Open Access

Sharp geometrical properties of a-rarefied sets via fixed point index for the Schrödinger operator equations

Fixed Point Theory and Applications20152015:89

https://doi.org/10.1186/s13663-015-0342-1

Received: 23 January 2015

Accepted: 28 May 2015

Published: 16 June 2015

Abstract

In this paper, we use the theory of fixed point index for the Schrödinger operator equations to obtain a geometrical property of a-rarefied sets at infinity on cones. Meanwhile, we give an example to show that the reverse of this property is not true.

Keywords

Schrödinger operator equations rarefied set Poisson-Sch integral Green-Sch potential

1 Introduction and main theorem

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 \(\overline{S}\), respectively. 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 \({\mathbf{R}}^{n}\).

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 \(x_{n}=r\cos\theta_{1}\).

Let D be an arbitrary domain in \({\mathbf{R}}^{n}\) and \(\mathscr{A}_{a}\) denote the class of nonnegative radial potentials \(a(P)\), i.e. \(0\leq a(P)=a(r)\), \(P=(r,\Theta)\in D\), such that \(a\in L_{\mathrm{loc}}^{b}(D)\) with some \(b> {n}/{2}\) if \(n\geq4\) and with \(b=2\) if \(n=2\) or \(n=3\).

If \(a\in\mathscr{A}_{a}\), then the Schrödinger operator
$$\mathit{Sch}_{a}=-\Delta+a(P)I=0, $$
where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space \(C_{0}^{\infty}(D)\) to an essentially self-adjoint operator on \(L^{2}(D)\) (see [1], Chapter 11). We will denote it by \(\mathit{Sch}_{a}\) as well. This last one has a Green-Sch function \(G_{D}^{a}(P,Q)\). Here \(G_{D}^{a}(P,Q)\) is positive on D and its inner normal derivative \(\partial G_{D}^{a}(P,Q)/{\partial n_{Q}}\geq0\), where \({\partial}/{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into D.
We call a function \(u\not\equiv-\infty\) that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator \(\mathit{Sch}_{a}\) if its values belong to the interval \([-\infty,\infty)\) and at each point \(P\in D\) with \(0< r< r(P)\) the generalized mean-value inequality (see [2])
$$u(P)\leq\int_{\partial{B(P,r)}}u(Q)\frac{\partial G_{B(P,r)}^{a}(P,Q)}{\partial n_{Q}}\, d\sigma(Q) $$
is satisfied, where \(G_{B(P,r)}^{a}(P,Q)\) is the Green-Sch function of \(\mathit{Sch}_{a}\) in \(B(P,r)\) and \(d\sigma(Q)\) is a surface measure on the sphere \(\partial{B(P,r)}\).

If −u is a subfunction, then we call u a superfunction. If a function u is both subfunction and superfunction, it is, clearly, continuous and is called a generalized harmonic function (with respect to the Schrödinger operator \(\mathit{Sch}_{a}\)).

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 \(\Xi\times\Omega\). 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. We denote the set \(I\times\Omega\) with an interval on R by \(C_{n}(\Omega;I)\).

We shall say that a set \(H\subset C_{n}(\Omega)\) has a covering \(\{r_{j}, R_{j}\}\) if there exists a sequence of balls \(\{B_{j}\}\) with centers in \(C_{n}(\Omega)\) such that \(H\subset\bigcup_{j=0}^{\infty} B_{j}\), where \(r_{j}\) is the radius of \(B_{j}\) and \(R_{j}\) is the distance from the origin to the center of \(B_{j}\). For positive functions \(h_{1}\) and \(h_{2}\), we say that \(h_{1}\lesssim h_{2}\) if \(h_{1}\leq Mh_{2}\) for some constant \(M>0\). If \(h_{1}\lesssim h_{2}\) and \(h_{2}\lesssim h_{1}\), we say that \(h_{1}\approx h_{2}\).

From now on, we always assume \(D=C_{n}(\Omega)\). For the sake of brevity, we shall write \(G_{\Omega}^{a}(P,Q)\) instead of \(G_{C_{n}(\Omega)}^{a}(P,Q)\). Throughout this paper, let c denote various positive constants, because we do not need to specify them. Moreover, ϵ appearing in the expression in the following all sections will be a sufficiently small positive number.

Let Ω be a domain on \({\mathbf{S}}^{n-1}\) with smooth boundary. Consider the Dirichlet problem
$$\begin{aligned}& (\Lambda_{n}+\lambda)\varphi=0 \quad \text{on } \Omega, \\& \varphi=0 \quad \text{on } \partial{\Omega}, \end{aligned}$$
where \(\Lambda_{n}\) is the spherical part of the Laplace operator \(\Delta_{n}\):
$$\Delta_{n}=\frac{n-1}{r}\frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial r^{2}}+ \frac{\Lambda_{n}}{r^{2}}. $$
We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by \(\varphi(\Theta)\). In order to ensure the existence of λ and a smooth \(\varphi(\Theta)\). 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.
Solutions of an ordinary differential equation
$$ -Q''(r)-\frac{n-1}{r}Q'(r)+ \biggl( \frac{\lambda}{r^{2}}+a(r) \biggr)Q(r)=0,\quad 0< r< \infty. $$
(1.1)
It is well known (see, for example, [3]) that if the potential \(a\in \mathscr{A}_{a}\), then (1.1) has a fundamental system of positive solutions \(\{V,W\}\) such that V and W are increasing and decreasing, respectively (see [47]).

We will also consider the class \(\mathscr{B}_{a}\), consisting of the potentials \(a\in\mathscr{A}_{a}\) such that there exists the finite limit \(\lim_{r\rightarrow\infty}r^{2} a(r)=k\in[0,\infty)\), and moreover, \(r^{-1}|r^{2} a(r)-k|\in L(1,\infty)\). If \(a\in \mathscr{B}_{a}\), then the (sub)superfunctions are continuous (see [8]).

In the rest of paper, we assume that \(a\in\mathscr{B}_{a}\) and we shall suppress this assumption for simplicity.

Denote
$$\iota_{k}^{\pm}=\frac{2-n\pm\sqrt{(n-2)^{2}+4(k+\lambda)}}{2}, $$
then the solutions to (1.1) have the asymptotics (see [9])
$$ V(r)\approx r^{\iota_{k}^{+}}, \qquad W(r)\approx r^{\iota _{k}^{-}}, \quad \text{as } r\rightarrow\infty. $$
(1.2)
Let ν be any positive measure on cones such that the Green-Sch potential
$$G_{\Omega}^{a} \nu(P)=\int_{C_{n}(\Omega)}G_{\Omega}^{a}(P,Q) \, d\nu(Q)\not\equiv +\infty $$
for any \(P\in C_{n}(\Omega)\). Then the positive measure \(\nu'\) on \({\mathbf{R}}^{n}\) is defined by
$$d\nu'(Q)=\left \{ \textstyle\begin{array}{l@{\quad}l} W(t) \varphi(\Phi)\, d\nu(Q), & Q=(t,\Phi)\in C_{n}(\Omega; (1,+\infty)) , \\ 0,& Q\in{\mathbf{R}}^{n}-C_{n}(\Omega; (1,+\infty)). \end{array}\displaystyle \right . $$
The Poisson-Sch integral \(PI_{\Omega}^{a} \mu(P)\not\equiv+\infty\) (\(P\in C_{n}(\Omega)\)) of μ on cones is defined as follows:
$$PI_{\Omega}^{a} \mu(P)=\frac{1}{c_{n}}\int _{S_{n}(\Omega)}PI_{\Omega}^{a}(P,Q)\, d\mu(Q), $$
where
$$PI_{\Omega}^{a}(P,Q)=\frac{\partial G_{\Omega}^{a}(P,Q)}{\partial n_{Q}},\qquad c_{n}= \left \{ \textstyle\begin{array}{l@{\quad}l} 2\pi, & n=2, \\ (n-2)s_{n}, & n\geq3, \end{array}\displaystyle \right . $$
μ is a positive measure on \(\partial{C_{n}(\Omega)}\) and \({\partial}/{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into cones. Then the positive measure \(\mu'\) on \({\mathbf{R}}^{n}\) is defined by
$$d\mu'(Q)=\left \{ \textstyle\begin{array}{ll} t^{-1}W(t)\frac{\partial\varphi(\Phi)}{\partial n_{\Phi}}\, d\mu(Q), & Q=(t,\Phi)\in S_{n}(\Omega; (1,+\infty)) , \\ 0,&Q;\in{\mathbf{R}}^{n}-S_{n}(\Omega; (1,+\infty)). \end{array}\displaystyle \right . $$

Remark

We remark that the total masses of \(\mu'\) and \(\nu'\) are finite (see [2], Lemma 5 and [6], Lemma 4).

Let \(0\leq\alpha\leq n\) and λ be any positive measure on \({\mathbf{R}}^{n}\) having finite total mass. For each \(P=(r,\Theta)\in {\mathbf{R}}^{n}-\{O\}\), the maximal function \(M(P;\lambda,\alpha)\) with respect to \(\mathit{Sch}_{a}\) is defined by
$$M(P;\lambda,\alpha)=\sup_{ 0< \rho< \frac{r}{2}}\lambda\bigl(B(P,\rho)\bigr)V( \rho)W(\rho)\rho^{\alpha-2}. $$
The set
$$\bigl\{ P=(r,\Theta)\in{\mathbf{R}}^{n}-\{O\}; M(P;\lambda, \alpha)V^{-1}(r)W^{-1}(r)r^{2-\alpha}>\epsilon\bigr\} $$
is denoted by \(E(\epsilon; \lambda, \alpha)\).

The following Theorems A and B give a way to estimate the Green-Sch potential and the Poisson-Sch integrals with measures on \(C_{n}(\Omega)\) and \(S_{n}(\Omega)\), respectively.

Theorem A

Let ν be a positive measure on \(C_{n}(\Omega)\) such that \(G_{\Omega}^{a} \nu(P)\not\equiv +\infty\) (\(P=(r,\Theta)\in C_{n}(\Omega)\)) holds. Then for a sufficiently large L we have
$$\bigl\{ P\in C_{n}\bigl(\Omega; (L, +\infty)\bigr); G_{\Omega}^{a}\nu(P)\geq V(r)\bigr\} \subset E\bigl(\epsilon; \mu',1\bigr). $$

Theorem B

Let μ be a positive measure on \(S_{n}(\Omega)\) such that \(PI_{\Omega}^{a} \mu(P)\not\equiv+\infty\) (\(P=(r,\Theta)\in C_{n}(\Omega)\)). Then for a sufficiently large L we have
$$\bigl\{ P\in C_{n}\bigl(\Omega; (L, +\infty)\bigr); PI_{\Omega}^{a} \mu(P)\geq V(r)\bigr\} \subset E\bigl(\epsilon; \mu',1\bigr). $$
It is known that the Martin boundary of \(C_{n}(\Omega)\) is the set \(\partial{C_{n}(\Omega)}\cup\{\infty\}\), each of which is a minimal Martin boundary point. For \(P\in C_{n}(\Omega)\) and \(Q\in \partial{C_{n}(\Omega)}\cup\{\infty\}\), the Martin kernel can be defined by \(M_{\Omega}^{a}(P,Q)\). If the reference point P is chosen suitably, then we have
$$ M_{\Omega}^{a}(P,\infty)=V(r)\varphi(\Theta)\quad \text {and} \quad M_{\Omega}^{a}(P,O)=cW(r)\varphi(\Theta) $$
for any \(P=(r,\Theta)\in C_{n}(\Omega)\).

In [7, 10], Xue and Zhao-Yamada introduce the notations of a-thin (with respect to the Schrödinger operator \(\mathit{Sch}_{a}\)) at a point and a-rarefied sets at infinity (with respect to the Schrödinger operator \(\mathit{Sch}_{a}\)), which generalized the earlier notations obtained by Miyamoto, Hoshida, Brelot (see [1114]).

Definition 1

(see [7])

A set H in \({\mathbf{R}}^{n}\) is said to be a-thin at a point Q if there is a fine neighborhood E of Q which does not intersect \(H\backslash\{Q\}\). Otherwise H is said to be not a-thin at Q on cones.

Definition 2

(see [10])

A subset H of \(C_{n}(\Omega)\) is said to be a-rarefied at infinity on cones, if there exists a positive superfunction \(v(P)\) on cones such that
$$ \inf_{P\in C_{n}(\Omega)}\frac{v(P)}{M_{\Omega}^{a}(P,\infty)}\equiv0 $$
(1.3)
and
$$ H\subset\bigl\{ P=(r,\Theta)\in C_{n}(\Omega); v(P)\geq V(r)\bigr\} . $$
(1.4)
Let H be a bounded subset of \(C_{n}(\Omega)\). Then \(\hat{R}_{M_{\Omega}^{a}(\cdot,\infty)}^{H}\) is bounded on cones and the greatest generalized harmonic minorant of \(\hat{R}_{M_{\Omega}^{a}(\cdot,\infty)}^{H}\) is zero. We see from the Riesz decomposition theorem (see [6], Theorem 2) that there exists a unique positive measure \(\lambda_{H}^{a}\) on cones such that (see [7], p.6)
$$ \hat{R}_{M_{\Omega}^{a}(\cdot,\infty)}^{H}(P)=G_{\Omega }^{a} \lambda_{H}^{a}(P) $$
for any \(P\in C_{n}(\Omega)\) and \(\lambda_{H}^{a}\) is concentrated on \(I_{H}\), where
$$I_{H}=\bigl\{ P\in C_{n}(\Omega); H \text{ is not } a \text{-thin at } P\bigr\} . $$
We denote the total mass \(\lambda_{H}^{a}(C_{n}(\Omega))\) of \(\lambda_{H}^{a}\) by \(\lambda_{\Omega}^{a}(H)\).

Recently, GX Xue (see [7], Theorem 2.5) gave a criterion for a subset H of \(C_{n}(\Omega)\) to be a-rarefied set at infinity.

Theorem C

A subset H of \(C_{n}(\Omega)\) is a-rarefied at infinity on cones if and only if
$$ \sum_{j=0}^{\infty}W\bigl(2^{j}\bigr) \lambda_{H_{j}}^{a}\bigl(C_{n}(\Omega)\bigr)< \infty, $$
where \(H_{j}=H\cap C_{n}(\Omega;[2^{j},2^{j+1}))\) and \(j=0,1,2,\ldots\) .

Our aim in this paper is to characterize the geometrical property of a-rarefied sets at infinity.

Theorem 1

If a subset H of \(C_{n}(\Omega)\) is a-rarefied at infinity on cones, then H has a covering \(\{r_{j},R_{j}\}\) (\(j=0,1,2,\ldots\)) satisfying
$$ \sum_{j=0}^{\infty} \biggl( \frac{r_{j}}{R_{j}} \biggr)\frac {V(R_{j})}{V(r_{j})}\frac{W(R_{j})}{W(r_{j})}< \infty. $$
(1.5)

Next, we immediately have the following result from Theorem 1.

Corollary 1

Let \(v(P)\) be positive superfunction on cones. Then \(v(P)V^{-1}(r)\) uniformly converges to \(c_{\infty}(v,a)\varphi(\Theta)\) as \(r\rightarrow\infty\) outside a set which has a covering \(\{r_{j},R_{j}\}\) (\(j=0,1,2,\ldots\)) satisfying (1.5), where
$$c_{\infty}(v, a)=\inf_{P\in C_{n}(\Omega)}\frac{v(P)}{M^{a}_{\Omega}(P, \infty)}. $$

Finally, we prove the following result.

Theorem 2

If a subset H of \(C_{n}(\Omega)\) has a covering \(\{r_{j},R_{j}\}\) (\(j=0,1,2,\ldots\)) satisfying (1.5), then it is possible that H is not a-rarefied at infinity on cones.

2 Main lemmas

Lemma 1

Let λ be any positive measure on \({\mathbf{R}}^{n}\) having finite total mass. Then \(E(\epsilon; \lambda, 1)\) has a covering \(\{r_{j},R_{j}\}\) (\(j=1,2,\ldots\)) satisfying
$$\sum_{j=1}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)\frac {V(R_{j})W(R_{j})}{V(r_{j})W(r_{j})}< \infty. $$

Proof

Set
$$E_{j}(\epsilon;\lambda, 1)= \bigl\{ P=(r,\Theta)\in E(\epsilon; \lambda, 1):2^{j}\leq r< 2^{j+1}\bigr\} \quad (j=2,3,4,\ldots). $$
If \(P=(r,\Theta)\in E_{j}(\epsilon; \lambda, 1)\), then there exists a positive number \(\rho(P)\) such that
$$\biggl(\frac{\rho(P)}{r} \biggr)\frac{V(r)W(R)}{V(\rho(P))W(\rho (P))}\approx \biggl( \frac{\rho(P)}{r} \biggr)^{n-1}\leq \frac{\lambda(B(P,\rho(P)))}{\epsilon}. $$
Since \(E_{j}(\epsilon; \lambda, 1)\) can be covered by the union of a family of balls \(\{B(P_{j,i},\rho_{j,i}):P_{j,i}\in E_{k}(\epsilon; \lambda, 1)\}\) (\(\rho_{j,i}=\rho(P_{j,i})\)). By the Vitali lemma (see [15]), there exists \(\Lambda_{j} \subset E_{j}(\epsilon; \lambda, 1)\), which is at most countable, such that \(\{B(P_{j,i},\rho_{j,i}):P_{j,i}\in\Lambda_{j} \}\) are disjoint and \(E_{j}(\epsilon; \lambda, 1) \subset \bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},5\rho_{j,i})\). So
$$\bigcup_{j=2}^{\infty}E_{j}( \epsilon; \lambda, 1) \subset \bigcup_{j=2}^{\infty}\bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i},5 \rho_{j,i}). $$
On the other hand, note that
$$\bigcup_{P_{j,i}\in\Lambda_{j}} B(P_{j,i}, \rho_{j,i}) \subset\bigl\{ P=(r,\Theta):2^{j-1}\leq r< 2^{j+2}\bigr\} , $$
so that
$$\begin{aligned} \sum_{P_{j,i} \in \Lambda_{j}} \biggl(\frac{5\rho_{j,i}}{|P_{j,i}|} \biggr) \frac{ V(|P_{j,i}|)W(|P_{j,i}|) }{V(\rho_{j,i})W(\rho_{j,i}) } \approx& \sum_{P_{j,i} \in\Lambda_{j}} \biggl( \frac{5\rho _{j,i}}{|P_{j,i}|} \biggr)^{n-1} \\ \leq&5;^{n-1}\sum_{P_{j,i}\in\Lambda_{j}}\frac{\lambda (B(P_{j,i},\rho_{j,i}))}{\epsilon} \\ \leq& \frac{5^{n-1}}{\epsilon} \lambda\bigl(C_{n}\bigl(\Omega; \bigl[2^{j-1},2^{j+2} \bigr)\bigr)\bigr). \end{aligned}$$
Hence we obtain
$$\begin{aligned} \sum_{j=1}^{\infty}\sum _{P_{j,i} \in \Lambda_{j}} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr)\frac{ V(|P_{j,i}|)W(|P_{j,i}|) }{V(\rho_{j,i})W(\rho_{j,i}) } \approx& \sum_{j=1}^{\infty}\sum _{P_{j,i} \in\Lambda_{j}} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr)^{n-1} \\ \leq&\sum_{j=1}^{\infty}\frac{ \lambda(C_{n}(\Omega ; [2^{j-1},2^{j+2} )))}{\epsilon} \\ \leq& \frac{3\lambda({\mathbf{R}}^{n})}{\epsilon}. \end{aligned}$$
Since \(E(\epsilon; \lambda, 1)\cap\{P=(r,\Theta)\in{\mathbf{R}}^{n}; r\geq4\}=\bigcup_{j=2}^{\infty}E_{j}(\epsilon;\lambda, 1)\). Then \(E(\epsilon; \lambda, 1)\) 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
$$\sum_{j,i} \biggl(\frac{\rho_{j,i}}{|P_{j,i}|} \biggr) \frac{ V(|P_{j,i}|)W(|P_{j,i}|) }{V(\rho_{j,i})W(\rho_{j,i}) }\approx\sum_{j,i} \biggl( \frac{\rho_{j,i}}{|P_{j,i}|} \biggr)^{n-1}\leq\frac {3\lambda({\mathbf{R}}^{n})}{\epsilon}+6^{n-\alpha}< + \infty, $$
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

Since H is a-rarefied at infinity on cones, by Definition 2 there exists a positive superfunction \(v(P)\) on cones such that (1.3) and (1.4) hold.

For this \(v(P)\) there exists a unique positive measure \(\mu''\) on \(S_{n}(\Omega)\) and a unique positive measure \(\nu''\) on cones such that (see [2], Theorem 3)
$$ v(P)=c_{0}(v, a)M_{\Omega}^{a}(P,O)+G_{\Omega}^{a} \nu''(P)+PI_{\Omega}^{a} \mu''(P), $$
(3.1)
where
$$c_{0}(v, a)=\inf_{P\in C_{n}(\Omega)}\frac{v(P)}{M^{a}_{\Omega}(P, O)}. $$
Let us denote
$$\begin{aligned}& H_{1}=\biggl\{ P=(r,\Theta)\in C_{n}(\Omega); c_{0}(v, a)M_{\Omega}^{a}(P,O)\geq\frac{V(r)}{3} \biggr\} , \\& H_{2}=\biggl\{ P=(r,\Theta)\in C_{n}(\Omega); G_{\Omega}^{a}\nu''(P)\geq \frac {V(r)}{3}\biggr\} \end{aligned}$$
and
$$H_{3}=\biggl\{ P=(r,\Theta)\in C_{n}(\Omega); PI_{\Omega}^{a}\mu''(P)\geq \frac{V(r)}{3}\biggr\} , $$
respectively.
Then we see from (1.4) that
$$ H\subset H_{1}\cup H_{2} \cup H_{3}. $$
(3.2)
For each \(H_{i}\) (\(i=1,2,3\)), we know that it has a covering. It is evident from the boundedness of \(H_{1}\) that \(H_{1}\) has a covering \(\{r_{1},R_{1}\}\) satisfying
$$ \frac{r_{1}}{R_{1}}< +\infty. $$
(3.3)
When we apply Theorems A and B with the measures μ and ν defined by \(\mu=3\mu''\) and \(\nu=3\nu''\), respectively, we can find two positive constants L and ϵ such that
$$H_{2}\cap C_{n}\bigl(\Omega; (L, +\infty)\bigr)\subset E \bigl(\epsilon; \mu',1\bigr) $$
and
$$H_{3}\cap C_{n}\bigl(\Omega; (L, +\infty)\bigr)\subset E \bigl(\epsilon; \nu',1\bigr), $$
respectively.
By Lemma 1, these sets \(E(\epsilon; \mu',1)\) and \(E(\epsilon; \nu',1)\) have coverings \(\{r_{j}^{(2)},R_{j}^{(2)}\}\) (\(j=1,2,\ldots\)) and \(\{r_{j}^{(3)},R_{j}^{(3)}\}\) (\(j=1,2,\ldots\)) satisfying
$$ \sum_{j=1}^{\infty} \biggl( \frac{r_{j}^{(2)}}{R_{j}^{(2)}} \biggr)\frac {V(R_{j}^{(2)})W(R_{j}^{(2)})}{V(r_{j}^{(2)})W(r_{j}^{(2)})}< +\infty $$
(3.4)
and
$$ \sum_{j=1}^{\infty} \biggl( \frac{r_{j}^{(3)}}{R_{j}^{(3)}} \biggr)\frac {V(R_{j}^{(3)})W(R_{j}^{(3)})}{V(r_{j}^{(3)})W(r_{j}^{(3)})}< +\infty , $$
(3.5)
respectively.

Then \(H_{2}\) and \(H_{3}\) also have coverings \(\{r_{j}^{(2)},R_{j}^{(2)}\}\) (\(j=1,2,\ldots\)) and \(\{r_{j}^{(3)},R_{j}^{(3)}\}\) (\(j=1,2,\ldots\)) satisfying (3.4) and (3.5), respectively.

Thus by rearranging coverings \(\{r_{1},R_{1}\}\), \(\{r_{j}^{(2)},R_{j}^{(2)}\}\) (\(j=1,2,\ldots\)) and \(\{r_{j}^{(3)},R_{j}^{(3)}\}\) (\(j=1,2,\ldots\)), we know that the set H has a covering \(\{r_{j},R_{j}\}\) (\(j=0,1,2,\ldots\)) from (3.2) and satisfies (1.5) from (3.3), (3.4), and (3.5).

Thus we complete the proof of Theorem 1.

4 Proof of Theorem 2

Put
$$r_{j}=3\cdot2^{j-1}\cdot j^{\frac{1}{2-n}}\quad \text{and} \quad R_{j}=3\cdot 2^{j-1} \quad (j=1,2,3,\ldots). $$
A covering \(\{r_{j}, R_{j}\}\) satisfies
$$\sum_{j=1}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)\frac {V(R_{j})}{V(r_{j})}\frac{W(R_{j})}{W(r_{j})}\leq c\sum _{j=1}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)^{n-1}=c \sum_{j=1}^{\infty}j^{\frac{n-1}{2-n}}< + \infty $$
from (1.2).

Let \(C_{n}(\Omega')\) be a subset of \(C_{n}(\Omega)\), i.e. \(\overline{\Omega}'\subset\Omega\). Suppose that this covering is so located: there is an integer \(j_{0}\) such that \(B_{j}\subset C_{n}(\Omega')\) and \(R_{j}>2r_{j}\) for \(j\geq j_{0}\).

Next we shall prove that the set \(H=\bigcup_{j=j_{0}}^{\infty}B_{j}\) is not a-rarefied at infinity on \(C_{n}(\Omega)\). Since \(\varphi(\Theta)\geq c\) for any \(\Theta\in\Omega'\), we have \(M_{\Omega}^{a}(P,\infty)\geq cV(R_{j})\) for any \(P\in \overline{B}_{j}\), where \(j\geq j_{0}\). Hence we have
$$ \hat{R}_{M_{\Omega}^{a}(\cdot,\infty)}^{B_{j}}(P)\geq cV(R_{j}) $$
(4.1)
for any \(P\in\overline{B}_{j}\), where \(j\geq j_{0}\).
Take a measure δ on cones, \(\operatorname{supp} \delta\subset \overline{B}_{j}\), \(\delta(\overline{B}_{j})=1\) such that
$$ \int_{C_{n}(\Omega)}|P-Q|^{2-n}\, d\delta(P)=\bigl\{ \operatorname{Cap}(\overline{B}_{j})\bigr\} ^{-1} $$
(4.2)
for any \(Q\in\overline{B}_{j}\), where Cap denotes the Newton capacity. Since
$$G_{\Omega}^{a}(P,Q)\leq|P-Q|^{2-n} $$
for any \(P\in C_{n}(\Omega)\) and \(Q\in C_{n}(\Omega)\),
$$\begin{aligned} \bigl\{ \operatorname{Cap}(\overline{B}_{j})\bigr\} ^{-1} \lambda_{B_{j}}^{a}\bigl(C_{n}(\Omega)\bigr) =& \int \biggl(\int|P-Q|^{2-n}\, d\delta(P) \biggr)\, d\lambda _{B_{j}}^{a}(Q) \\ \geq& \int \biggl(\int G_{\Omega}^{a}(P,Q)\, d \lambda_{B_{j}}^{a}(Q) \biggr)\, d\delta(P) \\ =& \int\hat{R}_{M_{\Omega}^{a}(\cdot,\infty)}^{B_{j}}\, d\delta(P) \\ \geq& c V(R_{j})\delta(\overline{B}_{j})=c V(R_{j}) \end{aligned}$$
from (4.1) and (4.2). Hence we have (see [5], p.1517)
$$ \lambda_{B_{j}}^{a}\bigl(C_{n}(\Omega) \bigr)\geq c \operatorname{Cap}(\overline{B}_{j}) V(R_{j})\geq c r_{j}^{n-2}V(R_{j}). $$
(4.3)
If we observe \(\lambda_{H_{j}}^{a}(C_{n}(\Omega))=\lambda_{B_{j}}^{a}(C_{n}(\Omega))\), then we have by (1.2)
$$\sum_{j=j_{0}}^{\infty}W\bigl(2^{j}\bigr) \lambda_{H_{j}}^{a}\bigl(C_{n}(\Omega)\bigr)\geq c \sum_{j=j_{0}}^{\infty} \biggl(\frac{r_{j}}{R_{j}} \biggr)^{n-2}= c \sum_{j=j_{0}}^{\infty} \frac{1}{j}=+\infty, $$
from which it follows by Theorem C that H is not a-rarefied at infinity on cones.

Declarations

Acknowledgements

This work was completed while the second author was visiting the Department of Mathematical Sciences at the Columbia University, and he is grateful for the kind hospitality of the Department. This work was partially supported by NSF Grant DMS-0913205.

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
(2)
Mathematics Institute, Roskilde University

References

  1. Levin, B, Kheyfits, A: Asymptotic behavior of subfunctions of time-independent Schrödinger operator. In: Some Topics on Value Distribution and Differentiability in Complex and P-Adic Analysis, Chapter 11, pp. 323-397. Science Press, Beijing (2008) Google Scholar
  2. Qiao, L, Ren, YD: Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone. Monatshefte Math. 173(4), 593-603 (2014) MATHMathSciNetView ArticleGoogle Scholar
  3. Verzhbinskii, GM, Maz’ya, VG: Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I. Sib. Mat. Zh. 12, 874-899 (1971) View ArticleGoogle Scholar
  4. Qiao, L, Zhao, T: Boundary limits for fractional Poisson a-extensions of \(L^{p}\) boundary function in a cone. Pac. J. Math. 272(1), 227-236 (2014) MATHMathSciNetView ArticleGoogle Scholar
  5. Qiao, L, Pan, GS: Integral representations of generalized harmonic functions. Taiwan. J. Math. 17(5), 1503-1521 (2013) MATHMathSciNetGoogle Scholar
  6. Qiao, L, Pan, GS: Generalization of the Phragmén-Lindelöf theorems for subfunctions. Int. J. Math. 24(8), 1350062 (2013) MathSciNetView ArticleGoogle Scholar
  7. Xue, GX: A remark on the a-minimally thin sets associated with the Schrödinger operator. Bound. Value Probl. 2014, 133 (2014) View ArticleGoogle Scholar
  8. Simon, B: Schrödinger semigroups. Bull. Am. Math. Soc. 7, 447-526 (1982) MATHView ArticleGoogle Scholar
  9. Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977) MATHView ArticleGoogle Scholar
  10. Zhao, T, Yamada, AJ: Growth properties of Green-Sch potentials at infinity. Bound. Value Probl. 2014, 245 (2014) MathSciNetView ArticleGoogle Scholar
  11. Brelot, M: On Topologies and Boundaries in Potential Theory. Lecture Notes in Mathematics, vol. 175. Springer, Berlin (1971) MATHGoogle Scholar
  12. Miyamoto, I, Yoshida, H: On harmonic majorization of the Martin function at infinity in a cone. Czechoslov. Math. J. 55(130)(4), 1041-1054 (2005) MathSciNetView ArticleGoogle Scholar
  13. Miyamoto, I, Yoshida, H: On a-minimally thin sets at infinity in a cone. Hiroshima Math. J. 37(1), 61-80 (2007) MATHMathSciNetGoogle Scholar
  14. Yoshida, H: Harmonic majorant of a radial subharmonic function on a strip and their applications. Int. J. Pure Appl. Math. 30(2), 259-286 (2006) MATHMathSciNetGoogle Scholar
  15. Stein, EM: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970) MATHGoogle Scholar

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© Li and Ychussie 2015