Let
B
\mathfrak {B}
be a homothecy invariant collection of convex sets in
R
n
\mathbb {R}^{n}
. Given a measure
μ
\mu
, the associated weighted geometric maximal operator
M
B
,
μ
M_{\mathfrak {B}, \mu }
is defined by
M
B
,
μ
f
(
x
)
:=
sup
x
∈
B
∈
B
1
μ
(
B
)
∫
B
|
f
|
d
μ
.
\begin{align*} M_{\mathfrak {B}, \mu }f(x) := \sup _{x \in B \in \mathfrak {B}}\frac {1}{\mu (B)}\int _{B}|f|d\mu . \end{align*}
It is shown that, provided
μ
\mu
satisfies an appropriate doubling condition with respect to
B
\mathfrak {B}
and
ν
\nu
is an arbitrary locally finite measure, the maximal operator
M
B
,
μ
M_{\mathfrak {B}, \mu }
is bounded on
L
p
(
ν
)
L^{p}(\nu )
for sufficiently large
p
p
if and only if it satisfies a Tauberian condition of the form
ν
(
{
x
∈
R
n
:
M
B
,
μ
(
1
E
)
(
x
)
>
1
2
}
)
≤
c
μ
,
ν
ν
(
E
)
.
\begin{align*} \nu \big (\big \{x \in \mathbb {R}^{n} : M_{\mathfrak {B}, \mu }(\textbf {1}_E)(x) > \frac {1}{2} \big \}\big ) \leq c_{\mu , \nu }\nu (E). \end{align*}
As a consequence of this result we provide an alternative characterization of the class of Muckenhoupt weights
A
∞
,
B
A_{\infty , \mathfrak {B}}
for homothecy invariant Muckenhoupt bases
B
\mathfrak {B}
consisting of convex sets. Moreover, it is immediately seen that the strong maximal function
M
R
,
μ
M_{\mathfrak {R}, \mu }
, defined with respect to a product-doubling measure
μ
\mu
, is bounded on
L
p
(
ν
)
L^{p}(\nu )
for some
p
>
1
p > 1
if and only if
ν
(
{
x
∈
R
n
:
M
R
,
μ
(
1
E
)
(
x
)
>
1
2
}
)
≤
c
μ
,
ν
ν
(
E
)
\begin{align*} \nu \big (\big \{x \in \mathbb {R}^{n} : M_{\mathfrak {R}, \mu }(\textbf {1}_E)(x) > \frac {1}{2}\big \}\big ) \leq c_{\mu , \nu }\nu (E)\; \end{align*}
holds for all
ν
\nu
-measurable sets
E
E
in
R
n
\mathbb {R}^{n}
. In addition, we discuss applications in differentiation theory, in particular proving that a
μ
\mu
-weighted homothecy invariant basis of convex sets satisfying appropriate doubling and Tauberian conditions must differentiate
L
∞
(
ν
)
L^{\infty }(\nu )
.