The paper contains the proof of a weighted Fefferman-Stein inequality in a probabilistic setting. Suppose that
f
=
(
f
n
)
n
≥
0
f=(f_n)_{n\geq 0}
,
g
=
(
g
n
)
n
≥
0
g=(g_n)_{n\geq 0}
are martingales such that
g
g
is differentially subordinate to
f
f
, and let
w
=
(
w
n
)
n
≥
0
w=(w_n)_{n\geq 0}
be a weight, i.e., a nonnegative, uniformly integrable martingale. Denoting by
M
f
=
sup
n
≥
0
|
f
n
|
Mf=\sup _{n\geq 0}|f_n|
,
M
w
=
sup
n
≥
0
w
n
Mw=\sup _{n\geq 0}w_n
the maximal functions of
f
f
and
w
w
, we prove the weighted inequality
|
|
g
|
|
L
1
(
w
)
≤
C
|
|
M
f
|
|
L
1
(
M
w
)
,
\begin{equation*} ||g||_{L^1(w)}\leq C||Mf||_{L^1(Mw)}, \end{equation*}
where
C
=
3
+
2
+
4
ln
2
=
7.186802
…
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.