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

Abstract

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

v2: 16 pages, corrected following referee report