\Vert_4 \Vert KWK \Vert_2 \Vert K \Vert_4 \label{eq:u1bds}$$ because $K \in {\mathfrak{S}_{4}}$ \[or any ${\mathfrak{S}_{p}}$, $p > 2$, as a matter of fact\]: just notice that by , $$\begin{aligned}
\Vert K \Vert_4^4& = \sum_{j = 0}^{\infty} \frac{1}{\vert z - z_j \vert^2} \leq \\
&\leq  3/4 + 2 \sum_{k = 2}^{\infty} \left( \frac{2}{3} \right)^2 \cdot \frac{1}{k^2} < 20 < \left( \frac{11}{5} \right)^4,
\end{aligned} \label{eq:k4boundset}$$ so $$\Vert K \Vert_4 \leq \frac{11}{5}; \quad \Vert K \Vert_4^2 \leq 5. \label{eq:k4bound}$$ Therefore we can claim the following.

\[prop:normbds\] Under the assumptions , , $0 < \alpha < \frac{1}{2}$, suppose that $N_{*} = N_*(\alpha)$ is chosen in such a way that $$M(\alpha) \frac{\log en}{n^{2\alpha}} \leq \frac{1}{2} \quad \text{ for all } \quad n \geq N_*. \label{eq:nstardef}$$ Then for $n > N_*(\alpha)$ if $z \in {\partial (\mathcal{D}_n )}$ all the operators $U_j \in $ are of the trace class, their norms satisfy inequalities $$\begin{aligned}
\Vert U_j \Vert_1 &\leq 2 \left[ M(\alpha) \frac{\log en}{n^{2\alpha}}\right]^j, \quad j \geq 2, \label{eq:ujnorm}\\
\Vert U_1 \Vert_1 & = \Vert R^0 W R^0 \Vert_1 \leq \frac{5 M(\alpha) \log en}{n^{2\alpha}} \label{eq:u1norm}\end{aligned}$$ and the Neumann - Riesz series for the difference of two resolvents $$\begin{aligned}
R - R^0 &= \sum_{j = 1}^{\infty} U_j \label{eq:rdiff}\\
\intertext{converges 