Weighted variation inequalities for differential operators and singular integrals

We prove weighted strong q-variation inequalities with 2<q<∞ for differential and singular integral operators. For the first family of operators the weights used can be either Sawyer's one-sided A_p⁺ weights or Muckenhoupt's A_p weights according to whether the differential operators in consideration are one-sided or symmetric. We use only Muckenhoupt's A_p weights for the second family. All these inequalities hold equally in the vector-valued case, that is, for functions with values in ℓ^p for 1<p<∞. As application, we show variation inequalities for mean bounded positive invertible operators on L^p with positive inverses.