Boundary behaviors of modified Green’s function with respect to the stationary Schrödinger operator and its applications

Zhiqiang Li

doi:10.1186/s13661-015-0499-x

Research
Open Access

Boundary behaviors of modified Green’s function with respect to the stationary Schrödinger operator and its applications

Zhiqiang Li¹Email author

Boundary Value Problems20152015:242

https://doi.org/10.1186/s13661-015-0499-x

Received: 29 May 2015
Accepted: 26 November 2015
Published: 29 December 2015

Abstract

In this paper, we construct a modified Green’s function with respect to the stationary Schrödinger operator on cones. As applications, we not only obtain the boundary behaviors of generalized harmonic functions but also characterize the geometrical properties of the exceptional sets with respect to the Schrödinger operator.

Keywords

boundary behavior
modified Green’s function
stationary Schrödinger operator
cone

1 Introduction and results

Let R and ${\mathbf{R}}_{+}$ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by ${\mathbf{R}}^{n}$ ($n\geq2$) the n-dimensional Euclidean space. A point in ${\mathbf{R}}^{n}$ is denoted by $P=(X,x_{n})$, $X=(x_{1},x_{2},\ldots,x_{n-1})$. The Euclidean distance between two points P and Q in ${\mathbf{R}}^{n}$ is denoted by $|P-Q|$. Also $|P-O|$ with the origin O of ${\mathbf{R}}^{n}$ is simply denoted by $|P|$. The boundary and the closure of a set S in ${\mathbf {R}}^{n}$ are denoted by ∂ S and $\overline {\mathbf{S}}$, respectively.

We introduce a system of spherical coordinates $(r,\Theta)$, $\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})$, in ${\mathbf{R}}^{n}$ which are related to Cartesian coordinates $(x_{1},x_{2},\ldots,x_{n-1},x_{n})$ by $x_{n}=r\cos\theta_{1}$.

The unit sphere and the upper half unit sphere in ${\mathbf{R}}^{n}$ are denoted by ${\mathbf{S}}^{n-1}$ and ${\mathbf{S}}^{n-1}_{+}$, respectively. For simplicity, a point $(1,\Theta)$ on ${\mathbf{S}}^{n-1}$ and the set $\{\Theta; (1,\Theta)\in\Omega\}$ for a set Ω, $\Omega\subset{\mathbf{S}}^{n-1}$, are often identified with Θ and Ω, respectively. For two sets $\Xi\subset{\mathbf{R}}_{+}$ and $\Omega\subset{\mathbf{S}}^{n-1}$, the set $\{(r,\Theta)\in {\mathbf{R}}^{n}; r\in\Xi,(1,\Theta)\in\Omega\}$ in ${\mathbf{R}}^{n}$ is simply denoted by $\Xi\times\Omega$. In particular, the half space ${\mathbf {R}}_{+}\times{\mathbf{S}}^{n-1}_{+}=\{(X,x_{n})\in{\mathbf{R}}^{n}; x_{n}>0\}$ will be denoted by ${\mathbf{T}}_{n}$.

For $P\in{\mathbf{R}}^{n}$ and $r>0$, let $B(P,r)$ denote the open ball with center at P and radius r in ${\mathbf{R}}^{n}$. $S_{r}=\partial{B(O,r)}$. By $C_{n}(\Omega)$, we denote the set ${\mathbf{R}}_{+}\times\Omega$ in ${\mathbf{R}}^{n}$ with the domain Ω on ${\mathbf{S}}^{n-1}$. We call it a cone. Then $T_{n}$ is a special cone obtained by putting $\Omega={\mathbf{S}}^{n-1}_{+}$. We denote the sets $I\times\Omega$ and $I\times\partial{\Omega}$ with an interval on R by $C_{n}(\Omega;I)$ and $S_{n}(\Omega;I)$. By $S_{n}(\Omega ; r)$ we denote $C_{n}(\Omega)\cap S_{r}$. By $S_{n}(\Omega)$ we denote $S_{n}(\Omega; (0,+\infty))$, which is $\partial{C_{n}(\Omega)}-\{O\}$.

We shall say that a set $E\subset C_{n}(\Omega)$ has a covering $\{r_{j}, R_{j}\}$ if there exists a sequence of balls $\{B_{j}\}$ with centers in $C_{n}(\Omega)$ such that $E\subset\bigcup_{j=1}^{\infty} B_{j}$, where $r_{j}$ is the radius of $B_{j}$ and $R_{j}$ is the distance between the origin and the center of $B_{j}$.

Let $\mathscr{A}_{a}$ denote the class of non-negative radial potentials $a(P)$, i.e. $0\leq a(P)=a(r)$, $P=(r,\Theta)\in C_{n}(\Omega)$, such that $a\in L_{\mathrm{loc}}^{b}(C_{n}(\Omega))$ with some $b> {n}/{2}$ if $n\geq4$ and with $b=2$ if $n=2$ or $n=3$.

This article is devoted to the stationary Schrödinger equation

$$\operatorname{Sch}_{a}u(P)=-\Delta u(P)+a(P)u(P)=0\quad \text{for } P \in C_{n}(\Omega), $$

where Δ is the Laplace operator and $a\in\mathscr{A}_{a}$. These solutions are called generalized harmonic functions (associated with the operator $\operatorname{Sch}_{a}$). Note that they are (classical) harmonic functions in the case $a=0$. Under these assumptions the operator $\operatorname{Sch}_{a}$ can be extended in the usual way from the space $C_{0}^{\infty}(C_{n}(\Omega))$ to an essentially self-adjoint operator on $L^{2}(C_{n}(\Omega))$ (see [1]). We will denote it $\operatorname{Sch}_{a}$ as well. The latter has a Green-Sch function $G(\Omega;a)(P,Q)$. Here $G(\Omega;a)(P,Q)$ is positive on $C_{n}(\Omega)$ and its inner normal derivative $\partial G(\Omega;a)(P,Q)/{\partial n_{Q}}\geq0$. We denote this derivative by $\mathbb{PI}(\Omega;a)(P,Q)$, which is called the Poisson kernel with respect to the stationary Schrödinger operator. We remark that $G(\Omega;0)(P,Q)$ and $\mathbb{PI}(\Omega;0)(P,Q)$ are the Green’s function and Poisson kernel of the Laplacian in $C_{n}(\Omega)$, respectively.

Let $\Delta^{*}$ be a Laplace-Beltrami operator (spherical part of the Laplace) on $\Omega\subset{\mathbf{S}}^{n-1}$ and $\lambda_{j}$ ($j=1,2,3\ldots$ , $0<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\ldots$) be the eigenvalues of the eigenvalue problem for $\Delta^{*}$ on Ω (see, e.g., [2], p.41)

$$\begin{aligned}& \Delta^{*}\varphi(\Theta)+\lambda\varphi(\Theta)=0\quad \text{in } \Omega, \\& \varphi(\Theta)=0\quad \text{on } \partial{\Omega}. \end{aligned}$$

Corresponding eigenfunctions are denoted by $\varphi_{jv}$ ($1\leq v\leq v_{j}$), where $v_{j}$ is the multiplicity of $\lambda_{j}$. We set $\lambda_{0}=0$, normalize the eigenfunctions in $L^{2}(\Omega)$, and $\varphi_{1}=\varphi_{11}>0$.

In order to ensure the existence of $\lambda_{j}$ ($j=1,2,3\ldots$), we put a rather strong assumption on Ω: if $n\geq3$, then Ω is a $C^{2,\alpha}$-domain ($0<\alpha<1$) on ${\mathbf {S}}^{n-1}$ surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see [3], pp.88-89, for the definition of $C^{2,\alpha}$-domain). Then $\varphi_{jv}\in C^{2}(\overline{\Omega})$ ($j=1,2,3,\ldots$ , $1\leq v\leq v_{j}$) and ${\partial\varphi_{1}}/{\partial n}>0$ on ∂Ω (here and below, ${\partial}/{\partial n}$ denotes differentiation along the interior normal).

Hence the well-known estimates (see, e.g., [4], p.14) imply the following inequality:

$$ \sum_{v=1}^{v_{j}}\varphi_{jv}( \Theta)\frac{\partial\varphi _{jv}(\Phi)}{\partial n_{\Phi}}\leq M(n)j^{2n-1}, $$

(1.1)

where the symbol $M(n)$ denotes a constant depending only on n.

Let $V_{j}(r)$ ($j=1,2,3,\ldots$) and $W_{j}(r)$ ($j=1,2,3,\ldots$) stand, respectively, for the increasing and non-increasing, as $r\rightarrow+\infty$, solutions of the equation

$$ -Q''(r)-\frac{n-1}{r}Q'(r)+ \biggl( \frac{\lambda_{j}}{r^{2}}+a(r) \biggr)Q(r)=0,\quad 0< r< \infty, $$

(1.2)

normalized under the condition $V_{j}(1)=W_{j}(1)=1$ (see [5, 6]).

We shall also consider the class $\mathscr{B}_{a}$, consisting of the potentials $a\in\mathscr{A}_{a}$ such that there exists a finite limit $\lim_{r\rightarrow\infty}r^{2} a(r)=k\in[0,\infty)$, moreover, $r^{-1}|r^{2} a(r)-k|\in L(1,\infty)$. If $a\in \mathscr{B}_{a}$, then the g.h.f.s. are continuous (see [7]).

In the rest of this paper, we assume that $a\in\mathscr{B}_{a}$ and we shall suppress this assumption for simplicity. Further, we use the standard notations $u^{+}=\max(u,0)$, $u^{-}=-\min(u,0)$, $[d]$ is the integer part of d and $d=[d]+\{d\}$, where d is a positive real number.

Denote

$$\iota_{j,k}^{\pm}=\frac{2-n\pm\sqrt{(n-2)^{2}+4(k+\lambda_{j})}}{2} \quad (j=0,1,2,3\ldots). $$

It is well known (see [8]) that in the case under consideration the solutions to equation (1.2) have the asymptotics

$$ V_{j}(r)\sim d_{1} r^{\iota_{j,k}^{+}},\qquad W_{j}(r)\sim d_{2} r^{\iota _{j,k}^{-}}\quad \text{as } r \rightarrow\infty, $$

(1.3)

where $d_{1}$ and $d_{2}$ are some positive constants.

If $a\in\mathscr{A}_{a}$, it is well known that the following expansion holds for the Green’s function $G(\Omega;a)(P,Q)$ (see [9], Chapter 11):

$$ G(\Omega;a) (P,Q)=\sum_{j=0}^{\infty} \frac{1}{\chi'(1)}V_{j}\bigl(\min (r,t)\bigr)W_{j}\bigl( \max(r,t)\bigr) \Biggl( \sum_{v=1}^{v_{j}} \varphi_{jv}(\Theta )\varphi_{jv}(\Phi) \Biggr), $$

(1.4)

where $P=(r,\Theta)$, $Q=(t,\Phi)$, $r\neq t$ and $\chi'(s)=w (W_{1}(r),V_{1}(r) )|_{r=s}$ is their Wronskian. The series converges uniformly if either $r\leq s t$ or $t\leq s r$ ($0< s<1$). The expansion (1.4) can also be rewritten in terms of the Gegenbauer polynomials.

For a non-negative integer m and two points $P=(r,\Theta), Q=(t,\Phi)\in C_{n}(\Omega)$, we put

$$K(\Omega;a,m) (P,Q)=\left \{ \textstyle\begin{array}{l@{\quad}l} 0 & \mbox{if } 0< t< 1, \\ \widetilde{K}(\Omega;a,m)(P,Q) & \mbox{if } 1\leq t< \infty , \end{array}\displaystyle \right . $$

where

$$\widetilde{K}(\Omega;a,m) (P,Q)=\sum_{j=0}^{m} \frac{1}{\chi'(1)}V_{j}(r)W_{j}(t) \Biggl( \sum _{v=1}^{v_{j}}\varphi_{jv}(\Theta) \varphi_{jv}(\Phi) \Biggr). $$

If we modify the Green’s function with respect to the stationary Schrödinger operator on cones as follows:

$$G(\Omega;a,m) (P,Q)=G(\Omega;a) (P,Q)-K(\Omega;a,m) (P,Q) $$

for two points $P=(r,\Theta), Q=(t,\Phi)\in C_{n}(\Omega)$, then the modified Poisson kernel with respect to the stationary Schrödinger operator on cones can be defined by

$$\mathbb{PI}(\Omega;a,m) (P,Q)=\frac{\partial G(\Omega;a,m)(P,Q)}{\partial n_{Q}}. $$

We remark that

$$\mathbb{PI}(\Omega;a,0) (P,Q)=\mathbb{PI}(\Omega;a) (P,Q). $$

In this paper, we shall use the modified Poisson integrals with respect to the stationary Schrödinger operator defined by

$$\mathbb{PI}_{\Omega}^{a}(m,u) (P)= \int_{S_{n}(\Omega)}\mathbb {PI}(\Omega;a,m) (P,Q)u(Q)\, d \sigma_{Q}, $$

where $u(Q)$ is a continuous function on $\partial C_{n}(\Omega)$ and $d\sigma_{Q}$ is the surface area element on $S_{n}(\Omega)$.

If γ is a real number and $\gamma\geq0$ (resp. $\gamma<0$), we assume in addition that $1\leq p<\infty$,

$$\begin{aligned}& \iota_{[\gamma],k}^{+}+\{\gamma\}>\bigl(-\iota_{1,k}^{+}-n+2 \bigr)p+n-1 \\& \bigl(\mbox{resp. }-\iota_{[-\gamma],k}^{+}-\{-\gamma\}>\bigl(- \iota_{1,k}^{+}-n+2\bigr)p+n-1\bigr), \end{aligned}$$

in the case $p>1$,

$$\begin{aligned}& \frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}< \iota _{m+1,k}^{+}< \frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}+1 \\& \biggl( \mbox{resp. } \frac{-\iota_{[-\gamma],k}^{+}-\{-\gamma\}-n+1}{p}< \iota _{m+1,k}^{+}< \frac{-\iota_{[-\gamma],k}^{+}-\{-\gamma\}-n+1}{p}+1 \biggr), \end{aligned}$$

and in the case $p=1$,

$$\begin{aligned}& \iota_{[\gamma],k}^{+}+\{\gamma\}-n+1\leq\iota_{m+1,k}^{+}< \iota _{[\gamma],k}^{+}+\{\gamma\}-n+2 \\& \bigl( \mbox{resp. } -\iota_{[-\gamma],k}^{+}-\{-\gamma\}-n+1\leq \iota _{m+1,k}^{+}< -\iota_{[-\gamma],k}^{+}-\{- \gamma\}-n+2 \bigr). \end{aligned}$$

If these conditions all hold, we write $\gamma\in \mathscr{C}(k,p,m,n)$ (resp. $\gamma\in\mathscr{D}(k,p,m,n)$).

Let $\gamma\in\mathscr{C}(k,p,m,n)$ (resp. $\gamma\in \mathscr{D}(k,p,m,n)$) and u be functions on $\partial{C_{n}(\Omega)}$ satisfying

$$ \begin{aligned} &\int_{S_{n}(\Omega)}\frac{|u(t,\Phi)|^{p}}{1+t^{\iota_{[\gamma ],k}^{+}+\{\gamma\}}}\, d\sigma_{Q}< \infty \\ &\biggl(\mbox{resp. } \int _{S_{n}(\Omega)}\bigl\vert u(t,\Phi)\bigr\vert ^{p} \bigl(1+t^{\iota_{[-\gamma],k}^{+}+\{ -\gamma\}}\bigr)\, d\sigma_{Q}< \infty \biggr). \end{aligned} $$

(1.5)

For γ and u, we define the positive measure μ (resp. ν) on ${\mathbf{R}}^{n}$ by

$$\begin{aligned}& d\mu(Q)=\left \{ \textstyle\begin{array}{l@{\quad}l} |u(t,\Phi)|^{p}{t^{-\iota_{[\gamma],k}^{+}-\{\gamma\}}}\, d\sigma_{Q}, & Q=(t,\Phi)\in S_{n}(\Omega; (1,+\infty)) , \\ 0,& Q\in{\mathbf{R}}^{n}-S_{n}(\Omega; (1,+\infty)) \end{array}\displaystyle \right . \\& \left (\mbox{resp. }d\nu(Q)=\left \{ \textstyle\begin{array}{l@{\quad}l} |u(t,\Phi)|^{p}{t^{\iota_{[-\gamma],k}^{+}+\{-\gamma\}}}\, d\sigma _{Q}, & Q=(t,\Phi)\in S_{n}(\Omega; (1,+\infty)) , \\ 0,& Q\in{\mathbf{R}}^{n}-S_{n}(\Omega; (1,+\infty)) \end{array}\displaystyle \right . \right ). \end{aligned}$$

We remark that the total masses of μ and ν are finite.

Let $p>-1$, $\epsilon>0$, $0\leq\zeta\leq np$ and μ be any positive measure on ${\mathbf{R}}^{n}$ having finite mass. For each $P=(r,\Theta)\in{\mathbf{R}}^{n}-\{O\}$, the maximal function with respect to the stationary Schrödinger operator is defined by (see [10])

$$M(P;\mu,\zeta)=\sup_{ 0< \rho< \frac{r}{2}}\mu\bigl(B(P,\rho)\bigr) \bigl[V_{1}(\rho)W_{1}(\rho)\bigr]^{p}\rho ^{\zeta-2p}. $$

The set

$$\bigl\{ P=(r,\Theta)\in{\mathbf{R}}^{n}-\{O\}; M(P;\mu,\zeta) \bigl[V_{1}(\rho)W_{1}(\rho)\bigr]^{-p} \rho^{2p-\zeta}>\epsilon\bigr\} $$

is denoted by $E(\epsilon; \mu, \zeta)$.

Recently, Yoshida-Miyamoto (cf. [11], Theorem 1) gave the asymptotic behavior of $\mathbb{PI}_{\Omega}^{0}(m,u)(P)$ at infinity on cones.

Theorem A

If u is a continuous function on $\partial{C_{n}(\Omega)}$ satisfying

$$\int_{\partial{C_{n}(\Omega)}}\frac{|u(t,\Phi)|}{1+t^{\iota _{n,0}^{+}+m}}\, dQ< \infty, $$

then

$$\lim_{r\rightarrow\infty, P=(r,\Theta)\in T_{n}}\mathbb{PI}_{\Omega}^{0}(m,u) (P)=o\bigl(\iota_{m+1,0}^{+}\varphi _{1}^{1-n}( \Theta)\bigr). $$

Now we have the following.

Theorem 1

If $p>-1$, $\gamma\in \mathscr{C}(k,p,m,n)$ (resp. $\gamma\in\mathscr{D}(k,p,m,n)$) and u is a measurable function on $\partial{C_{n}(\Omega)}$ satisfying (1.5), then there exists a covering $\{r_{j},R_{j}\}$ of $E(\epsilon; \mu,\zeta)$ (resp. $E(\epsilon; \nu,\zeta)$) ($\subset C_{n}(\Omega)$) satisfying

$$ \sum_{j=0}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)^{2p-\zeta} \biggl[\frac {V_{j}(R_{j})}{V_{j}(r_{j})}\frac{W_{j}(R_{j})}{W_{j}(r_{j})} \biggr]^{p}< \infty $$

(1.6)

such that

$$\begin{aligned}& \lim_{r\rightarrow \infty, P=(r,\Theta)\in C_{n}(\Omega)-E(\epsilon;\mu,\zeta)}r^{\frac{-\iota_{[\gamma ],k}^{+}-\{\gamma\}+n-1}{p}}\varphi_{1}^{\frac{\zeta}{p}-1}( \Theta) \mathbb{PI}_{\Omega}^{a}(m,u) (P)=0 \end{aligned}$$

(1.7)

$$\begin{aligned}& \Bigl(\textit{resp. }\lim_{r\rightarrow\infty, P=(r,\Theta)\in C_{n}(\Omega)-E(\epsilon;\nu,\zeta)}r^{\frac{\iota_{[-\gamma ],k}^{+}+\{-\gamma\}+n-1}{p}} \varphi_{1}^{\frac{\zeta}{p}-1}(\Theta) \mathbb{PI}_{\Omega}^{a}(m,u) (P)=0 \Bigr). \end{aligned}$$

(1.8)

Remark

In the case that $a=0$, $p=1$, $\gamma=n+m$ and $\zeta=n$, then (1.6) is a finite sum, the set $E(\epsilon; \mu,n)$ is a bounded set and (1.7)-(1.8) hold in $C_{n}(\Omega)$. This is just the result of Theorem A.

As an application of modified Green’s function with respect to the stationary Schrödinger operator and Theorem 1, we give the solutions of the Dirichlet problem for the Schrödinger operator on $C_{n}(\Omega)$.

Theorem 2

If u is a continuous function on $\partial{C_{n}(\Omega)}$ satisfying

$$ \int_{S_{n}(\Omega)}\frac{|u(t,\Phi)|}{1+V_{m+1}(t)t^{n-1}}\, d\sigma _{Q}< \infty, $$

(1.9)

then the function $\mathbb{PI}_{\Omega}^{a}(m,u)(P)$ satisfies

$$\begin{aligned}& \mathbb{PI}_{\Omega}^{a}(m,u)\in C^{2} \bigl(C_{n}(\Omega)\bigr)\cap C^{0}\bigl(\overline{C_{n}( \Omega)}\bigr), \\& \operatorname{Sch}_{a} \mathbb{PI}_{\Omega}^{a}(m,u)=0 \quad \textit{in } C_{n}(\Omega), \\& \mathbb{PI}_{\Omega}^{a}(m,u)=u\quad \textit{on } \partial{C_{n}(\Omega),} \\& \lim_{r\rightarrow \infty, P=(r,\Theta)\in C_{n}(\Omega)}{r^{-\iota_{{m+1},k}^{+}}}\varphi_{1}^{n-1}( \Theta) \mathbb{PI}_{\Omega}^{a}(m,u) (P)=0. \end{aligned}$$

2 Lemmas

Throughout this paper, let M denote various constants independent of the variables in questions, which may be different from line to line.

Lemma 1

$$\begin{aligned} (\mathrm{i})&\quad \mathbb{PI}(\Omega;a) (P,Q)\leq M r^{\iota_{1,k}^{-}}t^{\iota_{1,k}^{+}-1} \varphi_{1}(\Theta) \\ (\mathrm{ii})&\quad \bigl(\textit{resp. }\mathbb{PI}(\Omega;a) (P,Q)\leq M r^{\iota _{1,k}^{+}}t^{\iota_{1,k}^{-}-1}\varphi_{1}(\Theta)\bigr) \end{aligned}$$

for any $P=(r,\Theta)\in C_{n}(\Omega)$ and any $Q=(t,\Phi)\in S_{n}(\Omega)$ satisfying $0<\frac{t}{r}\leq\frac{4}{5}$ (resp. $0<\frac{r}{t}\leq\frac{4}{5}$);

$$\begin{aligned} (\mathrm{iii})&\quad \mathbb{PI}(\Omega;0) (P,Q)\leq M\frac{\varphi _{1}(\Theta)}{t^{n-1}}+M \frac{r\varphi_{1}(\Theta)}{|P-Q|^{n}} \end{aligned}$$

for any $P=(r,\Theta)\in C_{n}(\Omega)$ and any $Q=(t,\Phi)\in S_{n}(\Omega; (\frac{4}{5}r,\frac{5}{4}r))$.

Proof

(i) and (ii) are obtained by Levin (see [9], Chapter 11). (iii) follows from the work of Azarin (see [12], Lemma 4 and Remark). □

Lemma 2

(see [9], p.356)

For a non-negative integer m, we have

$$ \bigl\vert \mathbb{PI}(\Omega;a,m) (P,Q)\bigr\vert \leq M(n,m,s)V_{m+1}(r) \frac{W_{m+1}(t)}{t}\varphi_{1}(\Theta)\frac {\partial \varphi_{1}(\Phi)}{\partial n_{\Phi}} $$

(2.1)

for any $P=(r,\Theta)\in C_{n}(\Omega)$ and $Q=(t,\Phi)\in S_{n}(\Omega)$ satisfying $r\leq s t$ ($0< s<1$), where $M(n,m,s)$ is a constant dependent on n, m, and s.

Lemma 3

Let $p>-1$ and μ be any positive measure on ${\mathbf{R}}^{n}$ having finite total mass. Then $E(\epsilon; \mu, \zeta)$ has a covering $\{r_{j},R_{j}\}$ ($j=1,2,\ldots$) satisfying

$$\sum_{j=1}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)^{2p-\zeta} \biggl[\frac {V_{j}(R_{j})}{V_{j}(r_{j})}\frac{W_{j}(R_{j})}{W_{j}(r_{j})} \biggr]^{p}< \infty. $$

Proof

Set

$$E_{j}(\epsilon; \mu, \zeta)= \bigl(P=(r,\Theta)\in E(\epsilon; \mu, \zeta):2^{j}\leq r< 2^{j+1}\bigr)\quad (j=2,3,4,\ldots). $$

If $P=(r,\Theta)\in E_{j}(\epsilon; \mu, \zeta)$, then there exists a positive number $\rho(P)$ such that

$$\biggl(\frac{\rho(P)}{r} \biggr)^{2p-\zeta} \biggl[\frac {V_{j}(r)}{V_{j}(\rho(P))} \frac{W_{j}(r)}{W_{j}(\rho(P))} \biggr]^{p}\sim \biggl(\frac{\rho(P)}{r} \biggr)^{np-\zeta}\leq \frac{\mu(B(P,\rho(P)))}{\epsilon}. $$

Here $E_{j}(\epsilon; \mu, \zeta)$ can be covered by the union of a family of balls $(B(P_{j,i},\rho_{j,i}):P_{j,i}\in E_{j}(\epsilon; \mu, \zeta))$ ($\rho_{j,i}=\rho(P_{j,i})$). By the Vitali lemma (see [13]), there exists $\Lambda_{j} \subset E_{j}(\epsilon; \mu, \zeta)$, which is at most countable, such that $(B(P_{j,i},\rho_{j,i}):P_{j,i}\in\Lambda_{j} )$ are disjoint and $E_{j}(\epsilon; \mu, \zeta) \subset \bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},5\rho_{j,i})$.

So

$$\bigcup_{j=2}^{\infty}E_{j}( \epsilon; \mu, \zeta) \subset \bigcup_{j=2}^{\infty}\bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},5 \rho_{j,i}). $$

On the other hand, note that $\bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},\rho_{j,i}) \subset(P=(r,\Theta):2^{j-1}\leq r<2^{j+2}) $, so that

$$\begin{aligned} \sum_{P_{j,i} \in \Lambda_{j}} \biggl(\frac{5\rho_{j,i}}{|P_{j,i}|} \biggr)^{2p-\zeta } \biggl[\frac{V_{j}(|P_{j,i}|)}{V_{j}(5\rho_{j,i})}\frac {W_{j}(|P_{j,i}|)}{W_{j}(5\rho_{j,i})} \biggr]^{p} \sim& \sum_{P_{j,i} \in\Lambda_{j}} \biggl( \frac{5\rho _{j,i}}{|P_{j,i}|} \biggr)^{np-\zeta} \\ \leq&5^{np-\zeta}\sum_{P_{j,i}\in\Lambda_{j}}\frac{\mu (B(P_{j,i},\rho_{j,i}))}{\epsilon} \\ \leq& \frac{5^{np-\zeta}}{\epsilon} \mu\bigl(C_{n}\bigl(\Omega; \bigl[2^{j-1},2^{j+2} \bigr)\bigr)\bigr). \end{aligned}$$

Hence we obtain

$$\begin{aligned} \sum_{j=1}^{\infty}\sum _{P_{j,i} \in \Lambda_{j}} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr)^{2p-\zeta } \biggl[\frac{V_{j}(|P_{j,i}|)}{V_{j}(\rho_{j,i})}\frac {W_{j}(|P_{j,i}|)}{W_{j}(\rho_{j,i})} \biggr]^{p} \sim& \sum _{j=1}^{\infty}\sum_{P_{j,i} \in \Lambda_{j}} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr)^{np-\zeta} \\ \leq&\sum_{j=1}^{\infty}\frac{ \mu(C_{n}(\Omega ;[2^{j-1},2^{j+2})))}{\epsilon} \\ \leq& \frac{3\mu({\mathbf{R}}^{n})}{\epsilon}. \end{aligned}$$

Since $E(\epsilon; \mu, \zeta)\cap\{P=(r,\Theta)\in{\mathbf{R}}^{n}; r\geq4\}=\bigcup_{j=2}^{\infty}E_{j}(\epsilon; \mu, \zeta)$, $E(\epsilon; \mu, \zeta)$ is finally covered by a sequence of balls $(B(P_{j,i},\rho_{j,i}), B(P_{1},6))$ ($j=2,3,\ldots$ ; $i=1,2,\ldots$) satisfying

$$\sum_{j,i} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr)^{2p-\zeta} \biggl[\frac{V_{j}(|P_{j,i}|)}{V_{j}(\rho_{j,i})}\frac {W_{j}(|P_{j,i}|)}{W_{j}(\rho_{j,i})} \biggr]^{p} \sim\sum_{j,i} \biggl( \frac{\rho_{j,i}}{|P_{j,i}|} \biggr)^{np-\zeta}\leq \frac{3\mu({\mathbf{R}}^{n})}{\epsilon}+6^{np-\zeta}< + \infty, $$

where $B(P_{1},6)$ ($P_{1}=(1,0,\ldots,0)\in{\mathbf{R}}^{n}$) is the ball which covers $\{P=(r,\Theta)\in{\mathbf{R}}^{n}; r<4\}$. □

3 Proof of Theorem 1

We only prove the case $p>-1$ and $\gamma\geq0$, the remaining cases can be proved similarly.

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

$$ \int_{S_{n}(\Omega;(R_{\epsilon},\infty))}\frac{|u(Q)|^{p}}{1+t^{\iota _{[\gamma],k}^{+}+\{\gamma\}}}\, d\sigma_{Q}< \epsilon. $$

(3.1)

The relation $G(\Omega;a)(P,Q)\leq G(\Omega;0)(P,Q)$ implies the inequality (see [14])

$$ \mathbb{PI}(\Omega;a) (P,Q)\leq\mathbb{PI}(\Omega;0) (P,Q). $$

(3.2)

For $0< s<\frac{4}{5}$ and any fixed point $P=(r,\Theta)\in C_{n}(\Omega)-E(\epsilon; \mu, \zeta)$ satisfying $r>\frac{5}{4}R_{\epsilon}$, let $I_{1}=S_{n}(\Omega;(0,1))$, $I_{2}=S_{n}(\Omega;[1,R_{\epsilon}])$, $I_{3}=S_{n}(\Omega;(R_{\epsilon},\frac{4}{5}r])$, $I_{4}=S_{n}(\Omega;(\frac{4}{5}r,\frac{5}{4}r))$, $I_{5}=S_{n}(\Omega;[\frac{5}{4}r,\frac{r}{s}))$, $I_{6}=S_{n}(\Omega;[\frac{r}{s},\infty))$ and $I_{7}=S_{n}(\Omega;[1,\frac{r}{s}))$, we write

$$\begin{aligned} \mathbb{PI}_{\Omega}^{a}(m,u) (P) =& \sum _{i=1}^{6} \int_{I_{i}}\mathbb{PI}(\Omega;a,m) (P,Q)u(Q)\, d\sigma _{Q} \\ =& \sum_{i=1}^{5} \int_{I_{i}}\mathbb{PI}(\Omega;a) (P,Q)u(Q)\, d\sigma _{Q} \\ &{}- \int_{I_{7}}\frac{\partial{ \widetilde{K}(\Omega;a,m)(P,Q)}}{\partial n_{Q}}u(Q)\, d\sigma_{Q} \\ &{}+ \int_{I_{6}}\mathbb{PI}(\Omega;a,m) (P,Q)u(Q)\, d \sigma_{Q}, \end{aligned}$$

which yields

$$ \mathbb{PI}_{\Omega}^{a}(m,u) (P) \leq \sum _{i=1}^{7}U_{i}(P), $$

where

$$\begin{aligned}& U_{i}(P)= \int_{I_{i}}\bigl\vert \mathbb{PI}(\Omega;a) (P,Q)\bigr\vert \bigl\vert u(Q)\bigr\vert \, d\sigma_{Q}\quad (i=1,2,3,4,5), \\& U_{6}(P)= \int_{I_{6}}\bigl\vert \mathbb{PI}(\Omega;a,m) (P,Q)\bigr\vert \bigl\vert u(Q)\bigr\vert \, d\sigma_{Q}, \\& U_{7}(P)= \int_{I_{7}}\biggl\vert \frac{\partial{ \widetilde{K}(\Omega;a,m)(P,Q)}}{\partial n_{Q}}\biggr\vert \bigl\vert u(Q)\bigr\vert \, d\sigma_{Q}. \end{aligned}$$

If $\iota_{[\gamma],k}^{+}+\{\gamma\}>(-\iota_{1,k}^{+}-n+2)p+n-1$, then $(\iota_{1,k}^{+}-1+\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}}{p})q+n-1>0$. By (1.5), (3.1), Lemma 1(i), and Hölder’s inequality, we have the following growth estimates:

$$\begin{aligned}& U_{2}(P) \leq Mr^{\iota_{1,k}^{-}}\varphi_{1}(\Theta) \int_{I_{2}}t^{\iota _{1,k}^{+}-1}\bigl\vert u(Q)\bigr\vert \, d \sigma_{Q} \\& \hphantom{U_{2}(P)}\leq Mr^{\iota_{1,k}^{-}}\varphi_{1}(\Theta) \biggl( \int_{I_{2}}\frac {|u(Q)|^{p}}{t^{\iota_{[\gamma],k}^{+}+\{\gamma\}}}\, d\sigma_{Q} \biggr)^{\frac{1}{p}} \biggl( \int_{I_{2}}t^{(\iota_{1,k}^{+}-1+\frac{\iota _{[\gamma],k}^{+}+\{\gamma\}}{p})q}\, d\sigma_{Q} \biggr)^{\frac {1}{q}} \\& \hphantom{U_{2}(P)}\leq M r^{\iota_{1,k}^{-}}R_{\epsilon}^{\iota_{1,k}^{+}+n-2+\frac{\iota _{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}} \varphi_{1}(\Theta), \end{aligned}$$

(3.3)

$$\begin{aligned}& U_{1}(P)\leq M r^{\iota_{1,k}^{-}}\varphi_{1}( \Theta), \end{aligned}$$

(3.4)

$$\begin{aligned}& U_{3}(P)\leq M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\} -n+1}{p}}\varphi_{1}( \Theta). \end{aligned}$$

(3.5)

If $\iota_{m+1,k}^{+}>\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}$, then $(\iota_{1,k}^{-}-1+\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}}{p})q+n-1<0$. We obtain by (3.1), Lemma 1(ii), and Hölder’s inequality

$$\begin{aligned} U_{5}(P) \leq& Mr^{\iota_{1,k}^{+}}\varphi_{1}(\Theta) \int_{S_{n}(\Omega;[\frac {5}{4}r,\infty))}t^{\iota_{1,k}^{-}-1}\bigl\vert u(Q)\bigr\vert \, d \sigma_{Q} \\ \leq& Mr^{\iota_{1,k}^{+}}\varphi_{1}(\Theta) \biggl( \int_{S_{n}(\Omega ;[\frac{5}{4}r,\infty))}\frac{|u(Q)|^{p}}{t^{\iota_{[\gamma ],k}^{+}+\{\gamma\}}}\, d\sigma_{Q} \biggr)^{\frac{1}{p}} \\ &{}\times\biggl( \int _{S_{n}(\Omega;[\frac{5}{4}r,\infty))}t^{(\iota_{1,k}^{-}-1+\frac {\iota_{[\gamma],k}^{+}+\{\gamma\}}{p})q}\, d\sigma_{Q} \biggr)^{\frac {1}{q}} \\ \leq& M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}}\varphi _{1}(\Theta). \end{aligned}$$

(3.6)

By (3.2) and Lemma 1(iii), we consider the inequality

$$U_{4}(P)\leq U_{4}'(P)+U_{4}''(P), $$

where

$$U_{4}'(P)=M\varphi_{1}(\Theta) \int_{I_{4}}t^{1-n}\bigl\vert u(Q)\bigr\vert \, d \sigma_{Q},\qquad U_{4}''(P)=Mr \varphi_{1}(\Theta) \int_{I_{4}}\frac {|u(Q)|}{|P-Q|^{n}}\, d\sigma_{Q}. $$

We first have

$$\begin{aligned} U_{4}'(P) =& M\varphi_{1}(\Theta) \int_{I_{4}}t^{\iota_{1,k}^{+}+\iota _{1,k}^{-}-1}\bigl\vert u(Q)\bigr\vert \, d \sigma_{Q} \\ \leq& Mr^{\iota_{1,k}^{+}}\varphi_{1}(\Theta) \int_{S_{n}(\Omega;(\frac {4}{5}r,\infty))}t^{\iota_{1,k}^{-}-1}\bigl\vert u(Q)\bigr\vert \, d \sigma_{Q} \\ \leq& M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}} \varphi_{1}(\Theta), \end{aligned}$$

(3.7)

which is similar to the estimate of $U_{5}(P)$.

Next, we shall estimate $U_{4}''(P)$.

Take a sufficiently small positive number c such that $I_{4}\subset B(P,\frac{1}{2}r)$ for any $P=(r,\Theta)\in\Pi(c)$ (see [15]), where

$$\Pi(c)=\Bigl\{ P=(r,\Theta)\in C_{n}(\Omega); \inf_{z\in\partial\Omega } \bigl\vert (1,\Theta)-(1,z)\bigr\vert < c, 0< r< \infty\Bigr\} , $$

and divide $C_{n}(\Omega)$ into two sets $\Pi(c)$ and $C_{n}(\Omega)-\Pi(c)$.

If $P=(r,\Theta)\in C_{n}(\Omega)-\Pi(c)$, then there exists a positive $c'$ such that $|P-Q|\geq{c'}r$ for any $Q\in S_{n}(\Omega)$, and hence

$$\begin{aligned} U_{4}''(P) \leq&M\varphi_{1}( \Theta) \int_{I_{4}}t^{1-n}\bigl\vert u(Q)\bigr\vert \, d \sigma_{Q} \\ \leq& M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}} \varphi_{1}(\Theta), \end{aligned}$$

(3.8)

which is similar to the estimate of $U_{4}'(P)$.

We shall consider the case $P=(r,\Theta)\in\Pi(c)$. Now put

$$H_{i}(P)=\bigl\{ Q\in I_{4}; 2^{i-1}\delta(P)\leq \vert P-Q\vert < 2^{i}\delta(P)\bigr\} , $$

where $\delta(P)=\inf_{Q\in\partial{C_{n}(\Omega)}}|P-Q|$.

Since $S_{n}(\Omega)\cap\{Q\in{\mathbf{R}}^{n}: |P-Q|< \delta(P)\}=\varnothing$, we have

$$U_{4}''(P)=M\sum _{i=1}^{i(P)} \int_{H_{i}(P)}r\varphi_{1}(\Theta)\frac {|u(Q)|}{|P-Q|^{n}}\, d \sigma_{Q}, $$

where $i(P)$ is a positive integer satisfying $2^{i(P)-1}\delta(P)\leq\frac{r}{2}<2^{i(P)}\delta(P)$.

Since $r\varphi_{1}(\Theta)\leq M\delta(P)$ ($P=(r,\Theta)\in C_{n}(\Omega)$), similar to the estimate of $U_{4}'(P)$, we obtain

$$\begin{aligned}& \int_{H_{i}(P)}r\varphi_{1}(\Theta)\frac{\vert u(Q)\vert }{\vert P-Q\vert ^{n}}\,d \sigma _{Q} \\& \quad \leq 2^{(1-i)n}\varphi_{1}(\Theta){\delta(P)}^{\frac{\zeta -np}{p}} \int_{H_{i}(P)}{\delta(P)}^{\frac{\zeta}{p}-n}\bigl\vert u(Q)\bigr\vert \,d\sigma _{Q} \\& \quad \leq M\varphi_{1}^{1-\frac{\zeta}{p}}(\Theta){\delta(P)}^{\frac {\zeta-np}{p}} \int_{H_{i}(P)}r^{1-\frac{\zeta}{p}}\bigl\vert u(Q)\bigr\vert \,d \sigma_{Q} \\& \quad \leq Mr^{n-\frac{\zeta}{p}}\varphi_{1}^{1-\frac{\zeta}{p}}(\Theta ){ \delta(P)}^{\frac{\zeta-np}{p}} \int_{H_{i}(P)}t^{1-n}\bigl\vert u(Q)\bigr\vert \,d \sigma_{Q} \\& \quad \leq M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n-\zeta+1}{p}+n}\varphi _{1}^{1-\frac{\zeta}{p}}( \Theta) \biggl(\frac{ \mu(H_{i}(P))}{(2^{i}\delta(P))^{\zeta}} \biggr)^{\frac{1}{p}} \end{aligned}$$

for $i=0,1,2,\ldots,i(P)$.

Since $P=(r,\Theta)\notin E(\epsilon; \mu, \zeta)$, we have

$$\begin{aligned} \frac{\mu(H_{i}(P))}{\{2^{i}\delta(P)\}^{np-\zeta}} \lesssim& \mu\bigl(B\bigl(P,2^{i}\delta(P)\bigr) \bigr)\bigl[V_{1}\bigl(2^{i}\delta(P)\bigr)W_{1} \bigl(2^{i}\delta (P)\bigr)\bigr]^{p}\bigl[2^{i} \delta(P)\bigr]^{\zeta-2p} \\ \lesssim& M(P; \mu, \zeta) \\ \leq& \epsilon\bigl[V_{1}(r)W_{1}(r)\bigr]^{p}r^{\zeta-2p} \\ \leq& \epsilon r^{\zeta-np}\quad \bigl(i=0,1,2,\ldots,i(P)-1\bigr) \end{aligned}$$

and

$$\frac{\mu(H_{i(P)}(P))}{\{2^{i}\delta(P)\}^{np-\zeta}}\lesssim\mu \biggl(B\biggl(P,\frac{r}{2}\biggr)\biggr) \biggl[V_{1}\biggl(\frac{r}{2}\biggr)W_{1}\biggl( \frac{r}{2}\biggr)\biggr]^{p} \biggl(\frac {r}{2} \biggr)^{\zeta-2p} \leq\epsilon r^{\zeta-np}. $$

So

$$ U_{4}''(P)\leq M \epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}} \varphi _{1}^{1-\frac{\zeta}{p}}(\Theta). $$

(3.9)

We only consider $U_{7}(P)$ in the case $m\geq1$, since $U_{7}(P)\equiv0$ for $m=0$. By the definition of $\widetilde{K}(\Omega;a,m)$, (1.1), and Lemma 2, we see (see [16])

$$U_{7}(P)\leq\frac{M}{\chi'(1)}\sum_{j=0}^{m} j^{2n-1}q_{j}(r), $$

where

$$q_{j}(r)=V_{j}(r)\varphi_{1}(\Theta) \int_{I_{7}}\frac {W_{j}(t)|u(Q)|}{t}\,d\sigma_{Q}. $$

To estimate $q_{j}(r)$, we write

$$q_{j}(r)\leq q_{j}'(r)+q_{j}''(r), $$

where

$$\begin{aligned}& q_{j}'(r)=V_{j}(r)\varphi_{1}( \Theta) \int_{I_{2}}\frac {W_{j}(t)|u(Q)|}{t}\,d\sigma_{Q}, \\& q_{j}''(r)=V_{j}(r) \varphi_{1}(\Theta) \int _{S_{n}(\Omega;(R_{\epsilon},\frac{r}{s}))}\frac {W_{j}(t)|u(Q)|}{t}\,d\sigma_{Q}. \end{aligned}$$

If $\iota_{m+1,k}^{+}<\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}+1$, then $(-\iota_{m+1,k}^{+}-n+2+\frac{\iota_{[\gamma],k}^{+}+\{\gamma\} }{p})q+n-1>0$. Notice that

$$V_{j}(r)\frac{V_{m+1}(t)}{V_{j}(t)t}\leq M\frac{V_{m+1}(r)}{r}\leq Mr^{\iota_{m+1,k}^{+}-1} \quad \biggl(t\geq1, R_{\epsilon}< \frac{r}{s} \biggr). $$

Thus, by (1.3), (1.5), and Hölder’s inequality we conclude

$$\begin{aligned} q_{j}'(r) =& V_{j}(r)\varphi_{1}( \Theta) \int_{I_{2}}\frac {|u(Q)|}{V_{j}(t)t^{n-1}}\,d\sigma_{Q} \\ \leq& MV_{j}(r)\varphi_{1}(\Theta) \int_{I_{2}}\frac{V_{m+1}(t)}{t^{\iota _{m+1,k}^{+}}}\frac{|u(Q)|}{V_{j}(t)t^{n-1}}\,d \sigma_{Q} \\ \leq& r^{\iota_{m+1,k}^{+}-1}\varphi_{1}(\Theta) \biggl( \int_{I_{2}}\frac {|u(Q)|^{p}}{t^{\iota_{[\gamma],k}^{+}+\{\gamma\}}}\,d\sigma_{Q} \biggr)^{\frac{1}{p}} \biggl( \int_{I_{2}}t^{(-\iota_{m+1,k}^{+}-n+2+\frac {\iota_{[\gamma],k}^{+}+\{\gamma\}}{p})q}\,d\sigma_{Q} \biggr)^{\frac {1}{q}} \\ \leq& M r^{\iota_{m+1,k}^{+}-1}R_{\epsilon}^{-\iota_{m+1,k}^{+}+1+\frac{\iota _{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}}\varphi_{1}( \Theta). \end{aligned}$$

Analogous to the estimate of $q_{j}'(r)$, we have

$$q_{j}''(r)\leq M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\} -n+1}{p}} \varphi_{1}(\Theta). $$

Thus we can conclude that

$$q_{j}(r)\leq M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}}\varphi _{1}( \Theta), $$

which yields

$$ U_{7}(P)\leq M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}}\varphi _{1}( \Theta). $$

(3.10)

If $\iota_{m+1,k}^{+}>\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}$, then $(-\iota_{m+1,k}^{+}-n+1+\frac{\iota_{[\gamma],k}^{+}+\{\gamma\} }{p})q+n-1<0$. By (3.1), Lemma 2, and Hölder’s inequality we have

$$\begin{aligned} U_{6}(P) \leq&MV_{m+1}(r)\varphi_{1}(\Theta) \int_{I_{6}}\frac {|u(Q)|}{V_{m+1}(t)t^{n-1}}\,d\sigma_{Q} \\ \leq& MV_{m+1}(r)\varphi_{1}(\Theta) \biggl( \int_{I_{6}}\frac {|u(Q)|^{p}}{t^{\iota_{[\gamma],k}^{+}+\{\gamma\}}}\,d\sigma_{Q} \biggr)^{\frac{1}{p}} \biggl( \int_{I_{6}}t^{(-\iota_{m+1,k}^{+}-n+1+\frac {\iota_{[\gamma],k}^{+}+\{\gamma\}}{p})q}\,d\sigma_{Q} \biggr)^{\frac {1}{q}} \\ \leq&M\epsilon r^{\frac{\iota_{[\gamma],k}^{+}+\{\gamma\}-n+1}{p}}\varphi _{1}(\Theta). \end{aligned}$$

(3.11)

Combining (3.3)-(3.11), we see that if $R_{\epsilon}$ is sufficiently large and ϵ is sufficiently small, then

$$\mathbb{PI}_{\Omega}^{a}(m,u) (P)=o\bigl(r^{\frac{\iota_{[\gamma],k}^{+}+\{ \gamma\}-n+1}{p}} \varphi_{1}^{1-\frac{\zeta}{p}}(\Theta)\bigr) $$

as $r\rightarrow\infty$, where $P=(r,\Theta)\in C_{n}(\Omega; (R_{\epsilon},+\infty))-E(\epsilon; \mu, \zeta)$. Finally, there exists an additional finite ball $B_{0}$ covering $C_{n}(\Omega; (0,R_{\epsilon}])$, which, together with Lemma 3, gives the conclusion of Theorem 1.

4 Proof of Theorem 2

For any fixed $P=(r,\Theta)\in C_{n}(\Omega)$, take a number satisfying $R>\max(1,\frac{r}{s})$ ($0< s<\frac{4}{5}$).

By (1.9) and Lemma 2, we have

$$\begin{aligned}& \int_{S_{n}(\Omega;(R,\infty))}\bigl\vert \mathbb{PI}(\Omega ;a,m) (P,Q)\bigr\vert \bigl\vert u(Q)\bigr\vert \,d\sigma_{Q} \\& \quad \leq V_{m+1}(r)\varphi_{1}(\Theta) \int_{S_{n}(\Omega;(R,\infty ))}\frac{\vert u(Q)\vert }{V_{m+1}(t)t^{n-1}}\,d\sigma_{Q} \\& \quad \leq M V_{m+1}(r)\varphi_{1}(\Theta) \\& \quad < \infty. \end{aligned}$$

Then $\mathbb{PI}_{\Omega}^{a}(m,u)(P)$ is absolutely convergent and finite for any $P\in C_{n}(\Omega)$. Thus $\mathbb{PI}_{\Omega}^{a}(m, u)(P)$ is a generalized harmonic function on $C_{n}(\Omega)$.

Now we study the boundary behavior of $\mathbb{PI}_{\Omega}^{a}(m,u)(P)$. Let $Q'=(t',\Phi')\in \partial{C_{n}(\Omega)}$ be any fixed point and l be any positive number satisfying $l>\max(t'+1,\frac{4}{5}R)$.

Set $\chi_{S(l)}$, the characteristic function of $S(l)=\{Q=(t,\Phi)\in\partial{C_{n}(\Omega)},t\leq l\}$ and write

$$\begin{aligned} \mathbb{PI}_{\Omega}^{a}(m,u) (P) =& \biggl( \int_{S_{n}(\Omega;(0,1))}+ \int_{S_{n}(\Omega;[1,\frac{5}{4}l])}+ \int_{S_{n}(\Omega;(\frac {5}{4}l,\infty))} \biggr)\mathbb{PI}(\Omega;a,m) (P,Q)u(Q)\,d \sigma_{Q} \\ =& U'(P)-U''(P)+U'''(P), \end{aligned}$$

where

$$\begin{aligned}& U'(P)= \int_{S_{n}(\Omega;(0,\frac{5}{4}l])}\mathbb{PI}(\Omega ;a) (P,Q)u(Q)\,d \sigma_{Q}, \\& U''(P)= \int_{S_{n}(\Omega;[1,\frac{5}{4}l])}\frac{\partial{ K(\Omega;a,m)(P,Q)}}{\partial n_{Q}}u(Q)\,d\sigma_{Q}, \\& U'''(P)= \int_{S_{n}(\Omega;(\frac{5}{4}l,\infty))} \mathbb {PI}(\Omega;a,m) (P,Q)u(Q)\,d \sigma_{Q}. \end{aligned}$$

Notice that $U'(P)$ is the Poisson a-integral of $u(Q)\chi_{S(\frac{5}{4}l)}$, we have $\lim_{P\rightarrow Q',P\in C_{n}(\Omega)}U'(P)=u(Q')$. Since $\lim_{\Theta\rightarrow \Phi'}\varphi_{jv}(\Theta)=0$ ($j=1,2,3\ldots$ ; $1\leq v\leq v_{j}$) as $P=(r,\Theta)\rightarrow Q'=(t',\Phi')\in S_{n}(\Omega)$, we have $\lim_{P\rightarrow Q',P\in C_{n}(\Omega)}U''(P)=0$ from the definition of the kernel function $K(\Omega;a,m)(P,Q)$. $U'''(P)=O(V_{m+1}(r)\varphi_{1}(\Theta))$ and therefore it tends to zero.

So the function $\mathbb{PI}_{\Omega}^{a}(m,u)(P)$ can be continuously extended to $\overline{C_{n}(\Omega)}$ such that

$$\lim_{P\rightarrow Q',P\in C_{n}(\Omega)}\mathbb{PI}_{\Omega}^{a}(m,u) (P)=u\bigl(Q'\bigr) $$

for any $Q'=(t',\Phi')\in \partial{C_{n}(\Omega)}$ from the arbitrariness of l, which with Theorem 1 gives the conclusion of Theorem 2.

Declarations

Acknowledgements

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

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)

Institute of Management Science and Engineering, Henan University, Kaifeng, 475001, China

References

Reed, M, Simon, B: Methods of Modern Mathematical Physics, vol. 3. Academic Press, New York (1970) Google Scholar
Rosenblum, G, Solomyak, M, Shubin, M: Spectral Theory of Differential Operators. VINITI, Moscow (1989) Google Scholar
Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977) View ArticleMATHGoogle Scholar
Muller, C: Spherical Harmonics. Lect. Notes in Math., vol. 17. Springer, Berlin (1966) Google Scholar
Xue, GX: A remark on the a-minimally thin sets associated with the Schrödinger operator. Bound. Value Probl. 2014, 133 (2014) View ArticleGoogle Scholar
Zhao, TH, Yang, ZH, Chu, YM: Monotonicity properties of a function involving the psi function with applications. J. Inequal. Appl. 2015, 193 (2015) MathSciNetView ArticleGoogle Scholar
Simon, B: Schrödinger semigroups. Bull. Am. Math. Soc. 7, 447-526 (1982) View ArticleMATHGoogle Scholar
Hartman, P: Ordinary Differential Equations. Wiley, New York (1964) MATHGoogle Scholar
Escassut, A, Tutschke, W, Yang, CC: Some Topics on Value Distribution and Differentiability in Complex and P-Adic Analysis. Science Press, Beijing (2008) MATHGoogle Scholar
Yanagishita, M: On the behavior at infinity for non-negative superharmonic functions in a cone. In: Potential Theory in Matsue. Adv. Stud. Pure Math., vol. 44, pp. 403-413. Math. Soc. Japan, Tokyo (2006) Google Scholar
Yoshida, H, Miyamoto, I: Solutions of the Dirichlet problem on a cone with continuous data. J. Math. Soc. Jpn. 50(1), 71-93 (1998) MathSciNetView ArticleMATHGoogle Scholar
Azarin, VS: Generalization of a theorem of Hayman on subharmonic functions in an m-dimensional cone. Am. Math. Soc. Transl. (2) 80, 119-138 (1969) MATHGoogle Scholar
Stein, EM: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970) MATHGoogle Scholar
Ancona, A: First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains. J. Anal. Math. 72, 45-92 (1997) MathSciNetView ArticleMATHGoogle Scholar
Miyamoto, I, Yoshida, H: On a-minimally thin sets at infinity in a cone. Hiroshima Math. J. 37(1), 61-80 (2007) MathSciNetMATHGoogle Scholar
Yoshida, H: A boundedness criterion for subharmonic function. J. Lond. Math. Soc. 24(2), 148-160 (1981) View ArticleMATHGoogle Scholar