Variation Inequalities for One-Sided Singular Integrals and Related Commutators

Abstract

We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ > 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones.