Let ${\bf R}^{n}(n\geq2)$ be the $n$-dimensional Euclidean space and $\bf S$ its an open set. The boundary and the closure of $\bf S$ are denoted by $\partial{\bf S}$ and $\overline{\bf S}$, respectively. In cartesian coordinate a point $P$ is denoted by $(X, x_n), $ where $X=(x_1, x_2, \cdots, x_{n-1}).$ For $P$ and $Q$ in ${\bf R}^{n}$, let $|P|$ be the Euclidean norm of $P$ and $|P-Q|$ the Euclidean distance. The unit sphere and the upper half unit sphere are denoted by ${\bf S}^{n-1}$ and ${\bf S}_{+}^{n-1}$, respectively. For $P\in {\bf R}^{n}$ and $r>0$, let $B(P, r)$ be the open ball of radius $r$ centered at $P$ in ${\bf R}^{n}$, then $S_{r}=\partial{B(O, r)}$. Furthermore, denote by $dS_{r}$ the $(n-1)$-dimensional volume elements induced by the Euclidean metric on $S_{r}$.

In the paper we are mainly concerned with some properties for the generalized Martin function associated with the stationary Schrödinger operator in a cone. Our aim is to give precise characterization for majorization of the generalized Martin functions in a cone. Deng et al. (see [17] and [23]) ever considered the growth for the potential functions in the half space. However, Miyamoto et al. (see [10, 11] and [12]) focused on the potential theories in a cone. Levin and Kheyfits (see [9]) paid attention to the problems associated with the stationary Schrödinger operator in a cone. In addition, Long and Qiao et al. (see [7, 8, 13-15] and [16]) considered some related problems about Dirichlet problem for the stationary Schrödinger operator at $\infty$ with respect to a cone as well as Levin and Kheyfits (see [9]). Based on the above statement, we will mainly generalize some results from Miyamoto and Yoshida (see [10]) to the stationary Schrödinger operator's setting. Unfortunately we don't have Riesz-Herglotz type theorem as the classical results which needed in the proof. To get over this difficulty, here we will depend on the generalized Martin representation theorem (see [8]). For the better statements about our results, we will introduce some notations and background materials below.

Relative to system of spherical coordinates, the Laplace operator $\Delta$ may be written by

$ \Delta=\frac{n-1}{r}\frac{\partial}{\partial r}+\frac{\partial^2}{\partial r^2}+\frac{\Delta^*}{r^2}, $

where the explicit form of the Beltrami operator $\Delta^*$ is given by Azarin (see [1]).

For an arbitrary domain $D$ in ${\bf R}^{n}$, $\mathcal{A}_D$ denotes the class of nonnegative radial potentials $a(P)$, i.e., $0\leq a(P)=a(r)$, $P=(r, \Theta)\in D$, such that $a\in L_{\rm loc}^{b}(D)$ with some $b> {n}/{2}$ if $n\geq4$ and with $b=2$ if $n=2$ or $n=3$.

If $a\in \mathcal{A}_D$, then the stationary Schrödinger operator with a potential $a(\cdot)$

$ \mathcal{L}_a=-\Delta+a(\cdot)I $

can be extended in the usual way from the space $C_0^{\infty}(D)$ to an essentially self-adjoint operator on $L^{2}(D), $ where $\Delta$ is the Laplace operator and $I$ the identical operator(see [18, Chap.13]). Then $\mathcal{L}_a$ has a Green $a$-function $G_{D}^{a}(\cdot, \cdot)$. Here $G_{D}^{a}(\cdot, \cdot)$ is positive on $D$ and its inner normal derivative $\partial G_{D}^{a}(\cdot, \cdot)/{\partial n_Q}$ is not negative, where ${\partial}/{\partial n_Q}$ denotes the differentiation at $Q$ along the inward normal into $D$. We write this derivative by $PI_{D}^{a}(\cdot, \cdot)$, which is called the Poisson $a$-kernel with respect to $D$. Denote by $G_{D}^{0}(\cdot, \cdot)$ the Green function of Laplacian.

For simplicity, a point $(1, \Theta)$ on ${\bf S}^{n-1}$ and the set $\{\Theta; (1, \Theta)\in \Omega\}$ for a set $\Omega$ ($\Omega\subset {\bf S}^{n-1}$) are often identified with $\Theta$ and $\Omega$, respectively. For two sets $\Xi\subset {\bf R}_+$ and $\Omega\subset {\bf S}^{n-1}, $ the set $\{(r, \Theta)\in{\bf R}^{n}; r\in\Xi, (1, \Theta)\in \Omega\}$ in ${\bf R}^{n}$ is simply denoted by $\Xi\times \Omega.$ In particular, the half space ${\bf R}_{+}\times {\bf S}_{+}^{n-1}=\{(X, x_n)\in{\bf R}^{n}; x_n>0\}$ will be denoted by ${\bf T}_n$. By $C_n(\Omega)$ we denote the set ${\bf R}_+\times \Omega$ in ${\bf R}^{n}$ with the domain $\Omega$ on ${\bf S}^{n-1}$ and call it a cone. We mean the sets $I\times\Omega$ and $I\times \partial{\Omega}$ with an interval on $\bf R_+$ by $C_n(\Omega;I)$ and $S_n(\Omega;I)$, and $C_n(\Omega)\cap S_{r}$ by $C_n(\Omega; r)$. By $S_n(\Omega)$ we denote $S_n(\Omega; (0, +\infty))$, which is $\partial{C_n(\Omega)}\setminus\{O\}.$ From now on, we always assume $D=C_n(\Omega)$ and write $G_{\Omega}^{a}(\cdot, \cdot)$ instead of $G_{C_n(\Omega)}^{a}(\cdot, \cdot)$.

Let $\Omega$ be a domain on ${\bf S}^{n-1}$ with smooth boundary and $\lambda$ the least positive eigenvalue for $-\Delta^\ast$ on $\Omega$ (see [19, p. 41]),

$ \begin{eqnarray*}&&(\Delta^\ast+\lambda)\varphi(\Theta)=0 \hspace {3mm} \textrm{on} \hspace {2mm} \Omega, \\ &&\varphi(\Theta)=0 \hspace {3mm} \textrm{on} \hspace {2mm} \partial{\Omega}.\end{eqnarray*} $

The corresponding eigenfunction is denoted by $\varphi(\Theta)$ satisfying $\displaystyle\int_\Omega\varphi^2(\Theta)dS_1=1$. In order to ensure the existence of $\lambda$ and $\varphi(\Theta)$, we put a rather strong assumption on $\Omega$: if $n\geq3, $ then $\Omega$ is a $C^{2, \alpha}$-domain $(0 < \alpha < 1)$ on ${\bf S}^{n-1}$ surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see [6, p. 88-89] for the definition of $C^{2, \alpha}$-domain).

Solutions of an ordinary differential equation

$ -Q''(r)-\frac{n-1}{r}Q'(r)+\left( \frac{\lambda}{r^2}+a(r)\right )Q(r)=0 \; {\rm for}\; 0 < r < \infty $

(1.1)

are known (see [22] for more references) that if the potential $a\in \mathcal{A}_D$. We know the equation (1.3) has a fundamental system of positive solutions $\{V, W\}$ such that $V$ is nondecreasing with

$ 0\leq V(0+)\leq V(r) \hspace {2mm} \textrm{as} \hspace {2mm} r\rightarrow+\infty $

(1.2)

and $W$ is monotonically decreasing with

$ +\infty=W(0+)>W(r)\searrow0 \hspace {2mm} \textrm{as} \hspace {2mm} r\rightarrow+\infty. $

(1.3)

We remark that both $V(r)\varphi(\Theta)$ and $W(r)\varphi(\Theta)$ are $a$-harmonic on $C_n(\Omega)$ and vanish continuously on $S_n(\Omega)$.

We will also consider the class $\mathcal{B}_D$, consisting of the potentials $a\in \mathcal{A}_D$ such that there exists the finite limit $\lim\limits_{r\rightarrow\infty}r^2a(r)=\kappa\in[0, \infty)$, moreover, $r^{-1}|r^2 a(r)-\kappa|\in L(1, \infty)$. If $a\in \mathcal{B}_D$, then the (super)subfunctions are continuous (e.g. see [20]). For simplicity, in the rest of paper we assume that $a\in\mathcal{B}_D$.

Denote

$ \iota_{\kappa}^{\pm}=\frac{2-n\pm\sqrt{(n-2)^2+4(\kappa+\lambda)}}{2}, $

then the solutions $V(r)$ and $W(r)$ to equation (1.1) normalized by $V(1)=W(1)=1$ have the asymptotic (see [6])

$ V(r)\approx r^{\iota_{\kappa}^{+}}, W(r)\approx r^{\iota_{\kappa}^{-}} \; \textrm{as}\; r\rightarrow\infty $

(1.4)

and

$ \chi=\iota_{\kappa}^{+}-\iota_{\kappa}^{-}=\sqrt{(n-2)^2+4(\kappa+\lambda)}, \chi'=\omega(V(r), W(r))\mid_{r=1}, $

where $\chi'$ is their Wronskian at $r=1$.

Remark 1 If $a=0$ and $\Omega={\bf S}_{+}^{n-1}$, then $\iota_{0}^{+}=1$, $\iota_{0}^{-}=1-n$ and $\varphi(\Theta)=(2n s_n^{-1})^{1/2}\cos\theta_1, $ where $s_n$ is the surface area $2\pi^{n/2}\{\Gamma(n/2)\}^{-1}$ of ${\bf S}^{n-1}$.

The function $M^a_{\Omega}$ defined on $C_n(\Omega)\times C_n(\Omega)\setminus\{(P_0, P_0)\}$ by

$ M^{a}_{\Omega}(P, Q)=\frac{G^{a}_{\Omega}(P, Q)}{G^{a}_{\Omega}(P_0, Q)} $

is called the generalized Martin kernel of $C_n(\Omega)$ (relative to $P_0$). If $Q=P_0$, the above quotient is interpreted as 0 (for a=0, refer to Armitage and Gardiner [3]).

The rest of the paper is organized as follows: in Section 2, we shall give our main theorems; in Section 3, some necessary lemmas are given; in Section 4, we shall prove the main results.

It is known that the Martin boundary $\vartriangle$ of $C_n(\Omega)$ is the set $\partial C_n(\Omega)\cup\{\infty\}$. When we denote the Martin kernel associated with the stationary Schrödinger operator by $M^a_{\Omega}(P, Q) (P\in C_n(\Omega), Q\in\partial C_n(\Omega)\cup\{\infty\})$ with respect to a reference point chosen suitably, for any $P\in C_n(\Omega), $ we see

$ M^a_{\Omega}(P, \infty)=V(r)\varphi(\Theta), M^a_{\Omega}(P, O)=\kappa W(r)\varphi(\Theta), $

(2.1)

where $O$ denotes the origin of $\bf R^n$ and $\kappa$ is a positive constant.

For a set $E\subset D$ and $\ell\in (0, 1)$, put

$ {E_\ell } = \bigcup\limits_{P \in E} B (P,\ell d(P)), $

where $d(P)=\inf\limits_{Q\in D^{c}}\mid P-Q\mid$. Next we start to sate our main theorems.

Theorem 1 Let $E$ be a set in $C_n(\Omega)$ satisfying $\overline{E}\cap\partial C_n(\Omega)=\phi$. If $E_\ell$ with a positive number $\ell(0 < \ell < 1)$ is $a$-minimally thin at $\infty$, then there exists a positive generalized harmonic function $u(P)$ on $C_n(\Omega)$ such that

$ \mathop {\inf }\limits_{P \in {C_n}(\Omega )} \frac{{u(P)}}{{M_\Omega ^a(P,\infty )}}{\rm{ < }}\mathop {\inf }\limits_{P \in E} \frac{{u(P)}}{{M_\Omega ^a(P,\infty )}}. $

For $E\subseteq C_n(\Omega)$ and a fixed point $Q\in\partial C_n(\Omega)$, $E$ is $a$-minimally thin at $Q$ if and only if $\widehat{R}^E_{M^a_{\Omega}(\cdot, Q)}\neq M^a_{\Omega}(\cdot, Q)$, where $\widehat{R}^E_{M^a_{\Omega}(\cdot, Q)}$ is the regularized reduced function of $M^a_{\Omega}(\cdot, Q)$ relative to $E$ and a superfunction on $C_n(\Omega)$ (refer to [8]).

Following the Armitage and Kuran (see [4]) as well as Miyamoto et al. (see [10]), we call that set $E\subset D$ characterizes the positive generalized harmonic majorization of $M^a_{\Omega}(\cdot, Q)$, if every positive generalized harmonic function $\upsilon$ in $D$ which majorizes $M^a_{\Omega}(\cdot, Q)$ on $E$ can majorize $M^a_{\Omega}(\cdot, Q)$ on $D$, that is to say

$ \mathop {\inf }\limits_{P \in D} \frac{{\upsilon (P)}}{{M_\Omega ^a(P,Q)}} = \mathop {\inf }\limits_{P \in E} \frac{{\upsilon (P)}}{{M_\Omega ^a(P,Q)}}. $

Theorem 2 Let $E$ be a subset $C_n(\Omega)$. The following conditions on $E$ are equivalent:

(a)$E$ characterizes the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$;

(b) for any $\ell\in (0, 1)$, $E_{\ell}$ is not $a$-minimally thin at $\infty$;

(c)for some $\ell\in (0, 1)$, $E_{\ell}$ is not $a$-minimally thin at $\infty$.

Theorem 3 Let $E$ be a subset $C_n(\Omega)$. The following conditions on $E$ are equivalent:

(a) $E$ characterizes the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$;

(b) for any $\ell\in (0, 1)$,

$ \int_{E_{\ell}}\frac{V(1+r)W(1+r)dP}{(1+r)^2}=+\infty; $

(c) for some $\ell\in (0, 1)$,

$ \int_{E_{\ell}}\frac{V(1+r)W(1+r)dP}{(1+r)^2}=+\infty. $

A sequence ${P_m}\subset D$ is called to be separated if there exists a positive constant $C$ such that

$ \mid P_i-P_j\mid\geq Cd(P_i) (i, j=1, 2, \cdots, i\neq j) $

(see [2]). With Theorem 3, we have the corollary as follows.

Corollary 1 Let $\{P_m\}\subset C_n(\Omega)$ be a separated sequence such that

$ \mathop {\inf }\limits_m \mid {P_m}\mid > 0. $

The sequence $\{P_m\}$ characterizes the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$ if and only if

$ \sum\limits_{m = 1}^\infty {\frac{{d{{({P_m})}^n}V(\mid {P_m}\mid )W(\mid {P_m}\mid )}}{{\mid {P_m}{\mid ^2}}}} = + \infty . $

Remark 2 When $a=0$, the theorems and corollary above are due to Miyamoto et al. (see [10]). If $a=0$ and $\Omega={\bf S}_{+}^{n-1}$, Theorem 1, Theorem 2 and Theorem 3 are from the Dahlberg's results in upper-half space or Liapunov-Dini domain in $\mathbf{R}^{n}$ (see [5]), and Corollary 1 results from Armitage and Kuran (see [4]).

For our arguments we collect the following results.

Lemma 1 (see [13])

$ PI_\Omega ^a(P,Q) \approx {t^{ - 1}}V(t)W(r)\varphi (\Theta )\frac{{\partial \varphi (\Phi )}}{{\partial {n_\Phi }}}, $

(3.1)

$ \left( {{\rm{resp}}{\rm{.}}PI_\Omega ^a(P,Q) \approx V(r){t^{ - 1}}W(t)\varphi (\Theta )\frac{{\partial \varphi (\Phi )}}{{\partial {n_\Phi }}}} \right) $

(3.2)

for any $P=(r, \Theta)\in C_n(\Omega)$ and any $Q=(t, \Phi)\in S_n(\Omega)$ satisfying $0 < \frac{t}{r}\leq \frac{1}{2}$ $(\textrm{resp.} 0 < \frac{r}{t}\leq \frac{1}{2}), $

$ PI^0_{\Omega}(P, Q)\lesssim \frac{\varphi(\Theta)}{t^{n-1}}\frac{\partial \varphi(\Phi)}{\partial n_{\Phi}} +\frac{r\varphi(\Theta)}{|P-Q|^{n}}\frac{\partial \varphi(\Phi)}{\partial n_{\Phi}} $

(3.3)

for any $P=(r, \Theta)\in C_n(\Omega)$ and any $Q=(t, \Phi)\in S_n(\Omega; (\frac{1}{2}r, 2r)).$

Lemma 2 (see [13])

$ G_\Omega ^a(P,Q) \approx V(t)W(r)\varphi (\Theta )\frac{{\partial \varphi (\Phi )}}{{\partial {n_\Phi }}}, $

(3.4)

$ \left( {{\rm{resp}}{\rm{.}}G_\Omega ^a(P,Q) \approx V(t)W(r)\varphi (\Theta )\frac{{\partial \varphi (\Phi )}}{{\partial {n_\Phi }}}} \right) $

(3.5)

for any $P=(r, \Theta)\in C_n(\Omega)$ and any $Q=(t, \Phi)\in S_n(\Omega)$ satisfying $0 < \frac{t}{r}\leq \frac{1}{2}$ $(\textrm{resp.} 0 < \frac{r}{t}\leq \frac{1}{2});$

$ G^0_{\Omega}(P, Q)\lesssim \frac{\varphi(\Theta)}{t^{n-2}}\frac{\partial \varphi(\Phi)}{\partial n_{\Phi}} +\frac{rt\varphi(\Theta)}{|P-Q|^{n}}\frac{\partial \varphi(\Phi)}{\partial n_{\Phi}} $

(3.6)

for any $P=(r, \Theta)\in C_n(\Omega)$ and any $Q=(t, \Phi)\in S_n(\Omega; (\frac{1}{2}r, 2r)).$

Lemma 3 (The generalized Martin representation, see [7]) If $u$ is a positive a-harmonic function on $C_n(\Omega)$, then there exists a measure $\mu_u$ on $\vartriangle$, uniquely determined by $u$, such that

$ u(P)=\int_{\vartriangle}M^a_{\Omega}(P, Q)d\mu_u(Q) (P\in C_n(\Omega)), $

where $\vartriangle $ is the Martin boundary of $ C_n(\Omega)$.

It is well-known that a cube is of the form

$ [\ell_12^{-k}, (\ell_1+1)2^{-k}]\times\cdots\times[\ell_n2^{-k}, (\ell_n+1)2^{-k}], $

where $k, \ell_1, \cdots, \ell_n$ are integers. Now we introduce a family of so-called Whitney cubes of $C_n(\Omega)$ having the following properties:

(a) $\cup_j W_j=C_n(\Omega);$

(b) ${\rm int} W_j \cap {\rm int} W_k=\emptyset(j\neq k);$

(c) ${\rm diam} W_j \leq {\rm dist}(W_j, \mathbf{R}^n\setminus C_n(\Omega))\leq 4{\rm diam}W_j, $

where ${\rm int} S$, ${\rm diam} S$ and ${\rm dist}(S_1, S_2)$ stand for the interior of $S$, the diameter of $S$ and the distance between $S_1$ and $S_2$, respectively (see [21], P.167, Theorem 1).

Lemma 4 (see [10]) Let $\{W_i\}_{i\geq1}$ be a family of the Whitney cubes of $C_n(\Omega)$ with $\ell$. Let $E$ be a subset of $C_n(\Omega)$. Then there exists a subsequence $\{W_{i_m}\}_{i\geq1}$ of $\{W_i\}_{i\geq1}$ such that

(a) $\cup_{m}W_{i_m}\subset E_{\ell}$;

(b) $W_{i_m}\cap E_{\ell\diagup4}\neq\emptyset (m=1, 2, , \cdots)$, $E_{\ell\diagup4}\subset\cup_{m}W_{i_m}$.

Lemma 5 (see [8]) Let a Borel subset $E$ of $C_n(\Omega)$ be $a$-minimally thin at $\infty$ with respect to $C_n(\Omega)$. Then we see that

$ \int_{E}V(1+\mid P\mid)W(1+\mid P\mid)(1+\mid P\mid)^{-2}dP < \infty. $

(3.7)

If $E$ is a union of cubes from the Whitney cubes of $C_n(\Omega)$, then (3.7) is also sufficient for $E$ to be $a$-minimally thin at $\infty$ with respect to $C_n(\Omega)$.

Proof of Theorem 1 When $E$ is a bounded subset of $C_n(\Omega)$, we may assume that $u(P)$ is a constant function. Otherwise we will follow the same method as Dahlberg to make the required function.

Set $\ell\in(0, 1)$. We assume that $\{P_m\}$ is a sequence of points $P_m$ which are central points of cubes $W_{i_m}$ in Lemma 4. From the assumption on $E$, it follows that $\{P_m\}$ can not converge to any boundary point of $C_n(\Omega)$. Since $\{P_m\}\in E_{\ell}$ due to Lemma 4, we see that $\mid P_m\mid\rightarrow+\infty(m\rightarrow+\infty)$. Because $E_{\ell}$ is $a$-minimally thin at $\infty$ and

$ \int_{W_{i_m}}\frac{V(1+r)W(1+r)dP}{(1+r)^2}\approx\frac{d(P_m)^nV(\mid P_m\mid)W(\mid P_m\mid)}{\mid P_m\mid^2} (m=1, 2, \cdots), $

(4.1)

we get by Lemma 4 and Lemma 5 that

$ \sum\limits_{m = 1}^\infty {\frac{{d{{({P_m})}^n}V(\mid {P_m}\mid )W(\mid {P_m}\mid )}}{{\mid {P_m}{\mid ^2}}}} {\rm{ < }}\infty . $

Hence from (1.2)-(1.4) we can take a positive integer $N$ such that $d(P_m)\leq \frac{1}{N}\mid P_m\mid$ for each $m\geq N$.

Choose a point $Q_m=(t_m, \Phi_m)\in\partial C_n(\Omega)\setminus\{O\}$ such that

$ \mid P_m-Q_m\mid=d(P_m) (m=N, N+1, \cdots). $

Then we see that $\mid Q_m\mid\geq \frac{N-1}{N}\mid P_m\mid$ and hence $\mid Q_m\mid\rightarrow+\infty (m\rightarrow+\infty)$. Define $h_1(P)$ as follow:

$ h_1(P)=\sum^{\infty}_{m=N}PI^{a}_{\Omega}(P, Q_m)\frac{d(P_m)^nV(\mid P_m\mid)}{\mid P_m\mid} (P\in C_n(\Omega)), $

then $h_1$ is well defined, and hence is a positive generalized harmonic function on $C_n(\Omega)$ which is due to Lemma 4.

First we will prove that

$ \mathop {\inf }\limits_{P \in E} \frac{{{h_1}(P)}}{{M_\Omega ^a(P,\infty )}} > 0. $

(4.2)

Denote the Possion Kernel of the ball $B_m=B(P_m, d(P_m)))$ by $PI_{B_m}(P, Q)$ for $P\in B_m$ and $Q\in\partial B_m$. Since $PI^{a}_{\Omega}(P, Q_m)\thickapprox PI^{0}_{\Omega}(P, Q_m)$ (see 13]), we have

$ PI^{a}_{\Omega}(P, Q_m)\gtrsim PI_{B_m}(P, Q_m) (P\in B_m;m=N, N+1, \cdots) $

and hence

$ PI^{a}_{\Omega}(P_m, Q_m)\gtrsim PI_{B_m}(P_m, Q_m)=s_n^{-1}d(P_m)^{1-n} (m=N, N+1, \cdots). $

Because

$ \varphi(\Phi)\thickapprox d(P') (P'=(1, \Phi), \Phi\in\Omega), $

we get that

$ h_1(P_m)\geq PI^{a}_{\Omega}(P_m, Q_m)\frac{d(P_m)^{n}V(\mid P_m\mid)}{\mid P_m\mid}\gtrsim M^a_{\Omega}(P_m, \infty) (m=N, N+1, \cdots). $

(4.3)

For any $P\in E$, then exists a point $P_m$ such that

$ \mid P-P_m\mid < \frac{{\rm diam}(W_{i_m})}{2}\lesssim\delta d(P)\leqslant\frac{\delta}{2}\mid P_m\mid. $

When $2r\leq t$ or $r\geq 2t$ ($2\mid P_m\mid\leq t$ or $\mid P_m\mid\geq 2t$), by Lemma 2 and (1.2)-(1.4) we obtain that

$ \frac{G^{a}_{\Omega}(P, Q)}{G^{a}_{\Omega}(P_m, Q)}\geq C. $

Since

$ \ell^{n-2}G^{0}_{\Omega}(\ell P, \ell Q)=G^{0}_{\Omega}(P, Q) (P, Q\in C_n(\Omega)) $

and

$ G^{a}_{\Omega}(P, Q)\approx G^{0}_{\Omega}(P, Q) (P, Q\in C_n(\Omega)) $

(refer to [10] and [13]), we know that

$ \frac{G^{a}_{\Omega}(P, Q)}{G^{a}_{\Omega}(P_m, Q)}\succsim 1 $

for $\frac{t}{2}\leq r$ and $\mid P_m\mid\leq 2t$. From Lemma 3 (the generalized Martin representation), we may obtain that

$ h_1(P)\gtrsim h_1(P_m). $

(4.4)

By (2.1) and (1.2)-(1.4) we also see that

$ M^a_{\Omega}(P, \infty)\lesssim M^a_{\Omega}(P_m, \infty). $

(4.5)

With (4.4)-(4.5) and (4.3) we see that (4.2) holds.

Now, for a fixed ray $L$ which is in $C_n(\Omega)$ and starts from $O$, we will show

$ \lim_{\mid P\mid\rightarrow\infty, P\in L}\frac{h_1(P)}{M^a_{\Omega}(P, \infty)}=0. $

(4.6)

Set

$ H_m(P)=\frac{PI^{a}_{\Omega}(P, Q_m)\mid P_m\mid}{M^a_{\Omega}(P, \infty)W(\mid P_m\mid)} (P\in C_n(\Omega); m=N, N+1, \cdots). $

Then we have that

$ \frac{{{h_1}(P)}}{{M_\Omega ^a(P,\infty )}} = \sum\limits_{m = N}^\infty {{H_m}} (P)\frac{{d{{({P_m})}^n}V(\mid {P_m}\mid )W(\mid {P_m}\mid )}}{{\mid {P_m}{\mid ^2}}}. $

By Lemma 1 we see that

$ \mathop {\lim }\limits_{\mid P\mid \to \infty ,P \in L} {H_m}(P) = 0 $

for any fixed $m\geq N$. Hence, if we can show that

$ \mid H_m(P)\mid\leq C (P\in L; m=N, N+1, \cdots) $

(4.7)

for some constant $C$ independent of $m$, then we will get (4.6) from (4.1) and Lebesgue's dominated convergence theorem.

To prove (4.7), we divide the proof into three cases. When $2r\leq t_m$ or $r\geq 2t_m$, by Lemma 1 we see that

$ \mid H_m(P)\mid\leq C (P=(r, \Theta)\in C_n(\Omega); m=N, N+1, \cdots). $

Finally, when $\frac{t_m}{2}\leq r\leq 2t_m$, we have

$ t^{n-1}_{m}PI^{0}_{\Omega}(P, Q_m)\leq C' (P=(r, \Theta)\in L; m=N, N+1, \cdots) $

for some constant $C'$ (refer to [10,p.1051]). Since $PI^{a}_{\Omega}(P, Q_m)\thickapprox PI^{0}_{\Omega}(P, Q_m)$, by (1.2)-(1.4) we have

$ t_{m}PI^{a}_{\Omega}(P, Q_m)\leq C''V(t_m)W(t_m) (P=(r, \Theta)\in L; m=N, N+1, \cdots). $

So

$ \mid H_m(P)\mid\leq C (P=(r, \Theta)\in L; m=N, N+1, \cdots). $

At last, we put $\Upsilon=\max\limits_{1\leq m\leq N}M^a_{\Omega}(P_m, \infty)$ and $h(P)=H_1(P)+\Upsilon$ for any $P\in C_n(\Omega)$. Then we easily get from (4.2) and (4.6) that $h(P)$ is a positive generalized harmonic function on $C_n(\Omega)$ which is required in Theorem 1.

Proof of Theorem 2 (a)$\Rightarrow$(b). Let $C$ be a positive constant and set $E_{C}=\{P\in E: M^a_{\Omega}(P, \infty)\geq C\}$. Then $E_{C}$ satisfies that $\overline{E_{C}}\cap\partial C_n(\Omega)=\emptyset$. Since $E$ characterizes the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$, $E_{C}$ also characterizes the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$. Otherwise, there would exists a positive generalized harmonic function $\upsilon(P)$ on $C_n(\Omega)$ satisfying

$ A = \mathop {\inf }\limits_{P \in {C_n}(\Omega )} \frac{{\upsilon (P)}}{{M_\Omega ^a(P,\infty )}}{\rm{ < }}\mathop {\inf }\limits_{P \in {E_C}} \frac{{\upsilon (P)}}{{M_\Omega ^a(P,\infty )}} = B. $

Let $u(P)=\upsilon(P)+BC$ for any $P\in C_n(\Omega)$. Then $u(P)\geq BM^a_{\Omega}(P, \infty)$ for $P\in E$, and so

$ \mathop {\inf }\limits_{P \in {C_n}(\Omega )} \frac{{u(P)}}{{M_\Omega ^a(P,\infty )}} = A{\rm{ < }}\;B \le \mathop {\inf }\limits_{P \in {E_C}} \frac{{u(P)}}{{M_\Omega ^a(P,\infty )}}, $

which contradicts (a).

If we can show that $(E_{C})_{\ell}$ is not $a$-minimally thin at infinity when $\ell\in(0, 1)$, then for all $\ell\in(0, 1)$ the set $E_{\ell}$ also is not $a$-minimally thin at infinity, and hence (b) holds.

Suppose that for some $\ell\in(0, 1)$ the set $(E_{C})_{\ell}$ is $a$-minimally thin at infinity. Then from Theorem 1 there exists a positive generalized harmonic function $\upsilon(P)$ on $C_n(\Omega)$ satisfying

$ \mathop {\inf }\limits_{P \in {C_n}(\Omega )} \frac{{\upsilon (P)}}{{M_\Omega ^a(P,\infty )}}{\rm{ < }}\mathop {\inf }\limits_{P \in {E_C}} \frac{{\upsilon (P)}}{{M_\Omega ^a(P,\infty )}}. $

We see that $E_{C}$ characterizes the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$, so for all $\ell\in(0, 1)$ the set $(E_{C})_{\ell}$ is not $a$-minimally thin at infinity.

(c)$\Rightarrow$(a). Suppose that $E$ does not characterize the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$. Then there exists a positive generalized harmonic function $\upsilon(P)$ on $C_n(\Omega)$ such that

$ A = \mathop {\inf }\limits_{P \in {C_n}(\Omega )} \frac{{\upsilon (P)}}{{M_\Omega ^a(P,\infty )}}{\rm{ < }}\mathop {\inf }\limits_{P \in E} \frac{{\upsilon (P)}}{{M_\Omega ^a(P,\infty )}} = B. $

Put $h(P)=\upsilon(P)-AM^a_{\Omega}(P, \infty)$ for any $P\in C_n(\Omega)$. Then $h(P)$ is a positive generalized harmonic function on $C_n(\Omega)$ satisfying

$ \mathop {\inf }\limits_{P \in {C_n}(\Omega )} \frac{{h(P)}}{{M_\Omega ^a(P,\infty )}} = 0. $

(4.8)

For any $P\in E_{\ell}(\ell\in(0, 1)$ there exists a point $P'$ such that $\mid P-P'\mid < \ell d(P')$, and by the generalized Martin representation and the same proof as Theorem 1 we see that

$ \mathop {\inf }\limits_{P \in {E_\ell }} \frac{{h(P)}}{{M_\Omega ^a(P,\infty )}} \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} \mathop {\inf }\limits_{P \in E} \frac{{h(P)}}{{M_\Omega ^a(P,\infty )}} > 0. $

(4.9)

From (4.8) and (4.9) we obtain that

$ \mathop {\inf }\limits_{P \in {C_n}(\Omega )} \frac{{h(P)}}{{M_\Omega ^a(P,\infty )}} \le \mathop {\inf }\limits_{P \in {E_\ell }} \frac{{h(P)}}{{M_\Omega ^a(P,\infty )}} $

for the positive supfunction $h(P)$ on $C_n(\Omega)$. It follows that $E_{\ell}$ is $a$-minimally thin at infinity. This contradicts (c).

Proof of Theorem 3 (a)$\Rightarrow$(b). Assume that

$ \int_{E_{\ell}}\frac{V(1+r)W(1+r)dP}{(1+r)^2} < \infty $

for some $\ell\in(0, 1)$. Let $\{W_{i_m}\}_{m\geq1}$ be a subsequence of $\{W_{i}\}_{i\geq1}$ from Lemma 4. With (a) of Lemma 4 we obtain

$ \int_{\cup_{m} W_{i_m}}\frac{V(1+r)W(1+r)dP}{(1+r)^2} < \infty. $

Since $\cup_{m} W_{i_m}$ is a union of cubes from the Whitney cubes of $C_n(\Omega)$ with $\ell$, by Lemma 5 we see that $\cup_{m} W_{i_m}$ is $a$-minimally thin at infinity. Further, from Lemma 4 we know that $E_{\frac{\ell}{4}}$ is $a$-minimally thin at infinity.

On the other hand, since $E$ characterizes the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$, we see from the Theorem 2 that $E_{\frac{\ell}{4}}$ is not $a$-minimally thin at infinity, which contradicts the conclusion above.

(c)$\Rightarrow$(a). Suppose that $E$ does not characterize the positive generalized harmonic majorization of $M^a_{\Omega}(P, \infty)$. Then it follows from Theorem 2 that for any $\ell\in(0, 1)$$ $ is$ a $-minimally thin at infinity. So we see from Lemma 5 that for any $\ell\in(0, 1) $

$ \int_{E_{\ell}}\frac{V(1+r)W(1+r)dP}{(1+r)^2} < \infty, $

which contradicts (c).

Proof of Corollary 1 If$\{P_m\}$ is a separated sequence, then

$ B(P_i, \ell d(P_i))\cap B(P_j, \ell d(P_j))=\emptyset (i, j=1, 2, \cdots; i\neq j) $

for a sufficiently small $\ell\in(0, 1)$, and hence

$ \int_{E_{\ell}}\frac{V(1+r)W(1+r)dP}{(1+r)^2}\approx\sum_{m=1}^{\infty}\frac{d(P_m)^nV(\mid P_m\mid)W(\mid P_m\mid)}{\mid P_m\mid^2}. $

Following (c) of Theorem 3, Corollary 1 immediately holds.