A weighted maximal inequality for differentially subordinate martingales

The paper contains the proof of a weighted Fefferman-Stein inequality in a probabilistic setting. Suppose that $f=(f_n)_{n\geq 0}$, $g=(g_n)_{n\geq 0}$ are martingales such that $g$ is differentially subordinate to $f$, and let $w=(w_n)_{n\geq 0}$ be a weight, i.e., a nonnegative, uniformly integrable martingale. Denoting by $Mf=\sup _{n\geq 0}\vert f_n\vert$, $Mw=\sup _{n\geq 0}w_n$ the maximal functions of $f$ and $w$, we prove the weighted inequality

$\vert\vert g\vert\vert _{L^1(w)}\leq C\vert\vert Mf\vert\vert _{L^1(Mw)},$

where $C=3+\sqrt {2}+4\ln 2=7.186802\ldots$. The proof rests on the existence of a special function enjoying appropriate majorization and concavity.