Abstract
This paper is devoted to investigating the boundedness of the oscillation and variation operators for the commutators generated by Calderon-Zygmund singular integrals with Lipschitz functions in the weighted Lebesgue spaces and the endpoint spaces in dimension 1. Certain criterions of boundedness are given. As applications, the weighted $(L^{p}, L^{q})$ -estimates for the oscillation and variation operators on the iterated commutators of Hilbert transform and Hermitian Riesz transform, the $(L^{p}, \dot{\wedge}_{(\beta -1/p)})$ -bounds as well as the endpoint estimates for the oscillation and variation operators of the corresponding first order commutators are established.
Highlights
Let T = {Tε}ε be a family of operators such that the limit limε→ Tεf (x) = Tf (x) exists in some sense
We will study the behaviors of oscillation and variation operators for the families of commutators defined by ( . ) and ( . ) in Lebesgue spaces
For /β ≤ p ≤ ∞, we can establish the following un-weighted results only for the oscillation and variation operators related to the first order commutator
Summary
We will establish a criterion on the weighted (Lp, Lq)-type estimates of the oscillation and ρ-variation operators for the iterated commutators of Calderón-Zygmund singular integrals with Lipschitz functions for < β < and < p < /β with /q = /p – β. The corresponding boundedness of the oscillation and variation operators for the commutators of Hilbert transform and the Hermitian Riesz transforms will be given.
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