THE BOUNDEDNESS OF CALDERÓN-ZYGMUND OPERATORS OF TYPE THETA ON GENERALIZED WEIGHTED LORENTZ SPACES

Tóm tắt

In this paper, we consider Calderón-Zygmund operators of type θ(see Definition 1.3 and Definition 1.4 in Section 1) on generalized weighted Lorentz spaces ()puwΛ, where uis 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 Hardy-Littlewood maximal operator and the sharp maximal operator (see Lemma 2.3 in Section 2) by using Kolmogorov’s inequality, Holder’s inequality, and the conditions of standard kernels in the definition of Calderón-Zygmund operators of the type θ. Thanks to this significant pointwise estimate, we then prove that Calderón-Zygmund operators of type θ are bounded on the generalized weighted Lorentz spaces ()puwΛ (see Theorem 2.4) by employing the ideas and techniques related to maximal operators from the work of Carro et al., (2021). Our main results extend the ones of Carro et al., (2021).