RETRACTED ARTICLE: A remark on the a-minimally thin sets associated with the Schrödinger operator

Abstract

The aim of this paper is to give a new criterion for a-minimally thin sets at infinity with respect to the Schrödinger operator in a cone, which supplement the results obtained by T. Zhao.

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 ${\mathbf{R}}^n$ ($n\ge 2$) the n-dimensional Euclidean space. A point in ${\mathbf{R}}^n$ is denoted by $P=(X,x_n)$, $X=(x_1,x_2,\dots ,x_{n-1})$. 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. Further, intS, diamS, and $dist(S_1,S_2)$ stand for the interior of S, the diameter of S, and the distance between $S_1$ and $S_2$, respectively.

We introduce a system of spherical coordinates $(r,\mathrm{\Theta })$, $\mathrm{\Theta }=({\theta }_1,{\theta }_2,\dots ,{\theta }_{n-1})$, in ${\mathbf{R}}^n$ which are related to cartesian coordinates $(x_1,x_2,\dots ,x_{n-1},x_n)$ by $x_n=rcos{\theta }_1$.

Let D be an arbitrary domain in ${\mathbf{R}}^n$ and ${\mathcal{A}}_a$ denote the class of non-negative radial potentials $a(P)$, i.e. $0\le a(P)=a(r)$, $P=(r,\mathrm{\Theta })\in D$, such that $a\in L_{\mathrm{loc}}^b(D)$ with some $b>n/2$ if $n\ge 4$ and with $b=2$ if $n=2$ or $n=3$ (see [[1], p.354] and [2]).

If $a\in {\mathcal{A}}_a$, then the stationary Schrödinger operator

$$Sc{h}_a=-\mathrm{\Delta }+a(P)I=0,$$

where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space $C_0^{\mathrm{\infty }}(D)$ to an essentially self-adjoint operator on $L^2(D)$ (see [[1], Ch. 11]). We will denote it $Sc{h}_a$ as well. This last one has a Green a-function $G_D^a(P,Q)$. Here $G_D^a(P,Q)$ is positive on D and its inner normal derivative $\partial G_D^a(P,Q)/\partial n_Q\ge 0$, where $\partial /\partial n_Q$ denotes the differentiation at Q along the inward normal into D.

We call a function $u\not\equiv -\mathrm{\infty }$ that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator $Sc{h}_a$ if its values belong to the interval $[-\mathrm{\infty },\mathrm{\infty })$ and at each point $P\in D$ with $0<r<r(P)$ we have the generalized mean-value inequality (see [[1], Ch. 11])

$$u(P)\le {\int }_{S(P,r)}u(Q)\frac{\partial G_{B(P,r)}^a(P,Q)}{\partial n_Q}d\sigma (Q)$$

satisfied, where $G_{B(P,r)}^a(P,Q)$ is the Green a-function of $Sc{h}_a$ in $B(P,r)$ and $d\sigma (Q)$ is a surface measure on the sphere $S(P,r)=\partial B(P,r)$. If −u is a subfunction, then we call u a superfunction (with respect to the Schrödinger operator $Sc{h}_a$).

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 $(1,\mathrm{\Theta })$ on ${\mathbf{S}}^{n-1}$ and the set $\{\mathrm{\Theta };(1,\mathrm{\Theta })\in \mathrm{\Omega }\}$ for a set Ω, $\mathrm{\Omega }\subset {\mathbf{S}}^{n-1}$, are often identified with Θ and Ω, respectively. For two sets $\mathrm{\Xi }\subset {\mathbf{R}}_{+}$ and $\mathrm{\Omega }\subset {\mathbf{S}}^{n-1}$, the set $\{(r,\mathrm{\Theta })\in {\mathbf{R}}^n;r\in \mathrm{\Xi },(1,\mathrm{\Theta })\in \mathrm{\Omega }\}$ in ${\mathbf{R}}^n$ is simply denoted by $\mathrm{\Xi }\times \mathrm{\Omega }$. By $C_n(\mathrm{\Omega })$, we denote the set ${\mathbf{R}}_{+}\times \mathrm{\Omega }$ in ${\mathbf{R}}^n$ with the domain Ω on ${\mathbf{S}}^{n-1}$. We call it a cone. We denote the set $I\times \mathrm{\Omega }$ with an interval on R by $C_n(\mathrm{\Omega };I)$.

From now on, we always assume $D=C_n(\mathrm{\Omega })$. For the sake of brevity, we shall write $G_{\mathrm{\Omega }}^a(P,Q)$ instead of $G_{C_n(\mathrm{\Omega })}^a(P,Q)$. We shall also write $g_1\approx g_2$ for two positive functions $g_1$ and $g_2$, if and only if there exists a positive constant c such that $c^{-1}{g}_1\le g_2\le c{g}_1$.

Let Ω be a domain on ${\mathbf{S}}^{n-1}$ with smooth boundary. Consider the Dirichlet problem

$$\begin{array}{c}({\mathrm{\Lambda }}_n+\lambda )\phi =0\text{on }\mathrm{\Omega },\hfill \\ \phi =0\text{on }\partial \mathrm{\Omega },\hfill \end{array}$$

where ${\mathrm{\Lambda }}_n$ is the spherical part of the Laplace operata ${\mathrm{\Delta }}_n$

$${\mathrm{\Delta }}_n=\frac{n-1}{r}\frac{\partial }{\partial r}+\frac{{\partial }^2}{\partial r^2}+\frac{{\mathrm{\Lambda }}_n}{r^2}.$$

We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by $\phi (\mathrm{\Theta })$. In order to ensure the existence of λ and a smooth $\phi (\mathrm{\Theta })$, we put a rather strong assumption on Ω: if $n\ge 3$, 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).

For any $(1,\mathrm{\Theta })\in \mathrm{\Omega }$, we have (see [[4], pp.7-8])

$$\delta (P)\approx r\phi (\mathrm{\Theta }),$$

(1)

where $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$ and $\delta (P)=dist(P,\partial C_n(\mathrm{\Omega }))$.

Solutions of an ordinary differential equation (see [[5], p.217])

$$-Q^{″}(r)-\frac{n-1}{r}{Q}^{\prime }(r)+(\frac{\lambda }{r^2}+a(r))Q(r)=0,0<r<\mathrm{\infty }.$$

(2)

It is well known (see, for example, [6]) that if the potential $a\in {\mathcal{A}}_a$, then equation (2) has a fundamental system of positive solutions $\{V,W\}$ such that V and W are increasing and decreasing, respectively.

We will also consider the class ${\mathcal{B}}_a$, consisting of the potentials $a\in {\mathcal{A}}_a$ such that there exists the finite limit ${lim}_{r\to \mathrm{\infty }}{r}^{2}a(r)=k\in [0,\mathrm{\infty })$, and, moreover, $r^{-1}|r^{2}a(r)-k|\in L(1,\mathrm{\infty })$. If $a\in {\mathcal{B}}_a$, then the (sub)superfunctions are continuous (see [7]). In the rest of paper, we assume that $a\in {\mathcal{B}}_a$ 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 equation (2) have the asymptotic (see [3])

$$V(r)\approx r^{{\iota }_k^{+}},W(r)\approx r^{{\iota }_k^{-}},\text{as }r\to \mathrm{\infty }.$$

(3)

It is well known that the Martin boundary of $C_n(\mathrm{\Omega })$ is the set $\partial C_n(\mathrm{\Omega })\cup \{\mathrm{\infty }\}$, each of which is a minimal Martin boundary point. For $P\in C_n(\mathrm{\Omega })$ and $Q\in \partial C_n(\mathrm{\Omega })\cup \{\mathrm{\infty }\}$, the Martin kernel can be defined by $M_{\mathrm{\Omega }}^a(P,Q)$. If the reference point P is chosen suitably, then we have

$$M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })=V(r)\phi (\mathrm{\Theta })\text{and}{M}_{\mathrm{\Omega }}^a(P,O)=cW(r)\phi (\mathrm{\Theta }),$$

(4)

for any $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$.

In [[8], p.67], Zhao introduce the notations of a-thin (with respect to the Schrödinger operator $Sc{h}_a$) at a point, a-polar set (with respect to the Schrödinger operator $Sc{h}_a$) and a-minimal thin sets at infinity (with respect to the Schrödinger operator $Sc{h}_a$). 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 $H\mathrm{\setminus }\{Q\}$. Otherwise H is said to be not a-thin at Q on $C_n(\mathrm{\Omega })$. A set H in ${\mathbf{R}}^n$ is called a polar set if there is a superfunction u on some open set E such that $H\subset \{P\in E;u(P)=\mathrm{\infty }\}$. A subset H of $C_n(\mathrm{\Omega })$ is said to be a-minimal thin at $Q\in \partial C_n(\mathrm{\Omega })\cup \{\mathrm{\infty }\}$ on $C_n(\mathrm{\Omega })$, if there exists a point $P\in C_n(\mathrm{\Omega })$ such that

$${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,Q)}^H(P)\ne M_{\mathrm{\Omega }}^a(P,Q),$$

where ${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,Q)}^H$ is the regularized reduced function of $M_{\mathrm{\Omega }}^a(\cdot ,Q)$ relative to H (with respect to the Schrödinger operator $Sc{h}_a$).

Let H be a bounded subset of $C_n(\mathrm{\Omega })$. Then ${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^H(P)$ is bounded on $C_n(\mathrm{\Omega })$ and hence the greatest a-harmonic minorant of ${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^H$ is zero. When by $G_{\mathrm{\Omega }}^a\mu (P)$ we denote the Green a-potential with a positive measure μ on $C_n(\mathrm{\Omega })$, we see from the Riesz decomposition theorem that there exists a unique positive measure ${\lambda }_H^a$ on $C_n(\mathrm{\Omega })$ such that

$${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^H(P)=G_{\mathrm{\Omega }}^a{\lambda }_H^a(P)$$

for any $P\in C_n(\mathrm{\Omega })$ and ${\lambda }_H^a$ is concentrated on $I_H$, where

$$I_H=\{P\in C_n(\mathrm{\Omega });H\text{ is not a-thin at }P\}.$$

The Green a-energy ${\gamma }_{\mathrm{\Omega }}^a(H)$ (with respect to the Schrödinger operator $Sc{h}_a$) of ${\lambda }_H^a$ is defined by

$${\gamma }_{\mathrm{\Omega }}^a(H)={\int }_{C_n(\mathrm{\Omega })}{G}_{\mathrm{\Omega }}^a{\lambda }_H^{a}d{\lambda }_H^a.$$

Also, we can define a measure ${\sigma }_{\mathrm{\Omega }}^a$ on $C_n(\mathrm{\Omega })$

$${\sigma }_{\mathrm{\Omega }}^a(H)={\int }_H{\left(\frac{M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })}{\delta (P)}\right)}^{2}dP.$$

In [[8], Theorem 5.4.3], Long gave a criterion that characterizes a-minimally thin sets at infinity in a cone.

Theorem A A subset H of $C_n(\mathrm{\Omega })$ is a-minimally thin at infinity on $C_n(\mathrm{\Omega })$ if and only if

$$\sum _{j=0}^{\mathrm{\infty }}{\gamma }_{\mathrm{\Omega }}^a(H_j)W\left(2^j\right)V^{-1}\left(2^j\right)<\mathrm{\infty },$$

where $H_j=H\cap C_n(\mathrm{\Omega };[2^j,2^{j+1}))$ and $j=0,1,2,\dots$ .

In recent work, Zhao (see [[2], Theorems 1 and 2]) proved the following results. For similar results in the half space with respect to the Schrödinger operator, we refer the reader to the papers by Ren and Su (see [9, 10]).

Theorem B The following statements are equivalent.

• (I) A subset H of $C_n(\mathrm{\Omega })$ is a-minimally thin at infinity on $C_n(\mathrm{\Omega })$.

• (II) There exists a positive superfunction $v(P)$ on $C_n(\mathrm{\Omega })$ such that

  $$\underset{P\in C_n(\mathrm{\Omega })}{inf}\frac{v(P)}{M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })}=0$$

  (5)

  and

  $$H\subset \{P\in C_n(\mathrm{\Omega });v(P)\ge M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })\}.$$

• (III) There exists a positive superfunction $v(P)$ on $C_n(\mathrm{\Omega })$ such that even if $v(P)\ge c{M}_{\mathrm{\Omega }}^a(P,\mathrm{\infty })$ for any $P\in H$, there exists $P_0\in C_n(\mathrm{\Omega })$ satisfying $v(P_0)<c{M}_{\mathrm{\Omega }}^a(P_0,\mathrm{\infty })$.

Theorem C If a subset H of $C_n(\mathrm{\Omega })$ is a-minimally thin at infinity on $C_n(\mathrm{\Omega })$, then we have

$${\int }_H\frac{dP}{{(1+|P|)}^n}<\mathrm{\infty }.$$

(6)

Remark From equation (3), we immediately know that equation (6) is equivalent to

$${\int }_{H}V(1+|P|)W(1+|P|){(1+|P|)}^{-2}dP<\mathrm{\infty }.$$

(7)

This paper aims to show that the sharpness of the characterization of an a-minimally thin set in Theorem C. In order to do this, we introduce the Whitney cubes in a cone.

A cube is the form

$$[l_1{2}^{-j},(l_1+1)2^{-j}]\times \cdots \times [l_n{2}^{-j},(l_n+1)2^{-j}],$$

where j, $l_1,\dots ,l_n$ are integers. The Whitney cubes of $C_n(\mathrm{\Omega })$ are a family of cubes having the following properties:

• (I) ${\bigcup }_k{W}_k=C_n(\mathrm{\Omega })$.

• (II) $int{W}_j\cap int{W}_k=\mathrm{\varnothing }$ ($j\ne k$).

• (III) $diam{W}_k\le dist(W_k,{\mathbf{R}}^n\mathrm{\setminus }{C}_n(\mathrm{\Omega }))\le 4diam{W}_k$.

Theorem 1 If H is a union of cubes from the Whitney cubes of $C_n(\mathrm{\Omega })$, then equation (7) is also sufficient for H to be a-minimally thin at infinity with respect to $C_n(\mathrm{\Omega })$.

From the Remark and Theorem 1, we have the following.

Corollary 1 Let $v(P)$ be a positive superfunction on $C_n(\mathrm{\Omega })$ such that equation (5) holds. Then we have

$${\int }_{\{P\in C_n(\mathrm{\Omega });v(P)\ge M_{\mathrm{\Omega }}^a(P,\mathrm{\infty })\}}V(1+|P|)W(1+|P|){(1+|P|)}^{-2}dP<\mathrm{\infty }.$$

Corollary 2 Let H be a Borel measurable subset of $C_n(\mathrm{\Omega })$ satisfying

$${\int }_{H}V(1+|P|)W(1+|P|){(1+|P|)}^{-2}dP=+\mathrm{\infty }.$$

If $v(P)$ is a non-negative superfunction on $C_n(\mathrm{\Omega })$ and c is a positive number such that $v(P)\ge c{M}_{\mathrm{\Omega }}^a(P,\mathrm{\infty })$ for all $P\in H$, then $v(P)\ge c{M}_{\mathrm{\Omega }}^a(P,\mathrm{\infty })$ for all $P\in C_n(\mathrm{\Omega })$.

2 Lemmas

To prove our results, we need some lemmas.

Lemma 1 Let $W_k$ be a cube from the Whitney cubes of $C_n(\mathrm{\Omega })$. Then there exists a constant c independent of k such that

$${\gamma }_{\mathrm{\Omega }}^a(W_k)\le c{\sigma }_{\mathrm{\Omega }}^a(W_k).$$

Proof If we apply a result of Long (see [[8], Theorem 6.1.3]) for compact set ${\overline{W}}_k$, we obtain a measure μ on $C_n(\mathrm{\Omega })$, $supp\mu \subset {\overline{W}}_k$, $\mu ({\overline{W}}_k)=1$ such that

$$\{\begin{array}{ll}{\int }_{C_n(\mathrm{\Omega })}{|P-Q|}^{2-n}d\mu (Q)={\{Cap({\overline{W}}_k)\}}^{-1}& \text{if }n\ge 3,\\ {\int }_{C_2(\mathrm{\Omega })}log|P-Q|d\mu (Q)=logCap({\overline{W}}_k)& \text{if }n=2\end{array}$$

(8)

for any $P\in {\overline{W}}_k$. Also there exists a positive measure ${\lambda }_{{\overline{W}}_k}^a$ on $C_n(\mathrm{\Omega })$ such that

$${\hat{R}}_{M_{\mathrm{\Omega }}^a(\cdot ,\mathrm{\infty })}^{{\overline{W}}_k}(P)=G_{\mathrm{\Omega }}^a{\lambda }_{{\overline{W}}_k}^a(P)$$

(9)

for any $P\in C_n(\mathrm{\Omega })$.

Let $P_k=(r_k,{\mathrm{\Theta }}_k)$, ${\rho }_k$, $t_k$ be the center of $W_k$, the diameter of $W_j$, the distance between $W_k$ and $\partial C_n(\mathrm{\Omega })$, respectively. Then we have ${\rho }_k\le t_k\le 4{\rho }_k$ and ${\rho }_k\le r_k$. Then from equation (1) we have

$$r_k{M}_{\mathrm{\Omega }}^a(P,\mathrm{\infty })\approx V(r_k){\rho }_k$$

(10)

for any $P\in {\overline{W}}_k$. We can also prove that

$$G_{\mathrm{\Omega }}^a(P,Q)\gtrsim \{\begin{array}{ll}{|P-Q|}^{2-n}& \text{if }n\ge 3,\\ log\frac{{\rho }_k}{|P-Q|}& \text{if }n=2\end{array}$$

(11)

for any $P\in {\overline{W}}_k$ and any $Q\in {\overline{W}}_k$. Hence we obtain

$$r_k{\lambda }_{{\overline{W}}_k}^a(C_n(\mathrm{\Omega }))\lesssim \{\begin{array}{ll}V(r_k){\rho }_{k}Cap({\overline{W}}_k)& \text{if }n\ge 3,\\ V(r_k){\rho }_k{\{log\frac{{\rho }_k}{Cap({\overline{W}}_k)}\}}^{-1}& \text{if }n=2\end{array}$$

(12)

from equations (8), (9), (10), and (11). Since

$${\gamma }_{\mathrm{\Omega }}^a(W_k)=\int G_{\mathrm{\Omega }}^a{\lambda }_{{\overline{W}}_k}^{a}d{\lambda }_{{\overline{W}}_k}^a\le {\int }_{{\overline{W}}_k}{M}_{\mathrm{\Omega }}^a(P,\mathrm{\infty })d{\lambda }_{{\overline{W}}_k}^a(P)\lesssim r_k^{{\iota }_k^{+}-1}{\rho }_k{\lambda }_{{\overline{W}}_k}^a(C_n(\mathrm{\Omega }))$$

from equations (3), (9), and (10), we have from (12)

$${\gamma }_{\mathrm{\Omega }}^a(W_k)\lesssim \{\begin{array}{ll}{r}_k^{2{\iota }_k^{+}-2}{\rho }_k^{2}Cap({\overline{W}}_k)& \text{if }n\ge 3,\\ r_k^{2{\iota }_k^{+}-2}{\rho }_k^2{\{log\frac{{\rho }_k}{Cap({\overline{W}}_k)}\}}^{-1}& \text{if }n=2.\end{array}$$

(13)

Since

$$\{\begin{array}{ll}Cap({\overline{W}}_k)\approx {\rho }_k^{n-2}& \text{if }n\ge 3,\\ Cap({\overline{W}}_k)\approx {\rho }_k& \text{if }n=2,\end{array}$$

we obtain from equation (13)

$${\gamma }_{\mathrm{\Omega }}^a(W_k)\lesssim r_k^{2{\iota }_k^{+}-2}{\rho }_k^n.$$

(14)

On the other hand, we have from equation (1)

$${\sigma }_{\mathrm{\Omega }}^a(W_k)\approx r_k^{2{\iota }_k^{+}-2}{\rho }_k^n,$$

which, together with equation (14), gives the conclusion of Lemma 1. □

3 Proof of Theorem 1

Let $\{W_k\}$ be a family of cubes from the Whitney cubes of $C_n(\mathrm{\Omega })$ such that $H={\bigcup }_k{W}_k$. Let $\{W_{k,j}\}$ be a subfamily of $\{W_k\}$ such that $W_{k,j}\subset (H_{j-1}\cup H_j\cup H_{j+1})$, where $j=1,2,3,\dots$ .

Since ${\gamma }_{\mathrm{\Omega }}^a$ is a countably subadditive set function (see [[8], p.49]), we have

$${\gamma }_{\mathrm{\Omega }}^a(H_j)\lesssim \sum _k{\gamma }_{\mathrm{\Omega }}^a(W_{k,j})$$

(15)

for $j=1,2,\dots$ . Hence for $j=1,2,\dots$ we see from Lemma 1

$$\sum _k{\gamma }_{\mathrm{\Omega }}^a(W_{k,j})\lesssim \sum _k{\sigma }_{\mathrm{\Omega }}^a(W_{k,j}),$$

(16)

which, together with equation (1), gives

$$\begin{array}{rcl}\sum _k{\sigma }_{\mathrm{\Omega }}^a(W_{k,j})& \lesssim & ({\int }_{H_{j-1}}+{\int }_{H_j}+{\int }_{H_{j+1}})V^2(r)r^{-2}dP\\ \lesssim & ({\int }_{H_{j-1}}+{\int }_{H_j}+{\int }_{H_{j+1}})r^{2({\iota }_k^{+}-1)}dP\\ \lesssim & r^{2(j-1)({\iota }_k^{+}-1)}|H_{j-1}|+r^{2j({\iota }_k^{+}-1)}|H_j|+r^{2(j+1)({\iota }_k^{+}-1)}|H_{j+1}|\end{array}$$

(17)

for $j=1,2,\dots$ . Thus equations (15), (16), and (17) give

$${\gamma }_{\mathrm{\Omega }}^a(H_j)\lesssim r^{2(j-1)({\iota }_k^{+}-1)}|H_{j-1}|+r^{2j({\iota }_k^{+}-1)}|H_j|+r^{2(j+1)({\iota }_k^{+}-1)}|H_{j+1}|$$

for $j=1,2,\dots$ . Finally we obtain from equation (1)

$$\begin{array}{rcl}\sum _{j=0}^{\mathrm{\infty }}{\gamma }_{\mathrm{\Omega }}^a(H_j)W\left(2^j\right)V^{-1}\left(2^j\right)& \lesssim & {\gamma }_{\mathrm{\Omega }}^a(H_0)+\sum _{j=0}^{\mathrm{\infty }}{2}^{j(2{\iota }_k^{+}-2)}{2}^{-j({\iota }_k^{+}+{\iota }_k^{-})}|H_j|\\ \lesssim & {\gamma }_{\mathrm{\Omega }}^a(H_0)+\sum _{j=0}^{\mathrm{\infty }}{2}^{-2j}W\left(2^j\right)V^{-1}\left(2^j\right)|H_j|\\ \lesssim & {\gamma }_{\mathrm{\Omega }}^a(H_0)+{\int }_{H}V(1+|P|)W(1+|P|){(1+|P|)}^{-2}dP\\ <& \mathrm{\infty },\end{array}$$

which shows with Theorem A that H is a-minimally thin at infinity with respect to $C_n(\mathrm{\Omega })$.