For any
1
≤
p
>
∞
1\leq p>\infty
, we determine the optimal constant
C
p
C_p
such that the following holds. If
(
h
k
)
k
≥
0
(h_k)_{k\geq 0}
is the Haar system, then for any vectors
a
k
a_k
from a separable Hilbert space
H
\mathcal {H}
and
θ
k
∈
{
0
,
1
}
\theta _k\in \{0,1\}
,
k
=
0
,
1
,
2
,
…
k=0,\,1,\,2,\,\ldots
, we have
\[
|
|
∑
k
=
0
n
θ
k
a
k
h
k
|
|
p
,
∞
≤
C
p
|
|
∑
k
=
0
n
a
k
h
k
|
|
p
.
\left |\left |\sum _{k=0}^n \theta _ka_kh_k\right |\right |_{p,\infty }\leq C_p\left |\left |\sum _{k=0}^n a_kh_k\right |\right |_p.
\]
This is generalized to the weak type inequality
\[
|
|
g
|
|
p
,
∞
≤
C
p
|
|
f
|
|
p
,
||g||_{p,\infty }\leq C_p||f||_p,
\]
where
f
f
is an
H
\mathcal {H}
-valued martingale and
g
g
is its transform by a predictable sequence taking values in
[
0
,
1
]
[0,1]
. We extend this further to the estimate
\[
|
|
Y
|
|
p
,
∞
≤
C
p
|
|
X
|
|
p
,
||Y||_{p,\infty }\leq C_p||X||_p,
\]
valid for any two
H
\mathcal {H}
-valued continuous-time martingales
X
X
,
Y
Y
, such that
(
[
Y
,
X
−
Y
]
t
)
([Y,X-Y]_t)
is nondecreasing and nonnegative as a function of
t
t
.