Jump inequalities via real interpolation

TL;DR

This paper explores jump inequalities at the endpoint case using real interpolation spaces, providing new estimates for vector-valued martingales, operators, and Fourier multipliers, extending previous work in harmonic analysis.

Contribution

It introduces a novel interpretation of jump inequalities via real interpolation spaces, enabling endpoint estimates for various vector-valued operators and martingales.

Findings

Established endpoint jump estimates for vector-valued martingales.

Extended jump inequalities to doubly stochastic operators.

Applied sampling techniques to transfer results from continuous to discrete settings.

Abstract

Jump inequalities are the $r=2$ endpoint of L\'epingle's inequality for $r$-variation of martingales. Extending earlier work by Pisier and Xu we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vector-valued martingales and doubly stochastic operators as well as to pass via sampling from $\mathbb{R}^{d}$ to $\mathbb{Z}^{d}$ for jump estimates for Fourier multipliers.