Vector valued q-variation for ergodic averages and analytic semigroups

In this paper, we establish UMD lattice-valued variational inequalities for ergodic averages of contractively regular operators and analytic semigroups. These results generalize, on the one hand some scalar-valued variational inequalities in ergodic theory, on the other hand Xu's very recent result on UMD lattice-valued maximal inequality. As an application, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence for semigroups acting on UMD lattice-valued functions.