the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates here. So, we've got
\begin{align} ||(S-\lambda)^{-1}\begin{pmatrix}p\\q\end{pmatrix}||_{(C^0)^2}\leq \frac{K}{|\lambda|}||\begin{pmatrix}p\\q\end{pmatrix}||\ \ \ \ \ \ \ (A)\\ ||(S-\lambda)^{-1}\begin{pmatrix}p\\q\end{pmatrix}||_{(C^2)^2}\leq K ||\begin{pmatrix}p\\q\end{pmatrix}|| \ \ \ \ \ \ \ (B) \end{align}
Now the statement is: "Interpolating between estimate $(A)$ and $(B)$, we obtain \begin{align} ||(S-\lambda)^{-1}\begin{pmatrix}p\\q\end{pmatrix}||_{(C^1)^2}\leq \frac{K}{\sqrt{\lambda}}||\begin{pmatrix}p\\q\end{pmatrix}|| \ \ \ \ (C) \end{align}
Sure, I've got some interpretation of what is meant by that. But nevertheless it reads like "If we look at (A) and (B)...there is probably some way, that (C) is right too.". I think that way is not very mathematical. But maybe there is a theorem, that guarantees me to do that or I've only got a totally wrong point of view to that problem. Maybe anyone have an idea how to understand it in a better way.