Let G be a homogeneous group. In this paper, the authors establish several general theorems for the boundedness of sublinear
operators and commutators generated by linear operators and BMO(G) functions on the weighted Lebesgue space on G. The conditions of these theorems are satisfied by many important operators in analysis and these operators satisfy only
some weak conditions on the size of operators and are known to be bounded in the unweighted case.
Some of these theorems are best possible even when G is the Euclidean space. The authors also give some applications of their theorems to the boundedness on weighted spaces of
rough singular integrals, oscillatory integrals, parabolic singular integrals, their commutators and the maximal operators
associated with them.