Convolution singular integrals on Lipschitz surfaces
Authors: Li Chun , McIntosh Alan , Semmes Stephen 
Published: June 2010

We prove the

L
p

{L_p}

-boundedness of convolution singular integral operators on a Lipschitz surface
\[

Σ
=
{
g
(

x

)

e
0

+

x

∈

R

n
+
1

:

x

∈

R

n

}

\Sigma = \{ g({\mathbf {x}}){e_0} + {\mathbf {x}} \in {\mathbb {R}^{n + 1}}:{\mathbf {x}} \in {\mathbb {R}^n}\}

\]
where

g
g

is a Lipschitz function which satisfies

‖

∇
g

‖

∞

≤

tan

ω
>
∞

{\left \| {\nabla g} \right \|_\infty } \leq {\text {tan}}\omega > \infty

. Here we have embedded

R

n
+
1

{\mathbb {R}^{n + 1}}

in the Clifford algebra

R

(
n
)

{\mathbb {R}_{(n)}}

with identity

e
0

{e_0}

, and are considering convolution with right-monogenic functions

ϕ
\phi

which satisfy

|

ϕ
(
x
)

|

≤
C

|
x
|

−
n

\left | {\phi (x)} \right | \leq C{\left | x \right |^{ - n}}

on a sector
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript mu Superscript o Baseline equals StartSet x equals x 0 plus bold x element-of double-struck upper R Superscript n plus 1 Baseline colon StartAbsoluteValue x 0 EndAbsoluteValue greater-than StartAbsoluteValue bold x EndAbsoluteValue tangent mu EndSet">

S
μ
o

=
{
x
=

x
0

+

x

∈

R

n
+
1

:

|

x
0

|

>

|

x

|

tan

μ
}

S_\mu ^o = \{ x = {x_0} + {\mathbf {x}} \in {\mathbb {R}^{n + 1}}:\left | {{x_0}} \right | > \left | {\mathbf {x}} \right |{\text {tan}} \mu \}

\]
where

μ
>
ω

\mu > \omega

. Provided there exists an

L
∞

{L_\infty }

function

ϕ
_

\underline \phi

satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingBelow phi With bar left-parenthesis upper R right-parenthesis minus ModifyingBelow phi With bar left-parenthesis r right-parenthesis equals integral Underscript StartLayout 1st Row r greater-than StartAbsoluteValue x EndAbsoluteValue greater-than upper R 2nd Row x element-of double-struck upper R Superscript n Baseline EndLayout Endscripts phi left-parenthesis x right-parenthesis d x">

ϕ
_

(
R
)
−

ϕ
_

(
r
)
=

∫

r
>

|

x

|

>
R

x
∈

R

n

ϕ
(
x
)

d
x

\underline \phi (R) - \underline \phi (r) = \int _{\substack {r > |x| > R\\x \in {\mathbb {R}^{n}}}} \phi (x)\;dx

\]
, then the related convolution singular integral operator
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T Subscript left-parenthesis phi comma ModifyingBelow phi right-parenthesis With bar Baseline u right-parenthesis left-parenthesis x right-parenthesis equals limit Underscript epsilon right-arrow 0 plus Endscripts left-brace integral Underscript StartLayout 1st Row y element-of normal upper Sigma 2nd Row StartAbsoluteValue x minus y EndAbsoluteValue greater-than-or-equal-to epsilon EndLayout Endscripts phi left-parenthesis x minus y right-parenthesis n left-parenthesis y right-parenthesis u left-parenthesis y right-parenthesis d upper S Subscript y Baseline plus ModifyingBelow phi With bar left-parenthesis epsilon n left-parenthesis x right-parenthesis right-parenthesis u left-parenthesis x right-parenthesis right-brace">

(

T

(
ϕ
,

ϕ
)

_

u
)
(
x
)
=

lim

ε
→
0
+

{

∫

y
∈
Σ

|

x
−
y

|

≥
ε

ϕ
(
x
−
y
)
n
(
y
)
u
(
y
)

d

S
y

+

ϕ
_

(
ε
n
(
x
)
)
u
(
x
)
}

({T_{(\phi ,\underline {\phi )} }}u)(x) = \lim _{\varepsilon \to 0+}\left \{\int _{\substack {y \in \Sigma \\|x - y| \geq \varepsilon }} \phi (x - y)n(y)u(y)\;d{S_y} + \underline \phi (\varepsilon n(x))u(x) \right \}

\]
is bounded on

L
p

(
Σ
)

{L_p}(\Sigma )

for

1
>
p
>
∞

1 > p > \infty

.