I am reading this paper:A SIMPLIFIED PROOF OF OPTIMAL L2-EXTENSION THEOREM AND EXTENSIONS FROM NON-REDUCED SUBVARIETIES by Hosono. https://arxiv.org/pdf/1910.05782.pdf
In page 9,he wrote this inequality:
$$\lim_{t\to -\infty}\int_{\frac{t-\Diamond}{ca_k}<\log|\omega_k|^2< \frac{t-\Diamond+1}{ca_k}} |\omega_k|^{-2}\, d\lambda(\omega_k)$$
$$\leq C(e^{-(m_p-m_{p-1})s})\int_{\log|\omega_k|^2< \frac{s-\Diamond}{ca_k}} |\omega_k|^{2((m_p-m_{p-1})ca_k-1)}\, d\lambda(\omega_k).$$
Here,$\omega_k$ is a complex variable function,$c,a_k,\Diamond,m_p,m_{p-1}$ are all real numbers independent of $\omega_k$,we also know that $m_p>m_{p-1}>0$.
Here,the $C\in\mathbb R$ in RHS is also independent of $\omega_k$.
Here is my question:
1.According to this post: Laurent series, integral over the annulus, radii, I think I can calculate LHS:$\pi/ca_k$ ,which is independent of the limit.Am I correct?
2.How can I prove this ineuality involving the change of the integral domains?
Any advices will be appreciated!Thanks a lot!