CALDERÓN-ZYGMUND COMMUTATORS OF TYPE THETA ON GENERALIZED MORREY-LORENTZ SPACE

Abstract

In this paper, we consider Calderón-Zygmund commutators  of type  (see Definitions 1.3, 1.4, and 1.5 in Section 1) on generalized Morrey-Lorentz spaces  (see Definition 1.1). In this setting, we first establish the pointwise estimates for the Hardy-Littlewood maximal operator and the sharp maximal operator acting on Calderón-Zygmund operators of the type  and Calderón-Zygmund commutators of type  (see Lemma 2.4, Lemma 2.5 in Section 2) by using Kolmogorov’s inequality, Holder’s inequality, the conditions of standard kernels in the definition of Calderón-Zygmund operators of type , and the well-known consequence of John-Nirenberg inequality. Thanks to these significant pointwise estimates, we then prove that Calderón-Zygmund operators of type  are bounded on generalized Morrey-Lorentz spaces  (see Theorem 2.1) by modifying the ideas and techniques related to maximal operators proposed by Thai et al. (2022a, 2022b), Carro et al. (2021), and Liu et al. (2002). Futhermore, we deduce that commutators  of type  are also bounded on these spaces due to the pointwise estimate for the sharp maximal operator acting on commutators  of type  and the boundedness of  in .