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It is easy to see that when ρ = 1, and μρ(h) = μ1(h), then (2) is the classical Marcinkiewicz integral μ(h) introduced by Stein in [1]. It has been proved in [1] that if Ω ∈ Lipγ(Sn−1) (0 < γ ≤ 1) and Ω is continuous, then the operator μ(h) is of the type (q, q) for 1 < q ≤ 2 and of the weak type (1, 1). Benedek et al., [2] proved that if Ω ∈ C1(Sn−1), then μ(h) it is of type (q, q) for any 1 < q ≤ ∞. The Lp boundedness of the μ(h) has been studied in [1,3–5]. In 1960, Hörmander [4] introduced the parametrized Marcinkiewicz integral operators proved that if Ω ∈ Lipγ(Sn−1), 0 < γ ≤ 1, then it is of strong type (q, q) for 1 < q ≤ 2. Sakamoto and Yabuta [6] proved the boundedness of the operator μρ(h) on Lq(Rn). Shi and Jiang [7] considered the weighted Lq−boundedness of parametrized Marcinkiewicz integral operator and its higher order commutator. Note that the Littlewood-paley g-function played very important roles in harmonic analysis and the parameterized Marcinkiewick integral is a special case of the Littlewood-paley g-function. Many authors studied properties of μρ(h) on different function spaces, for examples [8–14].

Higer-order commutators of parametrized Marcinkewicz integrals on Herz spaces with variable exponent