Extreme cases of limit operator theory on metric spaces

The theory of limit operators was developed by Rabinovich, Roch and Silbermann to study the Fredholmness of band-dominated operators on ${\mathrm{\ell }}^p({\mathbb{Z}}^N)$ for $p\in \{0\}\cup [1,\mathrm{\infty }]$, and recently generalised to discrete metric spaces with property A by \v{S}pakula and Willett for $p\in (1,\mathrm{\infty })$. In this paper, we study the remained extreme cases of $p\in \{0,1,\mathrm{\infty }\}$ (in the metric setting) to fill the gaps.