c}1-\frac{1-\alpha }{p}<m<2-\frac{1-\alpha }{p}\mathit{\text{if }}p>1,\hfill \\ \alpha \le m<\alpha +1\mathit{\text{if }}p=1.\hfill \end{array}$$

If f is a continuous function on ∂H satisfying (1.1), then the function $N_m[f]$ is a solution of the Neumann problem on H with f and (1.5) holds.

The following result extends Theorem B, which is our result in the case $p=1$ and $\alpha =m$.

Theorem 3 Let $1\le p<\mathrm{\infty }$, $\alpha >1-p$, l be a positive integer and

$$\begin{array}{c}1-\frac{1-\alpha }{p}<m<2-\frac{1-\alpha }{p}\mathit{\text{if }}p>1,\hfill \\ \alpha \le m<\alpha +1\mathit{\text{if }}p=1.\hfill \end{array}$$

If f is a continuous function on ∂H satisfying (1.1) and h is a solution of the Neumann problem on H with f such that

$$\underset{|x|\to \mathrm{\infty },x\in H}{lim}{h}^{+}(x)=o(|x{|}^{l+[1+\frac{\alpha -1}{p}]}),$$

(1.6)

then

$$h(x)=N_m[f](x)+\mathrm{\Pi }\left(x^{\prime }\right)+\sum _{j=1}^{[\frac{l+[1+\frac{\alpha -1}{p}]}{2}]}\frac{{(-1)}^j}{(2j)!}{x}_n^{2j}{\mathrm{\Delta }}^j\mathrm{\Pi }\left(x^{\prime }\right)$$

(1.7)

for any $x=(x^{\prime },x_n)\in H$ and $\mathrm{\Pi }(x^{\prime })$ is a polynomial of $x^{\prime }\in {\mathbf{R}}^{n-1}$ of degree less than $l+[1+\frac{\alpha -1}{p}]$.

2 Lemmas

In our discussions, the following estimates for the kernel function $K_{n,m}(x,y^{\prime })$ are fundamental (see [[10], Lemma 4.2] and [[4], Lemmas 2.1 and 2.4]).

Lemma 1

• (1) If $1\le |y^{\prime }|\le \frac{|x|}{2}$, then $|K_{n,m}(x,y^{\prime })|\lesssim |x{|}^{m-1}|y^{\prime }{|}^{-n-m+3}$.

• (2) If $\frac{|x|}{2}<|y^{\prime }|\le \frac{3}{2}|x|$, then $|K_{n,m}(x,y^{\prime })|\lesssim |x-y^{\prime }{|}^{2-n}$.

• (3) If $\frac{3}{2}|x|<|y^{\prime }|\le 2|x|$, then $|K_{n,m}(x,y^{\prime })|\lesssim x_n^{2-n}$.

• (4) If $|y^{\prime }|\ge 2|x|$ and $|y^{\prime }|\ge 1$, then $|K_{n,m}(x,y^{\prime })|\lesssim |x{|}^m|y^{\prime }{|}^{2-n-m}$.

The following lemma is due to Qiao (see [4]).

Lemma 2 If $ϵ>0$, $\eta \ge 0$ and λ is a positive measure in ${\mathbf{R}}^n$ satisfying $\lambda ({\mathbf{R}}^n)<\mathrm{\infty }$, then $E(ϵ;\lambda ,\eta )$ has a covering $\{r_j,R_j\}$ ($j=1,2,\dots$) such that

$$\sum _{j=1}^{\mathrm{\infty }}{\left(\frac{r_j}{R_j}\right)}^{\eta }<\mathrm{\infty }.$$

Lemma 3 ([[9], Lemma 4])

Let p, β, α and m be defined as in Theorem  1. If f is a locally integral and upper semi-continuous function on ∂H satisfying (1.1), then

$$\underset{x\in H,x\to y^{\prime }}{lim\hspace{0.17em}sup}\frac{\partial }{\partial x_n}{N}_m[f](x)\le f\left(y^{\prime }\right)$$

for any fixed $y^{\prime }\in \partial H$.

Lemma 4 ([[2], Lemma 1])

If $h(x)$ is a harmonic polynomial of $x=(x^{\prime },x_n)\in H$ of degree m and $\partial h/\partial x_n$ vanishes on ∂H, then there exists a polynomial $\mathrm{\Pi }(x^{\prime })$ of degree m such that

$$h(x)=\{\begin{array}{ll}\mathrm{\Pi }(x^{\prime })+{\sum }_{j=1}^{[\frac{m}{2}]}\frac{{(-1)}^j}{(2j)!}{x}_n^{2j}{\mathrm{\Delta }}^j\mathrm{\Pi }(x^{\prime }),& m⩾2,\\ \mathrm{\Pi }(x^{\prime }),& m=0,1.\end{array}$$

3 Proof of Theorem 1

For any $ϵ>0$, there exists $R_{ϵ}>1$ such that

$${\int }_{\partial H(R_{ϵ},\mathrm{\infty })}\frac{|f(y^{\prime }){|}^p}{{(1+|y^{\prime }|)}^{n+\alpha -2}}d{y}^{\prime }<ϵ.$$

(3.1)

Take any point $x\in H(R_{ϵ},\mathrm{\infty })-E(ϵ;\mu ,(n-2)p-\beta )$ such that $|x|>2R_{ϵ}$, and write

$$\begin{array}{rcl}{N}_m[f](x)& =& ({\int }_{G_1}+{\int }_{G_2}+{\int }_{G_3}+{\int }_{G_4}+{\int }_{G_5})K_{n,m}(x,y^{\prime })f\left(y^{\prime }\right)d{y}^{\prime }\\ =& U_1(x)+U_2(x)+U_3(x)+U_4(x)+U_5(x),\end{array}$$

where

$$\begin{array}{c}{G}_1=\{y^{\prime }\in \partial H:|y^{\prime }|\le 1\},G_2=\{y^{\prime }\in \partial H:1<|y^{\prime }|\le \frac{|x|}{2}\},\hfill \\ G_3=\{y^{\prime }\in \partial H:\frac{|x|}{2}<|y^{\prime }|\le \frac{3}{2}|x|\},G_4=\{y^{\prime }\in \partial H:\frac{3}{2}|x|<|y^{\prime }|\le 2|x|\}\hfill \\ G_5=\{y^{\prime }\in \partial H:|y^{\prime }|\ge 2|x|\}.\hfill \end{array}$$

First note that

$$\begin{array}{rcl}|U_1(x)|& \lesssim & {\int }_{G_1}\frac{|f(y^{\prime })|}{|x-y^{\prime }{|}^{n-2}}d{y}^{\prime }\\ \lesssim & |x{|}^{2-n}{\int }_{G_1}|f\left(y^{\prime }\right)|d{y}^{\prime },\end{array}$$

so that

$$\underset{|x|\to \mathrm{\infty },x\in H}{lim}|x{|}^{-1+\frac{1-\alpha }{p}}{U}_1(x)=0.$$

(3.2)

If $m<2-\frac{1-\alpha }{p}$ and $\frac{1}{p}+\frac{1}{q}=1$, then $(3-n-m+\frac{n+\alpha -2}{p})q+n-1>0$. By Lemma 1(1), (3.1) and the Hölder inequality, we have

$$\begin{array}{rcl}|U_2(x)|& \lesssim & |x{|}^{m-1}{\int }_{G_2}|y^{\prime }{|}^{-n-m+3}\left|f\left(y^{\prime }\right)\right|d{y}^{\prime }\\ \lesssim & |x{|}^{m-1}{\left({\int }_{G_2}\frac{|f(y^{\prime }){|}^p}{|y^{\prime }{|}^{n+\alpha -2}}d{y}^{\prime }\right)}^{\frac{1}{p}}{\left({\int }_{G_2}\right|y^{\prime }{|}^{(-n-m+3+\frac{n+\alpha -2}{p})q}d{y}^{\prime })}^{\frac{1}{q}}\\ \lesssim & |x{|}^{1-\frac{1-\alpha }{p}}{\left({\int }_{G_2}\frac{|f(y^{\prime }){|}^p}{|y^{\prime }{|}^{n+\alpha -2}}d{y}^{\prime }\right)}^{\frac{1}{p}}.\end{array}$$

(3.3)

Put

$$U_2(x)=U_{21}(x)+U_{22}(x),$$

where

$$\begin{array}{c}{U}_{21}(x)={\int }_{G_2\cap B_{n-1}(R_{ϵ})}{K}_{n,m}(x,y^{\prime })f\left(y^{\prime }\right)d{y}^{\prime },\hfill \\ U_{22}(x)={\int }_{G_2\setminus B_{n-1}(R_{ϵ})}{K}_{n,m}(x,y^{\prime })f\left(y^{\prime }\right)d{y}^{\prime }.\hfill \end{array}$$

If $|x|\ge 2R_{ϵ}$, then

$$|U_{21}(x)|\lesssim R_{ϵ}^{2-m-\frac{1-\alpha }{p}}|x{|}^{m-1}.$$

Moreover, by (3.1) and (3.3), we get

$$|U_{22}(x)|\lesssim ϵ|x{|}^{1-\frac{1-\alpha }{p}}.$$

That is,

$$|U_2(x)|\lesssim ϵ|x{|}^{1-\frac{1-\alpha }{p}}.$$

(3.4)

By Lemma 1(3), (3.1) and the Hölder inequality, we have

$$|U_4(x)|\lesssim ϵx_n^{2-n}|x{|}^{n-1-\frac{1-\alpha }{p}}.$$

(3.5)

If $m>1-\frac{1-\alpha }{p}$, then $(2-n-m+\frac{n+\alpha -2}{p})q+n-1<0$. We obtain, by Lemma 1(4), (3.1) and the Hölder inequality,

$$\begin{array}{rcl}|U_5(x)|& \lesssim & |x{|}^m{\int }_{G_5}|y^{\prime }{|}^{-n-m+2}\left|f\left(y^{\prime }\right)\right|d{y}^{\prime }\\ \lesssim & |x{|}^m{\left({\int }_{G_5}\frac{|f(y^{\prime }){|}^p}{|y^{\prime }{|}^{n+\alpha -2}}d{y}^{\prime }\right)}^{\frac{1}{p}}{\left({\int }_{G_5}\right|y^{\prime }{|}^{(-n-m+2+\frac{n+\alpha -2}{p})q}d{y}^{\prime })}^{\frac{1}{q}}\\ \lesssim & ϵ|x{|}^{1-\frac{1-\alpha }{p}}.\end{array}$$

(3.6)

Finally, we shall estimate $U_3(x)$. Take a sufficiently small positive number b such that $\partial H[\frac{|x|}{2},\frac{3}{2}|x|]\subset B(x,\frac{|x|}{2})$ for any $x\in \mathrm{\Pi }(b)$, where

$$\mathrm{\Pi }(b)=\{x\in H;\underset{y^{\prime }\in \partial H}{inf}|\frac{x}{|x|}-\frac{y^{\prime }}{|y^{\prime }|}|<b\}$$

and divide H into two sets $\mathrm{\Pi }(b)$ and $H-\mathrm{\Pi }(b)$.

If $x\in H-\mathrm{\Pi }(b)$, then there exists a positive number $b^{\prime }$ such that $|x-y^{\prime }|\ge b^{\prime }|x|$ for any $y^{\prime }\in \partial H$, and hence

$$\begin{array}{rl}|U_3(x)|& \lesssim {\int }_{G_3}\left|y^{\prime }{|}^{2-n}\right|f\left(y^{\prime }\right)|d{y}^{\prime }\\ \lesssim |x{|}^m{\int }_{G_3}|y^{\prime }{|}^{2-n-m}\left|f\left(y^{\prime }\right)\right|d{y}^{\prime }\\ \lesssim ϵ|x{|}^{1-\frac{1-\alpha }{p}},\end{array}$$

which is similar to the estimate of $U_5(x)$.

We shall consider the case $x\in \mathrm{\Pi }(b)$. Now put

$$H_i(x)=\{y^{\prime }\in \partial H[\frac{|x|}{2},\frac{3}{2}|x|];2^{i-1}\delta (x)\le |x-y^{\prime }|<2^i\delta (x)\},$$

where $\delta (x)={inf}_{y^{\prime }\in H}|x-y^{\prime }|$.

Since $\partial H\cap \{y^{\prime }\in {\mathbf{R}}^{n-1}:|x-y^{\prime }|<\delta (x)\}=\mathrm{\varnothing }$, we have

$$U_3(x)=\sum _{i=1}^{i(x)}{\int }_{H_i(x)}\frac{|g(y^{\prime })|}{|x-y^{\prime }{|}^{n-2}}d{y}^{\prime },$$

where $i(x)$ is a positive integer satisfying $2^{i(x)-1}\delta (x)\le \frac{|x|}{2}<2^{i(x)}\delta (x)$.

Similar to the estimate of $U_5(x)$, we obtain

$$\begin{array}{r}{\int }_{H_i(x)}\frac{|g(y^{\prime })|}{|x-y^{\prime }{|}^{n-2}}d{y}^{\prime }\\ \lesssim {\int }_{H_i(x)}\frac{|g(y^{\prime })|}{{\{2^{i-1}\delta (x)\}}^{n-2}}d{y}^{\prime }\\ \lesssim \delta {(x)}^{\frac{\beta -(n-2)p}{p}}{\int }_{H_i(x)}\delta {(x)}^{\frac{(n-2)p-\beta }{p}-n+2}\left|g\left(y^{\prime }\right)\right|d{y}^{\prime }\\ \lesssim {cos}^{-\frac{\beta }{p}}\theta \delta {(x)}^{\frac{\beta -(n-2)p}{p}}{\int }_{H_i(x)}|x{|}^{-\frac{\beta }{p}}|g\left(y^{\prime }\right)|d{y}^{\prime }\\ \lesssim |x{|}^{n-2-\frac{\beta }{p}}{cos}^{-\frac{\beta }{p}}\theta \delta {(x)}^{\frac{\beta -(n-2)p}{p}}{\int }_{H_i(x)}|y^{\prime }{|}^{2-n}\left|g\left(y^{\prime }\right)\right|d{y}^{\prime }\\ \lesssim |x{|}^{n-1+\frac{\alpha -\beta -1}{p}}{cos}^{-\frac{\beta }{p}}\theta {\left(\frac{\mu (H_i(x))}{2^i\delta {(x)}^{(n-2)p-\beta }}\right)}^{\frac{1}{p}}\end{array}$$

for $i=0,1,2,\dots ,i(x)$.

Since $x\notin E(ϵ;\mu ,(n-2)p-\beta )$, we have

$$\frac{\mu (H_i(x))}{{\{2^i\delta (x)\}}^{(n-2)p-\beta }}\lesssim \frac{\mu (B_{n-1}(x,2^i\delta (x)))}{{\{2^i\delta (x)\}}^{(n-2)p-\beta }}\lesssim M(x;\mu ,(n-2)p-\beta )\lesssim ϵ|x{|}^{\beta -(n-2)p}$$

for $i=0,1,2,\dots ,i(x)-1$ and

$$\frac{\mu (H_{i(x)}(x))}{{\{2^i\delta (x)\}}^{(n-2)p-\beta }}\lesssim \frac{\mu (B_{n-1}(x,\frac{|x|}{2}))}{{(\frac{|x|}{2})}^{(n-2)p-\beta }}\lesssim ϵ|x{|}^{\beta -(n-2)p}.$$

So

$$|U_3(x)|\lesssim ϵ|x{|}^{1+\frac{\alpha -1}{p}}{sec}^{\frac{\beta }{p}}\theta .$$

(3.7)

Combining (3.2), (3.4)-(3.7), we obtain that if $R_{ϵ}$ is sufficiently large and ϵ is a sufficiently small number, then $N_m[f](x)=o(|x{|}^{1+\frac{\alpha -1}{p}}{sec}^{\frac{\beta }{p}}\theta )$ as $|x|\to \mathrm{\infty }$, where $x\in H(R_{ϵ},+\mathrm{\infty })-E(ϵ;\mu ,(n-2)p-\beta )$. Finally, there exists an additional finite ball $B_0$ covering $H(0,R_{ϵ}]$, which together with Lemma 2, gives the conclusion of Theorem 1.

4 Proof of Theorem 2

For any fixed $x\in H$, take a number R satisfying $R>max\{1,2|x|\}$. If $m>\frac{1-\alpha }{p}$, then $(2-n-m+\frac{n+\alpha -2}{p})q+n-1<0$. By (1.1), Lemma 1(4) and the Hölder inequality, we have

$$\begin{array}{r}{\int }_{\partial H(R,\mathrm{\infty })}\left|K_{n,m}(x,y^{\prime })\right|\left|f\left(y^{\prime }\right)\right|d{y}^{\prime }\\ \lesssim |x{|}^m{\int }_{\partial H(R,\mathrm{\infty })}|y^{\prime }{|}^{2-n-m}\left|f\left(y^{\prime }\right)\right|d{y}^{\prime }\\ \lesssim |x{|}^m{\left({\int }_{\partial H(R,\mathrm{\infty })}\frac{|f(y^{\prime }){|}^p}{|y^{\prime }{|}^{n+\alpha -2}}d{y}^{\prime }\right)}^{\frac{1}{p}}{\left({\int }_{\partial H(R,\mathrm{\infty })}\right|y^{\prime }{|}^{(-n-m+2+\frac{n+\alpha -2}{p})q}d{y}^{\prime })}^{\frac{1}{q}}\\ <\mathrm{\infty }.\end{array}$$

Hence $N_m[f](x)$ is absolutely convergent and finite for any $x\in H$. Thus $N_m[f](x)$ is harmonic on H.

To prove

$$\underset{x\to y^{\prime },x\in H}{lim}\frac{\partial }{\partial x_n}{N}_m[f](x)=f\left(y^{\prime }\right)$$

for any point $y^{\prime }\in \partial H$, we only need to apply Lemma 3 to $f(y)$ and $-f(y)$.

We complete the proof of Theorem 2.

5 Proof of Theorem 3

Consider the function $h^{\prime }(x)=h(x)-N_m[f](x)$. Then it follows from Theorems 2 and 3 that $h^{\prime }(x)$ is a solution of the Neumann problem on H with f and it is an even function of $x_n$ (see [[2], p.92]).

Since

$$0\le {\{h-N_m[f]\}}^{+}(x)\le h^{+}(x)+{\{N_m[f]\}}^{-}(x)$$

for any $x\in H$, and

$$\underset{|x|\to \mathrm{\infty },x\in H}{lim}{N}_m[f](x)=o(|x{|}^{1+\frac{\alpha -1}{p}})$$

from Theorem 2.

Moreover, (1.6) gives that

$$\underset{|x|\to \mathrm{\infty },x\in H}{lim}(h-N_m[f])(x)=o(|x{|}^{l+[1+\frac{\alpha -1}{p}]}).$$

This implies that $h^{\prime }(x)$ is a polynomial of degree less than $l+[1+\frac{\alpha -1}{p}]$ (see [[11], Appendix]), which gives the conclusion of Theorem 3 from Lemma 4.