Weighted boundedness of commutators of the Marcinkiewicz integrals

Denote by μΩ the n-dimensional Marcinkiewicz integral and by μbΩ the commutator generalized by μΩ and a Lipschitz function b. The boundedness of μbΩ on the weighted Lebesgue spaces and the weighted Hardy spaces are studied when the kernel Ω satisfies the Lipschitz condition. It is shown that μbΩ is bounded from Lpωp into Lqωq when ω∈A(p,q), 1pn/β and 1/p-1/q=β/n, and from Hpωp into Lqωq when ωn(n-β)∈A1, n/(n+β)p≤1 and 1/p-1/q=β/n.