Weighted variation inequalities for differential operators and singular integrals

We prove weighted strong $q$-variation inequalities with $2<q<\infty$ 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 that 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 $\el^\rho$ for $1<\rho<\infty$. As application, we show variation inequalities for mean bounded positive invertible operators on $L^p$ with positive inverses.