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Thank you. I tried to do something in this spirit. If $k_n(t) = P_n k(t) P_n$, I believe that $I - \hat{k}_n(z)$ is indeed invertible for all $\Re(z) \geq 0$, provided n is large enough. So by the standard Paley-Wiener's theorem, there exists $r_n \in L^1(\mathbb{R}_+)$ such that $r_n(t)= k_n(t) + \int_0^t{k_n(t-s) r_n(s) ds}$. Now my main problem is how to check that $sup_{n \in \mathbb{N}} \int_0^\infty{ \lVert r_n(t) \rVert dt} < \infty$ ? Did I miss something? I don't see how to mimic the proof of Newman here.