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

Abstract

AbstractLet and T be a δ‐Calderón–Zygmund operator. When and , it is well‐known (see the work by Harboure, Segovia, and Torrea [Illinois J. Math. 41 (1997), no. 4, 676–700]) that the commutator is not bounded from the Hardy space into the Lebesgue space if b is not a constant function. Let φ be a Musielak–Orlicz function satisfying that, for any , belongs to the Muckenhoupt weight class with the critical weight exponent and is an Orlicz function with the critical lower type . In this paper, we find a proper subspace of such that, if then is bounded from the Musielak–Orlicz Hardy space into the Musielak–Orlicz space . Conversely, if and the commutators of the classical Riesz transforms are bounded from into , then . 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].