Abstract: The authors introduce weak Herz spaces onnon-homogeneous spaces and establish some weak type estimates for a classof sublinear operators in these spaces. As an application, theauthors prove that the commutators generated by Calder\'on-Zygmundoperators with $\os$ functions for $r\ge1$ satisfy the same weak estimates, where $\os\subset\rb$ if $r>1$ and $\os=\rb$ if $r=1$.

Key words: Weak Herz space, Sublinear operator, Calder\''on-Zygmund operator, Multilinear commutator, $\os$ function, Non-homogeneous space