Sharp weak type estimates for a family of Zygmund bases
Authors: Hagelstein Paul , Stokolos Alex 
Published: September 2021

Let

B

\mathcal {B}

be the collection of rectangular parallelepipeds in

R

3

\mathbb {R}^3

whose sides are parallel to the coordinate axes and such that

B

\mathcal {B}

consists of parallelepipeds with side lengths of the form

s
,

2
j

s
,
t

s, 2^j s, t

, where

s
,
t
>
0

s, t > 0

and

j
j

lies in a nonempty subset

S
S

of the integers. In this paper, we prove the following:

If

S
S

is a finite set, then the associated geometric maximal operator

M

B

M_\mathcal {B}

satisfies the weak type estimate

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue StartSet x element-of double-struck upper R cubed colon upper M Subscript script upper B Baseline f left-parenthesis x right-parenthesis greater-than alpha EndSet EndAbsoluteValue less-than-or-equal-to upper C integral Underscript double-struck upper R cubed Endscripts StartFraction StartAbsoluteValue f EndAbsoluteValue Over alpha EndFraction left-parenthesis 1 plus log Superscript plus Baseline StartFraction StartAbsoluteValue f EndAbsoluteValue Over alpha EndFraction right-parenthesis">

|

{
x
∈

R

3

:

M

B

f
(
x
)
>
α
}

|

≤
C

∫

R

3

|

f

|

α

(
1
+

log
+

⁡

|

f

|

α

)

\begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha }\left (1 + \log ^+ \frac {|f|}{\alpha }\right )\; \end{equation*}

but does not satisfy an estimate of the form

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue StartSet x element-of double-struck upper R cubed colon upper M Subscript script upper B Baseline f left-parenthesis x right-parenthesis greater-than alpha EndSet EndAbsoluteValue less-than-or-equal-to upper C integral Underscript double-struck upper R cubed Endscripts phi left-parenthesis StartFraction StartAbsoluteValue f EndAbsoluteValue Over alpha EndFraction right-parenthesis">

|

{
x
∈

R

3

:

M

B

f
(
x
)
>
α
}

|

≤
C

∫

R

3

ϕ

(

|

f

|

α

)

\begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right ) \end{equation*}

for any convex increasing function

ϕ
:

[

0
,
∞
)
→
[
0
,
∞
)

\phi : \mathbb [0, \infty ) \rightarrow [0, \infty )

satisfying the condition

lim

x
→
∞

ϕ
(
x
)

x
(
log
⁡
(
1
+
x
)
)

=
0.

\begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))} = 0. \end{equation*}

On the other hand, if

S
S

is an infinite set, then the associated geometric maximal operator

M

B

M_\mathcal {B}

satisfies the weak type estimate

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue StartSet x element-of double-struck upper R cubed colon upper M Subscript script upper B Baseline f left-parenthesis x right-parenthesis greater-than alpha EndSet EndAbsoluteValue less-than-or-equal-to upper C integral Underscript double-struck upper R cubed Endscripts StartFraction StartAbsoluteValue f EndAbsoluteValue Over alpha EndFraction left-parenthesis 1 plus log Superscript plus Baseline StartFraction StartAbsoluteValue f EndAbsoluteValue Over alpha EndFraction right-parenthesis squared">

|

{
x
∈

R

3

:

M

B

f
(
x
)
>
α
}

|

≤
C

∫

R

3

|

f

|

α

(
1
+

log
+

⁡

|

f

|

α

)

2

\begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \frac {|f|}{\alpha } \left (1 + \log ^+ \frac {|f|}{\alpha }\right )^{2} \end{equation*}

but does not satisfy an estimate of the form

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue StartSet x element-of double-struck upper R cubed colon upper M Subscript script upper B Baseline f left-parenthesis x right-parenthesis greater-than alpha EndSet EndAbsoluteValue less-than-or-equal-to upper C integral Underscript double-struck upper R cubed Endscripts phi left-parenthesis StartFraction StartAbsoluteValue f EndAbsoluteValue Over alpha EndFraction right-parenthesis">

|

{
x
∈

R

3

:

M

B

f
(
x
)
>
α
}

|

≤
C

∫

R

3

ϕ

(

|

f

|

α

)

\begin{equation*} \left |\left \{x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right \}\right | \leq C \int _{\mathbb {R}^3} \phi \left (\frac {|f|}{\alpha }\right ) \end{equation*}

for any convex increasing function

ϕ
:

[

0
,
∞
)
→
[
0
,
∞
)

\phi : \mathbb [0, \infty ) \rightarrow [0, \infty )

satisfying the condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow normal infinity Endscripts StartFraction phi left-parenthesis x right-parenthesis Over x left-parenthesis log left-parenthesis 1 plus x right-parenthesis right-parenthesis squared EndFraction equals 0 period">

lim

x
→
∞

ϕ
(
x
)

x
(
log
⁡
(
1
+
x
)

)
2

=
0.

\begin{equation*} \lim _{x \rightarrow \infty }\frac {\phi (x)}{x (\log (1 + x))^2} = 0. \end{equation*}