Given a block diagonal (sparse) symmetric matrix $S \in \mathbb{R}^{n\times n}$ and a low-rank symmetric matrix $L$, construct $R=S-L$. I believe that $R$ is, in general, weak block diagonally dominant, as $R_{ii}=S_{ii}-L_{ii}$ for diagonal blocks and $R_{ij} = -L_{ij}$ for $i\neq j$.
For the 2-block case, this would look like:
$R=\left[\begin{array}{cc} S_{11}-L_{11} & -L_{12}\\ -L_{21} & S_{22}-L_{22} \end{array}\right]$
I found a helpful source that if $R$ is strict block diagonally dominant, Ahlberg-Nilson-Varah bounds the max norm:
$\|R^{-1}\|_\infty\leq \frac{1}{\min_i \|R_{ii}^{-1}\|_\infty^{-1}-\sum_{j\neq i}\|L_{ij}\|_\infty}$.
What I'd like to know is if there is a similar bound for the off-diagonal blocks. That is for
$\|(R^{-1})_{ij}\|_\infty\forall i\neq j$
either in the weak or strict cases of block diagonally dominance.