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

Let δ∈(0,1]$\delta \in (0,1]$ and T be a δ‐Calderón–Zygmund operator. When p∈(0,1]$p\in (0,1]$ and b∈BMO(Rn)$b\in {\rm BMO}(\mathbb {R}^n)$, 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]$[b, T]$ is not bounded from the Hardy space Hp(Rn)$H^p(\mathbb {R}^n)$ into the Lebesgue space Lp(Rn)$L^p(\mathbb {R}^n)$ if b is not a constant function. Let φ be a Musielak–Orlicz function satisfying that, for any (x,t)∈Rn×[0,∞)$(x,t)\in \mathbb {R}^n\times [0,\infty)$, φ(·,t)$\varphi (\cdot ,t)$ belongs to the Muckenhoupt weight class A∞(Rn)$A_\infty (\mathbb {R}^n)$ with the critical weight exponent q(φ)∈[1,∞)$q(\varphi)\in [1,\infty)$ and φ(x,·)$\varphi (x,\cdot)$ is an Orlicz function with the critical lower type i(φ)>q(φ)(1+δ/n)$i(\varphi)> q(\varphi)(1+\delta /n)$. In this paper, we find a proper subspace BMOφ(Rn)${\mathop \mathcal {BMO}_\varphi ({\mathbb {R}}^n)}$ of BMO(Rn)$\mathop \mathrm{BMO}(\mathbb {R}^n)$ such that, if b∈BMOφ(Rn),$b\in {\mathop \mathcal {BMO}_\varphi ({\mathbb {R}}^n),}$ then [b,T]$[b,T]$ is bounded from the Musielak–Orlicz Hardy space Hφ(Rn)$H^\varphi (\mathbb {R}^n)$ into the Musielak–Orlicz space Lφ(Rn)$L^\varphi (\mathbb {R}^n)$. Conversely, if b∈BMO(Rn)$b\in {\rm BMO}({\mathbb {R}}^n)$ and the commutators {[b,Rj]}j=1n$\lbrace [b,R_j]\rbrace _{j=1}^n$ of the classical Riesz transforms are bounded from Hφ(Rn)$H^\varphi ({\mathbb {R}}^n)$ into Lφ(Rn)$L^\varphi (\mathbb {R}^n)$, then b∈BMOφ(Rn)$b\in {\mathop \mathcal {BMO}_\varphi ({\mathbb {R}}^n)}$. Our results generalize some recent results by Huy and Ky [Vietnam J. Math. (2020). https://doi.org/10.1007/s10013‐020‐00406‐2] and Liang, Ky, and Yang [Proc. Amer. Math. Soc. 144 (2016), no. 12, 5171–5181]. [ABSTRACT FROM AUTHOR]