Endpoint boundedness of commutators on spaces of homogeneous type

Let (chi, d, mu) be a space of homogeneous type in the sense of Coifman and Weiss. Let K be a family of sublinear operators that include the well-known Calderon-Zygmund operator. The authors prove that the commutator [b, T] is bounded from a Hardy-type subspace H-b(1) b(chi) of H-at(1) (chi) to L-1(chi), where L-1(chi) denotes the Lebesgue space of all integrable functions, H1 at(chi) the atomic Hardy space of Coifman and Weiss with the dual space BMO(chi), b is a non-constant BMO(chi)-function and T is an element of kappa. Indeed, the space H-b(1)(chi) is the largest subspace in H-1 at(chi) that possesses this property. The approach taken in this article adopts the bilinear decomposition theory for the product of functions in H-at(1) (chi) and BMO(chi).