Weighted Variation Inequalities for Singular Integrals and Commutators in Rearrangement Invariant Banach and Quasi-Banach Spaces

Abstract

Let Ω be a homogeneous function of degree zero and enjoy the vanishing condition on the unit sphere S n−1 (n ≥ 2). Let TΩ be the convolution singular integral operator with kernel Ω(x)|x| −n . In this paper, when Ω ∈ L ∞ (S n−1 ), we consider the quantitative weighted bounds of the composite operators of TΩ on rearrangement invariant Banach function spaces. These spaces contain the classical Lorentz spaces and Orlicz spaces as special examples. Weighted boundedness of the composite operators on rearrangement invariant quasi-Banach spaces were also given.