For $\tau > 0$, let S($\tau$) = {E + i$\eta$ | $\tau^{-1} \leq E \leq \tau$, $\eta \geq N^{-1 + \tau}$}, and let $H$ be an $N\times N$ Wigner random matrix (i.i.d entries up to Hermitian condition). The resolvent is $G(z) = (H-z)^{-1}$ and what I am trying to clarify is that on $S(\tau)$, $G_{ij}(z)$ is $N^2$-Lipschitz, i.e. each entry of $G$ is $N^2$-Lipschitz as a function of $z$.
Here is what I have tried so far. Let $u_k$, $k = 1, ..., N$ be the normalized eigenvectors of $H$ with eigenvalues $\lambda_k$, i.e. $Hu_k = \lambda_k u_k$ and $||u_k||_2 = 1$. Then we can write $G(z) = \sum_{k=1}^N \frac{u_ku_k^*}{\lambda_k -z}$. So $G_{ij}(z) - G_{ij}(z') = \sum_{k=1}^N \frac{(u_ku_k^*)_{ij}}{(\lambda_k - z)(\lambda_k-z')}(z - z')$ Taking magnitude on both sides, I would like to have $\left|\sum_{k=1}^N \frac{(u_ku_k^*)_{ij}}{(\lambda_k - z)(\lambda_k-z')}\right| \leq N^2$. Since we are on the domain $S(\tau)$, $\left|\frac{1}{(\lambda_k - z)(\lambda_k - z')}\right| \leq \frac{1}{\eta^2} \leq N^2$ We also have $\sum_{k}(u_ku_k^*)_{ij} = \delta_{ij}$. So then I write $\sum_{k=1}^N \left|\frac{(u_ku_k^*)_{ij}}{(\lambda_k - z)(\lambda_k-z')}\right| \leq \sum_{k=1}^N \left|\frac{(u_ku_k^*)_{ij}}{\eta^2}\right|$ but I don't see how to proceed from here because the fact that $\sum_{k}(u_ku_k^*)_{ij} = \delta_{ij}$ depends on cancellation in the sum, and with absolute value it may be as large as $N$ which would be too large to make the bound I want. What can I do to get around this?