# 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 $x=\left({x}^{\prime },{x}_{n}\right)$, where ${x}^{\prime }=\left({x}_{1},{x}_{2},\dots ,{x}_{n-1}\right)\in {\mathbf{R}}^{n-1}$ 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 ${B}_{n}\left(x,r\right)$ denote the open ball with center at x and radius r in ${\mathbf{R}}^{n}$.

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

We shall say that a set $E\subset H$ has a covering $\left\{{r}_{j},{R}_{j}\right\}$ if there exists a sequence of balls $\left\{{B}_{j}\right\}$ 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, $\left[d\right]$ is the integer part of d and $d=\left[d\right]+\left\{d\right\}$, 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

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

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

${K}_{n}\left(x,{y}^{\prime }\right)=-\frac{{\beta }_{n}}{|x-{y}^{\prime }{|}^{n-2}},$

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

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

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

The Neumann integral is defined by

$N\left[f\right]\left(x\right)={\int }_{\partial H}{K}_{n}\left(x,{y}^{\prime }\right)f\left({y}^{\prime }\right)\phantom{\rule{0.2em}{0ex}}d{y}^{\prime },$

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

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

${\int }_{\partial H}\frac{f\left({y}^{\prime }\right)}{{\left(1+|{y}^{\prime }|\right)}^{n-2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }<\mathrm{\infty }.$

In this paper, we consider functions f satisfying

${\int }_{\partial H}\frac{|f\left({y}^{\prime }\right){|}^{p}}{{\left(1+|{y}^{\prime }|\right)}^{n+\alpha -2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }<\mathrm{\infty }$
(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

$d\mu \left({y}^{\prime }\right)=\left\{\begin{array}{ll}|f\left({y}^{\prime }\right){|}^{p}|{y}^{\prime }{|}^{-n-\alpha +2}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime },& {y}^{\prime }\in \partial H\left(1,+\mathrm{\infty }\right),\\ 0,& Q\in {\mathbf{R}}^{n}-\partial H\left(1,+\mathrm{\infty }\right).\end{array}$

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 $M\left(x;\mu ,\delta \right)$ is defined by

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

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

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

${L}_{n,m}\left(x,{y}^{\prime }\right)=\left\{\begin{array}{ll}-{\beta }_{n}{\sum }_{k=0}^{m-1}\frac{|x{|}^{k}}{|y{|}^{n+k-2}}{C}_{k}^{\frac{n-2}{2}}\left(\frac{x\cdot {y}^{\prime }}{|x||{y}^{\prime }|}\right),& |{y}^{\prime }|\ge 1\phantom{\rule{0.25em}{0ex}}m\ge 1,\\ 0,& |{y}^{\prime }|<1\phantom{\rule{0.25em}{0ex}}m\ge 1,\\ 0,& m=0\end{array}$

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

${K}_{n,m}\left(x,{y}^{\prime }\right)={K}_{n}\left(x,{y}^{\prime }\right)-{L}_{n,m}\left(x,{y}^{\prime }\right)\phantom{\rule{1em}{0ex}}\left(m\ge 0\right).$

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

Put

${N}_{m}\left[f\right]\left(x\right)={\int }_{\partial H}{K}_{n,m}\left(x,{y}^{\prime }\right)f\left({y}^{\prime }\right)\phantom{\rule{0.2em}{0ex}}d{y}^{\prime },$

where f is a continuous function on ∂H. Here, note that ${N}_{0}\left[f\right]\left(x\right)$ is nothing but the Neumann integral $N\left[f\right]\left(x\right)$.

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

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

$\underset{|x|\to \mathrm{\infty },x\in H}{lim}{N}_{m}\left[f\right]\left(x\right)=o\left(|x{|}^{m}{sec}^{n-2}\theta \right).$
(1.2)

The next result deals with a type of uniqueness of solutions for the Neumann problem on H (see [, 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

${\int }_{\partial H}\frac{|f\left({y}^{\prime }\right)|}{{\left(1+|{y}^{\prime }|\right)}^{n+m-2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }<\mathrm{\infty },$

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

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

then

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

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

${\mathrm{\Delta }}^{j}=\left(\frac{{\partial }^{2}}{\partial {x}_{1}^{2}}+\frac{{\partial }^{2}}{\partial {x}_{2}^{2}}+\cdots +\frac{{\partial }^{2}}{\partial {x}_{n-1}^{2}}\right)\phantom{\rule{1em}{0ex}}\left(j=1,2\dots \right)$

and $\mathrm{\Pi }\left({x}^{\prime }\right)$ 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 ${N}_{m}\left[f\right]$ at infinity and establish the following theorem.

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

If f is a measurable function on satisfying (1.1), then there exists a covering $\left\{{r}_{j},{R}_{j}\right\}$ of $E\left(ϵ;\mu ,\left(n-2\right)p-\beta \right)$ (H) satisfying

$\sum _{j=0}^{\mathrm{\infty }}{\left(\frac{{r}_{j}}{{R}_{j}}\right)}^{\left(n-2\right)p-\beta }<\mathrm{\infty }$
(1.3)

such that

$\underset{|x|\to \mathrm{\infty },x\in H-E\left(ϵ;\mu ,\left(n-2\right)p-\beta \right)}{lim}{N}_{m}\left[f\right]\left(x\right)=o\left(|x{|}^{1+\frac{\alpha -1}{p}}{sec}^{\frac{\beta }{p}}\theta \right).$
(1.4)

Corollary 1 Let $1, $n+\alpha -2>-\left(n-1\right)\left(p-1\right)$ and

$1-\frac{1-\alpha }{p}

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

$\underset{|x|\to \mathrm{\infty },x\in H}{lim}{N}_{m}\left[f\right]\left(x\right)=o\left(|x{|}^{1+\frac{\alpha -1}{p}}{sec}^{n-2}\theta \right).$
(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 ${N}_{m}\left[f\right]$ is a solution of the Neumann problem on H with f and (1.4) holds, where the exceptional set $E\left(ϵ;\mu ,\left(n-2\right)p-\beta \right)$ (H) has a covering $\left\{{r}_{j},{R}_{j}\right\}$ 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 $E\left(ϵ;\mu ,0\right)$ 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 $1\le p<\mathrm{\infty }$, $n+\alpha -2>-\left(n-1\right)\left(p-1\right)$ and

If f is a continuous function on ∂H satisfying (1.1), then the function ${N}_{m}\left[f\right]$ 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

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

$\underset{|x|\to \mathrm{\infty },x\in H}{lim}{h}^{+}\left(x\right)=o\left(|x{|}^{l+\left[1+\frac{\alpha -1}{p}\right]}\right),$
(1.6)

then

$h\left(x\right)={N}_{m}\left[f\right]\left(x\right)+\mathrm{\Pi }\left({x}^{\prime }\right)+\sum _{j=1}^{\left[\frac{l+\left[1+\frac{\alpha -1}{p}\right]}{2}\right]}\frac{{\left(-1\right)}^{j}}{\left(2j\right)!}{x}_{n}^{2j}{\mathrm{\Delta }}^{j}\mathrm{\Pi }\left({x}^{\prime }\right)$
(1.7)

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

## 2 Lemmas

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

Lemma 1

1. (1)

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

2. (2)

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

3. (3)

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

4. (4)

If $|{y}^{\prime }|\ge 2|x|$ and $|{y}^{\prime }|\ge 1$, then $|{K}_{n,m}\left(x,{y}^{\prime }\right)|\lesssim |x{|}^{m}|{y}^{\prime }{|}^{2-n-m}$.

The following lemma is due to Qiao (see ).

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

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

Lemma 3 ([, 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

$\underset{x\in H,x\to {y}^{\prime }}{lim sup}\frac{\partial }{\partial {x}_{n}}{N}_{m}\left[f\right]\left(x\right)\le f\left({y}^{\prime }\right)$

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

Lemma 4 ([, Lemma 1])

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

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

## 3 Proof of Theorem 1

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

${\int }_{\partial H\left({R}_{ϵ},\mathrm{\infty }\right)}\frac{|f\left({y}^{\prime }\right){|}^{p}}{{\left(1+|{y}^{\prime }|\right)}^{n+\alpha -2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }<ϵ.$
(3.1)

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

$\begin{array}{rcl}{N}_{m}\left[f\right]\left(x\right)& =& \left({\int }_{{G}_{1}}+{\int }_{{G}_{2}}+{\int }_{{G}_{3}}+{\int }_{{G}_{4}}+{\int }_{{G}_{5}}\right){K}_{n,m}\left(x,{y}^{\prime }\right)f\left({y}^{\prime }\right)\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ =& {U}_{1}\left(x\right)+{U}_{2}\left(x\right)+{U}_{3}\left(x\right)+{U}_{4}\left(x\right)+{U}_{5}\left(x\right),\end{array}$

where

$\begin{array}{c}{G}_{1}=\left\{{y}^{\prime }\in \partial H:|{y}^{\prime }|\le 1\right\},\phantom{\rule{2em}{0ex}}{G}_{2}=\left\{{y}^{\prime }\in \partial H:1<|{y}^{\prime }|\le \frac{|x|}{2}\right\},\hfill \\ {G}_{3}=\left\{{y}^{\prime }\in \partial H:\frac{|x|}{2}<|{y}^{\prime }|\le \frac{3}{2}|x|\right\},\phantom{\rule{2em}{0ex}}{G}_{4}=\left\{{y}^{\prime }\in \partial H:\frac{3}{2}|x|<|{y}^{\prime }|\le 2|x|\right\}\hfill \\ {G}_{5}=\left\{{y}^{\prime }\in \partial H:|{y}^{\prime }|\ge 2|x|\right\}.\hfill \end{array}$

First note that

$\begin{array}{rcl}|{U}_{1}\left(x\right)|& \lesssim & {\int }_{{G}_{1}}\frac{|f\left({y}^{\prime }\right)|}{|x-{y}^{\prime }{|}^{n-2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \lesssim & |x{|}^{2-n}{\int }_{{G}_{1}}|f\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime },\end{array}$

so that

$\underset{|x|\to \mathrm{\infty },x\in H}{lim}|x{|}^{-1+\frac{1-\alpha }{p}}{U}_{1}\left(x\right)=0.$
(3.2)

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

$\begin{array}{rcl}|{U}_{2}\left(x\right)|& \lesssim & |x{|}^{m-1}{\int }_{{G}_{2}}|{y}^{\prime }{|}^{-n-m+3}|f\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \lesssim & |x{|}^{m-1}{\left({\int }_{{G}_{2}}\frac{|f\left({y}^{\prime }\right){|}^{p}}{|{y}^{\prime }{|}^{n+\alpha -2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\right)}^{\frac{1}{p}}{\left({\int }_{{G}_{2}}|{y}^{\prime }{|}^{\left(-n-m+3+\frac{n+\alpha -2}{p}\right)q}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\right)}^{\frac{1}{q}}\\ \lesssim & |x{|}^{1-\frac{1-\alpha }{p}}{\left({\int }_{{G}_{2}}\frac{|f\left({y}^{\prime }\right){|}^{p}}{|{y}^{\prime }{|}^{n+\alpha -2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\right)}^{\frac{1}{p}}.\end{array}$
(3.3)

Put

${U}_{2}\left(x\right)={U}_{21}\left(x\right)+{U}_{22}\left(x\right),$

where

$\begin{array}{c}{U}_{21}\left(x\right)={\int }_{{G}_{2}\cap {B}_{n-1}\left({R}_{ϵ}\right)}{K}_{n,m}\left(x,{y}^{\prime }\right)f\left({y}^{\prime }\right)\phantom{\rule{0.2em}{0ex}}d{y}^{\prime },\hfill \\ {U}_{22}\left(x\right)={\int }_{{G}_{2}\setminus {B}_{n-1}\left({R}_{ϵ}\right)}{K}_{n,m}\left(x,{y}^{\prime }\right)f\left({y}^{\prime }\right)\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }.\hfill \end{array}$

If $|x|\ge 2{R}_{ϵ}$, then

$|{U}_{21}\left(x\right)|\lesssim {R}_{ϵ}^{2-m-\frac{1-\alpha }{p}}|x{|}^{m-1}.$

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

$|{U}_{22}\left(x\right)|\lesssim ϵ|x{|}^{1-\frac{1-\alpha }{p}}.$

That is,

$|{U}_{2}\left(x\right)|\lesssim ϵ|x{|}^{1-\frac{1-\alpha }{p}}.$
(3.4)

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

$|{U}_{4}\left(x\right)|\lesssim ϵ{x}_{n}^{2-n}|x{|}^{n-1-\frac{1-\alpha }{p}}.$
(3.5)

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

$\begin{array}{rcl}|{U}_{5}\left(x\right)|& \lesssim & |x{|}^{m}{\int }_{{G}_{5}}|{y}^{\prime }{|}^{-n-m+2}|f\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \lesssim & |x{|}^{m}{\left({\int }_{{G}_{5}}\frac{|f\left({y}^{\prime }\right){|}^{p}}{|{y}^{\prime }{|}^{n+\alpha -2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\right)}^{\frac{1}{p}}{\left({\int }_{{G}_{5}}|{y}^{\prime }{|}^{\left(-n-m+2+\frac{n+\alpha -2}{p}\right)q}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\right)}^{\frac{1}{q}}\\ \lesssim & ϵ|x{|}^{1-\frac{1-\alpha }{p}}.\end{array}$
(3.6)

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

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

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

If $x\in H-\mathrm{\Pi }\left(b\right)$, 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

$\begin{array}{rl}|{U}_{3}\left(x\right)|& \lesssim {\int }_{{G}_{3}}|{y}^{\prime }{|}^{2-n}|f\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \lesssim |x{|}^{m}{\int }_{{G}_{3}}|{y}^{\prime }{|}^{2-n-m}|f\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \lesssim ϵ|x{|}^{1-\frac{1-\alpha }{p}},\end{array}$

which is similar to the estimate of ${U}_{5}\left(x\right)$.

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

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

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

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

${U}_{3}\left(x\right)=\sum _{i=1}^{i\left(x\right)}{\int }_{{H}_{i}\left(x\right)}\frac{|g\left({y}^{\prime }\right)|}{|x-{y}^{\prime }{|}^{n-2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime },$

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

Similar to the estimate of ${U}_{5}\left(x\right)$, we obtain

$\begin{array}{r}{\int }_{{H}_{i}\left(x\right)}\frac{|g\left({y}^{\prime }\right)|}{|x-{y}^{\prime }{|}^{n-2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \phantom{\rule{1em}{0ex}}\lesssim {\int }_{{H}_{i}\left(x\right)}\frac{|g\left({y}^{\prime }\right)|}{{\left\{{2}^{i-1}\delta \left(x\right)\right\}}^{n-2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \phantom{\rule{1em}{0ex}}\lesssim \delta {\left(x\right)}^{\frac{\beta -\left(n-2\right)p}{p}}{\int }_{{H}_{i}\left(x\right)}\delta {\left(x\right)}^{\frac{\left(n-2\right)p-\beta }{p}-n+2}|g\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \phantom{\rule{1em}{0ex}}\lesssim {cos}^{-\frac{\beta }{p}}\theta \delta {\left(x\right)}^{\frac{\beta -\left(n-2\right)p}{p}}{\int }_{{H}_{i}\left(x\right)}|x{|}^{-\frac{\beta }{p}}|g\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \phantom{\rule{1em}{0ex}}\lesssim |x{|}^{n-2-\frac{\beta }{p}}{cos}^{-\frac{\beta }{p}}\theta \delta {\left(x\right)}^{\frac{\beta -\left(n-2\right)p}{p}}{\int }_{{H}_{i}\left(x\right)}|{y}^{\prime }{|}^{2-n}|g\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \phantom{\rule{1em}{0ex}}\lesssim |x{|}^{n-1+\frac{\alpha -\beta -1}{p}}{cos}^{-\frac{\beta }{p}}\theta {\left(\frac{\mu \left({H}_{i}\left(x\right)\right)}{{2}^{i}\delta {\left(x\right)}^{\left(n-2\right)p-\beta }}\right)}^{\frac{1}{p}}\end{array}$

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

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

$\frac{\mu \left({H}_{i}\left(x\right)\right)}{{\left\{{2}^{i}\delta \left(x\right)\right\}}^{\left(n-2\right)p-\beta }}\lesssim \frac{\mu \left({B}_{n-1}\left(x,{2}^{i}\delta \left(x\right)\right)\right)}{{\left\{{2}^{i}\delta \left(x\right)\right\}}^{\left(n-2\right)p-\beta }}\lesssim M\left(x;\mu ,\left(n-2\right)p-\beta \right)\lesssim ϵ|x{|}^{\beta -\left(n-2\right)p}$

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

$\frac{\mu \left({H}_{i\left(x\right)}\left(x\right)\right)}{{\left\{{2}^{i}\delta \left(x\right)\right\}}^{\left(n-2\right)p-\beta }}\lesssim \frac{\mu \left({B}_{n-1}\left(x,\frac{|x|}{2}\right)\right)}{{\left(\frac{|x|}{2}\right)}^{\left(n-2\right)p-\beta }}\lesssim ϵ|x{|}^{\beta -\left(n-2\right)p}.$

So

$|{U}_{3}\left(x\right)|\lesssim ϵ|x{|}^{1+\frac{\alpha -1}{p}}{sec}^{\frac{\beta }{p}}\theta .$
(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 ${N}_{m}\left[f\right]\left(x\right)=o\left(|x{|}^{1+\frac{\alpha -1}{p}}{sec}^{\frac{\beta }{p}}\theta \right)$ as $|x|\to \mathrm{\infty }$, where $x\in H\left({R}_{ϵ},+\mathrm{\infty }\right)-E\left(ϵ;\mu ,\left(n-2\right)p-\beta \right)$. Finally, there exists an additional finite ball ${B}_{0}$ covering $H\left(0,{R}_{ϵ}\right]$, 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 $R>max\left\{1,2|x|\right\}$. If $m>\frac{1-\alpha }{p}$, then $\left(2-n-m+\frac{n+\alpha -2}{p}\right)q+n-1<0$. By (1.1), Lemma 1(4) and the Hölder inequality, we have

$\begin{array}{r}{\int }_{\partial H\left(R,\mathrm{\infty }\right)}|{K}_{n,m}\left(x,{y}^{\prime }\right)||f\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \phantom{\rule{1em}{0ex}}\lesssim |x{|}^{m}{\int }_{\partial H\left(R,\mathrm{\infty }\right)}|{y}^{\prime }{|}^{2-n-m}|f\left({y}^{\prime }\right)|\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\\ \phantom{\rule{1em}{0ex}}\lesssim |x{|}^{m}{\left({\int }_{\partial H\left(R,\mathrm{\infty }\right)}\frac{|f\left({y}^{\prime }\right){|}^{p}}{|{y}^{\prime }{|}^{n+\alpha -2}}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\right)}^{\frac{1}{p}}{\left({\int }_{\partial H\left(R,\mathrm{\infty }\right)}|{y}^{\prime }{|}^{\left(-n-m+2+\frac{n+\alpha -2}{p}\right)q}\phantom{\rule{0.2em}{0ex}}d{y}^{\prime }\right)}^{\frac{1}{q}}\\ \phantom{\rule{1em}{0ex}}<\mathrm{\infty }.\end{array}$

Hence ${N}_{m}\left[f\right]\left(x\right)$ is absolutely convergent and finite for any $x\in H$. Thus ${N}_{m}\left[f\right]\left(x\right)$ is harmonic on H.

To prove

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

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

We complete the proof of Theorem 2.

## 5 Proof of Theorem 3

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

Since

$0\le {\left\{h-{N}_{m}\left[f\right]\right\}}^{+}\left(x\right)\le {h}^{+}\left(x\right)+{\left\{{N}_{m}\left[f\right]\right\}}^{-}\left(x\right)$

for any $x\in H$, and

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

from Theorem 2.

Moreover, (1.6) gives that

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

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

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## Acknowledgements

The authors are thankful to the referees for their helpful suggestions and necessary corrections in the completion of this paper.

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Ren, Y., Yang, P. Growth estimates for modified Neumann integrals in a half-space. J Inequal Appl 2013, 572 (2013). https://doi.org/10.1186/1029-242X-2013-572 