Vector valued $q$-variation for differential operators and semigroups II

TL;DR

This paper extends variational inequalities to vector-valued settings for differential operators and semigroups, providing new bounds and convergence rates in ergodic theory using UMD lattice properties.

Contribution

It introduces UMD lattice-valued variational inequalities for differential operators and semigroups, generalizing scalar inequalities and recent maximal inequality results.

Findings

Established UMD lattice-valued variational inequalities

Derived jump estimates and convergence rates

Generalized scalar-valued inequalities to vector-valued contexts

Abstract

In this paper, we establish UMD lattice-valued variational inequalities for differential operators, ergodic averages 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 a consequence, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence.