Characterizations of the UMD property via tail estimates for tangent processes

TL;DR

This paper characterizes the UMD property of Banach spaces using tail estimates for maximal functions of tangent processes, establishing equivalences with Lorentz norm inequalities across various settings.

Contribution

It provides a new characterization of the UMD property via tail inequalities and maximal function estimates for tangent processes, extending to multiple process settings.

Findings

UMD property characterized by tail inequalities for tangent processes.

Equivalence between tail estimates and Lorentz norm inequalities.

Applicable in discrete-time, continuous-time, and discontinuous process settings.

Abstract

We characterize the UMD property of a Banach space by tail inequalities for maximal functions of tangent conditionally symmetric processes. More precisely, we prove that a Banach space $V$ is UMD if and only if for some (equivalently, for all) $p\in(0,\infty)$ one has that \[ \mathbb P(\sup_{r\geq 0} \| N_r\|>t)\lesssim_{p,V}\Bigl(\frac{s^p}{t^p}+\mathbb P(\sup_{r\geq 0} \| M_r\|>s)\Bigr), \qquad s,t>0, \] for all tangent conditionally symmetric $V$-valued processes $M$ and $N$. We further show that this estimate is equivalent to suitable Lorentz norm inequalities for the associated maximal functions, and obtain analogous characterizations in the discrete-time, continuous-time, and purely discontinuous settings.