CALDERÓN-ZYGMUND COMMUTATORS ON GENERALIZED WEIGHTED LORENTZ SPACES

Tóm tắt

In this paper, we consider commutators [b, T] of Calderón-Zygmund operators of type θ (see Definition 1.2 and 1.3 in Section 1) on generalized weighted Lorentz spaces ()puwΛ, where u is a function that belongs to the class pA of Muckenhoupt weights on n and w is a function that belongs to the class ()pBu of Ariño-Muckenhoupt weights on ()0,∞ (see Section 1). In this setting, we first establish the pointwise estimate for the sharp maximal operator acting on Calderón-Zygmund commutators of type θ (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 θ are bounded on generalized weighted Lorentz spaces ()puwΛ (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).