Recently, I am reading D. Popovici's article $L^2$ extension for jets of holomorphic sections of a Hermitian line bundle, https://arxiv.org/pdf/math/0409170.pdf where some parts possibly confuse me.
I have two questions:
First, I wonder how to get the jet extension over the whole of $X$ in Main Theorem 0.1.4? In page 19, para -1, we have got the extension and the $L^2$ estimate of it on $X_c$. It seems that the classical method of passing to the limit with $c\rightarrow\infty$ doesn't work due to the dependence of the estimate on $X_c$.
Second, in Theorem 0.1.5, why can we get the holomorphic function (jet extension) $F_k$ and the estimate on $\Omega$? In my opinion, although now $\Omega\subset\mathbb C^n$ is a bounded pseudoconvex open set, we still need to work on a relatively compact set $\Omega_c$. As I have mentioned above, I don't think the process of $c\rightarrow\infty$ will work.
Any help would be appreciated. Thanks a lot!
