Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
Authors: Hagelstein Paul , Luque Teresa , Parissis Ioannis 
Published: April 2015

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper M Subscript German upper B comma mu Baseline f left-parenthesis x right-parenthesis colon equals sup Underscript x element-of upper B element-of German upper B Endscripts StartFraction 1 Over mu left-parenthesis upper B right-parenthesis EndFraction integral Underscript upper B Endscripts StartAbsoluteValue f EndAbsoluteValue d mu period EndLayout">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row nu left-parenthesis StartSet x element-of double-struck upper R Superscript n Baseline colon upper M Subscript German upper B comma mu Baseline left-parenthesis bold 1 Subscript upper E Baseline right-parenthesis left-parenthesis x right-parenthesis greater-than one half EndSet right-parenthesis less-than-or-equal-to c Subscript mu comma nu Baseline nu left-parenthesis upper E right-parenthesis period EndLayout">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row nu left-parenthesis StartSet x element-of double-struck upper R Superscript n Baseline colon upper M Subscript German upper R comma mu Baseline left-parenthesis bold 1 Subscript upper E Baseline right-parenthesis left-parenthesis x right-parenthesis greater-than one half EndSet right-parenthesis less-than-or-equal-to c Subscript mu comma nu Baseline nu left-parenthesis upper E right-parenthesis EndLayout">

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 )

.