Generalized commutators for Marcinkiewicz type integrals with variable kernels

Let A be a function with derivatives of order m and D ^γ A ∈ (L)\dot] _b\dot \Lambda _\beta (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L ^∞(? ⁿ ) × L ^s (S ^n?1) (s ≥ n/(n ? β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ _Ω ^A and its variation (m)\tilde] _W ^A\tilde \mu _\Omega ^A are bounded from L ^p (? ⁿ ) to L ^q (? ⁿ ), where 1 < p < n/β and 1/q = 1/p ? β/n. The authors also consider the boundedness of μ_Ω ^A and its variation (m)\tilde] _W ^A\tilde \mu _\Omega ^A on Hardy spaces.