A Singular Integral Operator And Its Exchange Weighted Estimates

On a space of homogeneous type, suppose that A is a Young function and M#D is a new sharp maximal function. First, we introduce LA-Hormander condition. For the operator T satisfying LA-Hormander condition, we obtain that the Lp(ω) norm of the operator can be dominated by the Lp(ω) norm of a maximal function associated to the complementary function of A and a maximal function, for any weight ω in A∞and0<p<∞. Second, for the above operator T we introduce LA,K-Hormander condition, which kernel satisfies LA,k-Hormander condition, we obtain that the LP(ω)) norm of k order commutators of singular integral operators T with non-smooth kernels with BMO function can be dominated by the Lp(ω) norm of a maximal function associated to A, for any weight ω in A∞, and0<p<∞.