Maximal operators associated with some singular submanifolds
Authors: Heo Yaryong , Hong Sunggeum , Yang Chan 
Published: December 2016

Let

U

\mathrm {U}

be a bounded open subset of

R

d

\mathbb {R}^d

and let

Ω
\Omega

be a Lebesgue measurable subset of

U

\mathrm {U}

. Let

γ
=
(

γ
1

,
⋯
,

γ
n

)
:

U

∖
Ω
→

R

n

\gamma =(\gamma _1, \cdots , \gamma _n) : \mathrm {U}\setminus \Omega \rightarrow \mathbb {R}^n

be a Lebesgue measurable function, and let

μ
\mu

be a Borel measure on

R

d
+
n

\mathbb {R}^{d+n}

defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle mu comma f mathematical right-angle equals integral Underscript double-struck upper R Superscript d Baseline Endscripts f left-parenthesis y comma gamma left-parenthesis y right-parenthesis right-parenthesis psi left-parenthesis y right-parenthesis chi Subscript normal upper U minus normal upper Omega Baseline left-parenthesis y right-parenthesis d y comma">

⟨
μ
,
f
⟩
=

∫

R

d

f
(
y
,
γ
(
y
)
)
ψ
(
y
)

χ

U

∖
Ω

(
y
)

d
y
,

\begin{equation*} \langle \mu , f \rangle =\int _{\mathbb {R}^d} f(y, \gamma (y)) \psi (y)\,\chi _{\mathrm {U}\setminus \Omega }(y)\; dy, \end{equation*}

where

ψ
\psi

is a smooth function supported in

U

\mathrm {U}

. In this paper we give some conditions under which the Fourier decay estimates
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue ModifyingAbove mu With caret left-parenthesis xi right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C left-parenthesis 1 plus StartAbsoluteValue xi EndAbsoluteValue right-parenthesis Superscript negative epsilon">

|

μ
^

(
ξ
)

|

≤
C
(
1
+

|

ξ

|

)

−
ϵ

|\widehat {\mu }(\xi )| \le C (1+|\xi |)^{-\epsilon }

hold for some

ϵ
>
0

\epsilon >0

. As a corollary we obtain the

L
p

L^p

-boundedness properties of the maximal operators

M

S

\mathrm {M}_{S}

associated with a certain class of possibly non-smooth

n
n

-dimensional submanifolds of

R

d
+
n

\mathbb {R}^{d+n}

, i.e.,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M Subscript upper S Baseline f left-parenthesis x right-parenthesis equals sup Underscript r greater-than 0 Endscripts r Superscript negative d Baseline integral Underscript StartAbsoluteValue y EndAbsoluteValue greater-than r Endscripts StartAbsoluteValue f left-parenthesis x minus left-parenthesis y comma gamma left-parenthesis y right-parenthesis right-parenthesis right-parenthesis EndAbsoluteValue chi Subscript double-struck upper R Sub Superscript d Subscript minus normal upper Omega Sub Subscript sym Subscript Baseline d y comma">

M

S

f
(
x
)
=

sup

r
>
0

r

−
d

∫

|

y

|

>
r

|

f

(

x
−
(
y
,
γ
(
y
)
)

)

|

χ

R

d

∖

Ω

sym

d
y
,

\mathrm {M}_Sf(x)=\sup _{r>0}\, r^{-d}\int _{|y|>r} \big {|}f\big {(}x-(y,\gamma (y))\big {)}\big {|} \,\chi _{\mathbb {R}^d \setminus \Omega _{\text {sym}}} \,dy,

\]
where

Ω

sym

\Omega _{\text {sym}}

is a radially symmetric Lebesgue measurable subset of

R

d

\mathbb {R}^d

,

γ
(
y
)
=
(

γ
1

(
y
)
,
⋯
,

γ
n

(
y
)
)

\gamma (y)=(\gamma _1(y), \cdots , \gamma _n(y))

,

γ
i

(
t
y
)
=

t

a
i

γ
i

(
y
)

\gamma _i(t y)=t^{a_i} \gamma _i(y)

for each

t
>
0

t>0

where

a
i

∈

R

a_i \in \mathbb {R}

, and the function

γ
i

:

R

d

∖

Ω

sym

→

R

\gamma _i : \mathbb {R}^d \setminus \Omega _{\text {sym}} \rightarrow \mathbb {R}

satisfies some singularity conditions over a certain subset of

R

d

\mathbb {R}^d

. Also we investigate the endpoint

(
p
a
r
a
b
o
l
i
c

H
1

,

L

1
,
∞

)

(parabolic\; H^1, L^{1,\infty })

mapping properties of the maximal operators

M

H

\mathrm {M}_H

associated with a certain class of possibly non-smooth hypersurfaces, i.e.,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M Subscript upper H Baseline f left-parenthesis x right-parenthesis equals sup Underscript r greater-than 0 Endscripts StartAbsoluteValue integral Underscript double-struck upper R Superscript d Baseline Endscripts f left-parenthesis x minus left-parenthesis y comma gamma left-parenthesis y right-parenthesis right-parenthesis right-parenthesis r Superscript negative d Baseline psi left-parenthesis r Superscript negative 1 Baseline y right-parenthesis d y EndAbsoluteValue comma">

M

H

f
(
x
)
=

sup

r
>
0

|

∫

R

d

f

(

x
−
(
y
,
γ
(
y
)
)

)

r

−
d

ψ
(

r

−
1

y
)

d
y
|

,

\mathrm {M}_Hf(x)=\sup _{r>0}\left |\int _{\mathbb {R}^d} f\big {(}x-(y,\gamma (y))\big {)} r^{-d} \psi (r^{-1}y)\,dy \right |,

\]
where the function

γ
:

R

d

→

R

\gamma : \mathbb {R}^d \rightarrow \mathbb {R}

satisfies some singularity conditions over a certain subset of

R

d

\mathbb {R}^d

and

γ
(
t
y
)
=

t
m

γ
(
y
)

\gamma (t y)=t^m \gamma (y)

for each

t
>
0

t>0

where

m
>
0

m>0

.