The Boundedness for Commutators of Anisotropic Calderón-Zygmund Operators

Let T be an anisotropic Calderón-Zygmund operator and φ : ℝn × [0, ∞) → [0, ∞) be an anisotropic Musielak-Orlicz function with φ(x, ·) being an Orlicz function and φ(·,t) being a Muckenhoupt A∞ (A) weight. In this paper, our goal is to study two boundedness theorems for commutators of anisotropic Calderón-Zygmund operators. Precisely, when b ∈ BMOw(ℝn, A) (a proper subspace of anisotropic bounded mean oscillation space BMO(ℝn, A)), the commutator [b, T] is bounded from anisotropic weighted Hardy space Hl(ℝn, A) to weighted Lebesgue space Lw1 (ℝn) and when b ∈ BMO(ℝn) (bounded mean oscillation space), the commutator [b, T] is bounded on Musielak-Orlicz space Lφ(ℝn), which are extensions of the isotropic setting.