Musielak–Orlicz Hardy space estimates for commutators of Calderón–Zygmund operators

Abstract

Let 𝛿 ∈ (0, 1] and 𝑇 be a 𝛿-Calderón-Zygmund operator. When 𝑝 ∈ (0, 1] and 𝑏 ∈ BMO(ℝ 𝑛 ), it is well-known (see the work by Harboure, Segovia, and Torrea [Illinois J. Math. 41 (1997), no. 4, 676-700]) that the commutator [𝑏, 𝑇] is not bounded from the Hardy space 𝐻 𝑝 (ℝ 𝑛 ) into the Lebesgue space 𝐿 𝑝 (ℝ 𝑛 ) if 𝑏 is not a constant function. Let 𝜑 be a Musielak-Orlicz function satisfying that, for any (𝑥, 𝑡) ∈ ℝ 𝑛 × [0, ∞), 𝜑(⋅, 𝑡) belongs to the Muckenhoupt weight class 𝐴 ∞ (ℝ 𝑛 ) with the critical weight exponent 𝑞(𝜑) ∈ [1, ∞) and 𝜑(𝑥, ⋅) is an Orlicz function with the critical lower type 𝑖(𝜑) > 𝑞(𝜑)(1 + 𝛿∕𝑛). In this paper, we find a proper subspace  𝜑 (ℝ 𝑛 ) of BMO(ℝ 𝑛 ) such that, if 𝑏 ∈  𝜑 (ℝ 𝑛 ), then [𝑏, 𝑇] is bounded from the Musielak-Orlicz Hardy space 𝐻 𝜑 (ℝ 𝑛 ) into the Musielak-Orlicz space 𝐿 𝜑 (ℝ 𝑛 ). Conversely, if 𝑏 ∈ BMO(ℝ 𝑛 ) and the commutators {[𝑏, 𝑅 𝑗 ]} 𝑛 𝑗=1 of the classical Riesz transforms are bounded from 𝐻 𝜑 (ℝ 𝑛 ) into 𝐿 𝜑 (ℝ 𝑛 ), then 𝑏 ∈  𝜑 (ℝ 𝑛 ). Our results generalize some recent results by Huy