Weighted oscillation and variation inequalities for singular integrals and commutators satisfying Hörmander type conditions

Abstract

This paper is devoted to investigating the weighted L p -mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish the weighted type (p, p) estimates for 1 < p < ∞ and the weighted weak type (1, 1) estimate for the oscillation and variation operators of singular integrals with kernels satisfying certain Hormander type conditions, which contain the Riesz transforms, singular integrals with more general homogeneous kernels satisfying the Lipschitz conditions and the classical Dini’s conditions as model examples. Meanwhile, we also obtain the weighted L p -boundeness for such operators associated to the family of commutators generated by the singular integrals above with BMO(Rd)-functions.