_{Q \in
{\mathcal Q}_1} V(\delta_H(Q^*)) \, \mu(Q^*)\,,
\end{eqnarray*}
where the last line follows from \eqref{e:asymp-V-I} and
\eqref{e:asymp-V-II}. Note that
 $Q^*\subset B(0,4)$ for every $Q\in {\mathcal Q}_1$ and clearly
$\delta_H(Q^*) \le \delta_H(y)$ for all $y\in Q^*$. Thus by first
using these observations and the fact that each $Q^*\in {\mathcal
Q}^*$ intersects at most $K$ other cubes from ${\mathcal Q}^*$, and
then using Theorem \ref{t:bhp}, we conclude that the last sum is
dominated by
\begin{eqnarray*}
(K+1) \int_{H\cap B(0, 4)} V(\delta_H(y))\, \mu(dy)&\le
&(K+1)\frac{C_3(4)
V(2)}{G^H(w_1,y_1)} \int_{H\cap B(0,4)} G^H(w_1,y)\, \mu(dy)\\
&\le &(K+1) \frac{C_3(4)V(2)}{G^H(w_1,y_1)}G^H\mu(w_1)<\infty\,,
\end{eqnarray*}
where $w_1=(\widetilde{0},10)$ and $y_1=(\widetilde{0},2)$.
 Therefore,
\begin{equation}\label{e6}
\int_{\{s_1>M^H (\cdot,0)/8\}\cap B(0, 1)}|w|^{-d}\, dw < \infty\,.
\end{equation}

Consider now $x \in \{s_2>M^H(\cdot, 0)/4\}$. For
$y\in H\setminus Q^*(x)$ it holds that
$|x-y|\ge 4^{-1} \delta_H(x)$,
and hence
$$
s_2(x)\le \int_{\{\delta_H(x)\le 4|x-y|\}
 \cap B(0,\kappa)} G^H(x,y)\, \mu(dy)\,.
$$
Let $\psi:H\to \partial H\setminus\{0\}$ be
a Borel function such that
$|y-\psi(y)|\le 2\delta_H(y)$ and define a measure $\mu'$ on
$\partial H\setminus\{0\}$ by
$$
\int_{\partial H\setminus\{0\}} f \, d\mu' =c_1
\int_{H\cap B(0,\kappa)} (f\circ \psi )(y)V(\delta_H(y))\, \mu(dy)\,
,
$$
where $c_1:=C_4(1+\kappa) C_1(1+\kappa)(11)^dC_5(11(1+\kappa))$.
Then by Theorem \ref{t:bhp},
$$
\mu'(\partial H\setminus\{0\})=c_1\int_{H\cap B(0,\kappa)}
V(\delta_H(y))\, \mu(dy)\le
c_1\frac{C_3(\kappa)
V(1)}{G^H(v_1, v_2)}G^H\mu(v_1)<\infty
$$
where $v_1=(\widetilde{0}, 3\kappa)$ and $v_2=(\widetilde{0},1)$. Hence, $\mu'$ is
positive and bounded.

We claim that
\begin{equation}\label{e:new}
s_2(x) \le M^H \mu'(x)  \quad \text{if } x \in \{s_2>M^H(\cdot,
0)/8\}\cap B(0, 1).
\end{equation}

Suppose that $y \in B(0, \kappa)\cap H$ and $x\in
\{s_2>M^H(\cdot,0)/8\}\cap B(0, 1)$ with
$\delta_H(x)\le 4|x-y|$.
Then
\begin{eqnarray*}
&&|x-\psi(y)|\le |x-y|+|y-\psi(y)|\le
|x-y|+2\delta_H(y)\\
&&\le |x-y|+2\delta_H(x)+2 |x-y|\le 11 |x-y| \le 11 (1+ \kappa).
\end{eqnarray*}
Thus by Theorem \ref{t:martin-estimate},
\begin{eqnarray*}
M^H(x,\psi(y))&\ge& C_5(
11(1+ \kappa))^{-1}
V(\delta_H(x))(1+|\psi(y)|^2)^{d/2}|x-\psi(y)|^{-d}\\
&\ge& \frac{1}{(11)^d C_5(11 (1+ \kappa))} V(\delta_H(x))|x-y|^{-d}\,.
\end{eqnarray*}
Note that by Theorem \ref{t:green} and Proposition \ref{p:GoverV}(i),
$$
G^H(x,y) \, \le \,  C_4(1+\kappa) \frac{V(\delta_H(x))
V(\delta_H(y))}{V(|x-y|)^2}G(x,y) \, \le \, C_4(1+\kappa)
C_1(1+\kappa) V(\delta_H(x)) V(\delta_H(y))|x-y|^{-d}\,.
$$
Hence we get
$$
G^H(x,y) \, \le \,  c_{1} V(\delta_H(y)) M^H(x,\psi(y)).
$$
Therefore, for $x \in \{s_2>M^H(\cdot,
0)/8\}\cap B(0, 1),$
\begin{eqnarray*}
s_2(x)&\le & \int_{\{\delta_H(x)\le 2|x-y|\}
\cap B(0,\kappa)} G^H(x,y)\, \mu(dy)\\
&\le& c_{1} \int_{\{\delta_H(x)\le 2|x-y|\} \cap B(0,\kappa)}
V(\delta_H(y)) M^H(x,\psi(y))\, \mu(dy)\\
&\le &c_{1} \int_{H\cap B(0,\kappa)} V(\delta_H(y))
M^H(x,\psi(y))\, \mu(dy)\\
& =&\int M^H(x,z)\, \mu'(dz) \, = \, M^H\mu'(x)\,.
\end{eqnarray*}
Hence, we have proved \eqref {e:new}, and so, by Lemma \ref{l:sjogren},
\begin{equation}\label{e7}
\int_{\{s_2>M^H (\cdot,0)/8\}\cap B(0,1)}|x|^{-d}\, dx \, \le \,
\int_{\{M^H \mu' > M^H (\cdot,0)/8\}\cap B(0,1)} |x|^{-d}\, dx
\, <\, \infty.
\end{equation}


Together with \eqref{s3}, (\ref{e6}) and (\ref{e7}) we obtain that
$$
\int_{A_1}|x|^{-d}\, dx \, \le \,  \int_{\{s_1(x)+ s_2(x)>M^H(\cdot,0)/4\}
\cap B(0,1)}|x|^{-d} dx \, \le \, \sum_{i=1}^2\int_{\{s_i>M^H
(\cdot,0)/8\}\cap B(0,1)} |x|^{-d} dx \, < \, \infty.
$$
{\hfill $\Box$ \bigskip}


\begin{thm}\label{t:main1}
Suppose that $X$ is a subordinate Brownian motion satisfying {\bf (H)}.
Let $A$ be a Borel subset of $H$ and assume that
\eqref{c:criterion1} holds. Then $A$ is not minimally thin in $H$
with respect to
$X$ at $z=0$.
\end{thm}


\noindent{\bf Proof:}  Assume that $A$ is minimally thin in $H$ with respect to
$X$
at 0. Then, using Proposition \ref{p:new} and multiplying $s$ by a
constant if necessary, there exists an excessive function
$s=G^H\mu+M^H \nu$ with