We establish the boundedness for commutators of parameterized Littlewood-Paley operators and area integrals on weighted Lebesgue spaces L p (ω) when 1 < p < ∞, where the kernel satisfies certain logarithmic type Lipschitz condition. Moreover, the weighted endpoint estimates when p = 1 are also obtained., respectively, where ρ > 0, λ > 1 and Γ(x) = {(t, y) ∈ R n+1M is the Hardy-Littlewood maximal operator and M ♯ is the Fefferman-Stein sharp function. The corresponding dyadic maximal operators are denoted by M ∆ δ and M ♯,∆ δ , respectively. A function A : [0, ∞) → [0, ∞) is said to be a Young function if it is continuous, convex and increasing satisfying A(0) = 0 and A(t) → ∞ as t → ∞. The complementary Young functionĀ(t) of the Young function A(t) is defined bȳ A(s) = sup 0 t<∞ [st − A(t)], 0 s < ∞.As an example, Φ m (t) = t(1 + log + t) m , 1 m < ∞, is a Young function with its complementaryΦ m (t) ≈ e t 1/m . If A is a Young function, then the Luxembury norm of f on a cube Q ⊂ R n is defined by
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