With a monotone family
F
=
{
S
α
}
,
S
α
⊂
R
n
F\, = \,\{ {S_\alpha }\} ,\,{S_\alpha }\, \subset \,{{\textbf {R}}^n}
, we associate the Hardy-Littlewood maximal function
M
F
f
(
x
)
=
sup
α
(
1
/
|
S
α
|
)
∫
S
α
+
x
|
f
|
{M_F}f(x)\, = \,{\sup _\alpha }(1/\left | {{S_\alpha }} \right |)\int _{{S_\alpha }\, + \,x} {\left | f \right |}
. In general,
M
F
{M_F}
is not weak type (1.1). However, if we replace in the denominator
S
α
{S_\alpha }
by
S
F
∗
=
{
x
−
y
:
x
,
y
∈
S
α
}
S_F^ {\ast } \, = \,\{ x\, - \,y:\,x,\,y\, \in \,{S_\alpha }\}
, and denote the resulting maximal function by
M
F
∗
M_F^ {\ast }
, then
M
F
∗
M_F^ {\ast }
is weak type (1, 1) with weak type constant 1.