CALDERÓN-ZYGMUND COMMUTATORS ON GENERALIZED WEIGHTED LORENTZ SPACES

Abstract

In this paper, we consider commutators [b, T] of Calderón-Zygmund operators of type  (see Definition 1.2 and 1.3 in Section 1) on generalized weighted Lorentz spaces , where  is a function that belongs to the class  of Muckenhoupt weights on  and  is a function that belongs to the class  of Ariño-Muckenhoupt weights on  (see Section 1). In this setting, we first establish the pointwise estimate for the sharp maximal operator acting on Calderón-Zygmund commutators of type  (see Lemma 2.2 in Section 2) by using Kolmogorov’s inequality, generalized Holder’s inequality in the sense of  Luxemburg norm (see Definition 2.1) and Young function (see Lemma 2.1), and the well-known John-Nirenberg inequality. In light of this significant estimate, we then indicate that Calderón-Zygmund commutators of type  are bounded on generalized weighted Lorentz spaces  (see Theorem 2.1) by exploiting the ideas and techniques concerning maximal operators from the study by Carro et al. (2021). Our aforementioned main results extend the ones of Carro et al. (2021).