$A_1$ Fefferman-Stein inequality for maximal functions of martingales in uniformly smooth spaces

TL;DR

This paper extends the Fefferman-Stein inequality to maximal functions of martingales in uniformly smooth Banach spaces, providing a new weighted inequality that links maximal functions and square functions.

Contribution

It establishes a Fefferman-Stein type inequality for martingales in uniformly smooth Banach spaces with weights, a novel generalization in this setting.

Findings

Proves a weighted inequality for martingale maximal functions in uniformly smooth spaces.

Shows the inequality relates the maximal function to the square function with weights.

Extends classical harmonic analysis results to a martingale and Banach space context.

Abstract

Let $f$ be a martingale with values in a uniformly $p$-smooth Banach space and $w$ any positive weight. We show that $\mathbb{E} (f^* \cdot w) \lesssim \mathbb{E}(S_p f \cdot w^*)$, where $\cdot^*$ is the martingale maximal operator and $S_p$ is the $\ell^p$ sum of martingale increments.