Advances in Pure Mathematics, 2011, 1, 334-339
doi:10.4236/apm.2011.16060 Published Online November 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
A Class of Singular Integral Operators Associated to
Surfaces of Revolution*
Muliang Cao1, Huoxiong Wu2#
1Guangzhou Branch of Institute of Electronic Technology, PLA Information Engineering University,
Guangzhou, China
2School of Mathematical Sciences, Xiamen University, Fujian, China
E-mail: ml_cao@163.com, #huoxwu@xmu.edu.cn
Received July 19, 2011; revised August 10, 2011; accepted August 20, 2011
Abstract
In this paper, the authors establish the Lp-mapping properties of a class of singular integral operators along
surfaces of revolution with rough kernels. The size condition on the kernels is optimal and much weaker than
that for the classical Calderon-Zygmund singular integral operators.
Keywords: Singular Integrals, Surface of Revolution, Maximal Operator, Rough Kernel
1. Introduction and Main Results
Let , , be the n-dimensional Euclidean space
and be the unit sphere in equipped with the
normalized Lebesgue measure Let
n
R
n
S
2n
1n
R
dd


.


,; n
yy yR
 
n
be the surface of revolution
generated by a suitable function . For
nonzero points
:0, R
x
R, we denote '.
x
xx Let
be a homogeneous function of degree zero
on and satisfy
n
1
 1
LS
n
R
 
1d
n
Syy

0, (1.1)
Suppose that is a radial measurable function. De-
fine the singular integral operator
h
,h
T
in 1n
R
along
by





,1
1
,
.., d
||
n
hn
n
n
R
Tfxx
hy y
pvfxy xyy
y

(1.2)
for all (the Schwartz function class on
1n
f
1n
R
),
where

1
,n
1nn
x
xRRR
.
Operators of the type (1.2) have been studied quite
extensively (see [1-13] and therein numerous references).
We refer the readers to see Stein-Wainger’s report [14,
15] for more background information. In 1996, Kim,
Wainger, Wright and Ziesler proved the following result.
Theorem A [10]. Let be convex, in-
creasing and
20,C

00
. Suppose that
1n
CS


n
R
is
a homogeneous function of degree zero on and sat-
isfies (1.1). Then ,1
T
is bounded on
1pn
LR
for
1p
.
By a minor modification of the proof in Theorem A,
one can show that the conclusion of Theorem A remains
valid if the condition is replaced by the
condition
1n
CS


1qn
LS
 for some (see [16, pp.
372-373], as well as [6]). Subsequently, this result was
improved and extended by many authors (see [1,2,5,6,
11,12] et al.). In particular, in 2001, Lu, Pan and Yang
gave the general theorem as follows.
1
R
q
Theorem B [11]. Let be a continu-
ously differentiable on
:0,

0,
and satisfy
0tCt

 for some
and small , where
is a constant independent of . Suppose that
t
C t
11n
HS
 and
0,hL
. Then ,h
T
is
bounded on
1pn
LR
for 1, provided that the
maximal operator
p
defined by
 
1
2
2
1
p
2
j
j
j
,s
j
u


, d
f
rs
f r

tst t

is bounded on
2p
LR for
1.p
hL
Actually, the condition can be weak-
ened to the case:
0,

1
0
0
{;
s
hsups
hR sh

d}, 1tt
 
with
11 12,1p2min
 (see [9]), and the size
condition on
in Theorem B is the best one, so far,
*The project was supported by the NSF of China (G11071200) and the
N
SF of Fujian Province of China (No. 2010J01013).
M. L. CAO ET AL.335
even if (see [8]).

0t

t
On the other hand, for , it is known
in [17], if , that
21
,hLRrdr

,h
T0
is bounded on
1pn
LR

1
S
for , provided , which
is optimal and much weaker than that for the classical
Calderon-Zygmund singular integral operators. It should
be noted that the spaces and
do not include in each other.
1
1n
HS
p
1
21
,Rrdr

n
R

1/2
logLL


1/2
logLL


1/2 1
log n
LLS
n
Inspired by Al-Salman’s work [17], we shall establish
the following main result in this paper.
Theorem 1. Let be a suitable func-
tion, which ensures that the integral in (1.2) exists in
principle-value sense when, say, (the Schwartz
function class on ). Suppose that
, is a homogenous function of de-
gree zero on and satisfies (1.1). If
, then
:0, R

f
1n

1n
S

1n
R
hL

1)
 
21
21
,n
h
Tf
.
n
LR
LR f
C
2)

 
1
1
,,
pn
pn
hLR
LR
Tf Cf
for provided that the lower dimensional
maximal operator
1p,
M
defined by


1
/2
0
,sup, d
R
R
R
M
frsRfrtstt

(1.3)
is bounded on for 1
2p
LR
p
.
In Theorem 1, if

t
is continuously differentiable
on and satisfy
0,
 
0tCt

 for some
and small , where is a constant independent of ,
in particular, if , then the integral in
(1.2) exists in principle-value sense when, say,
t C

t
1n
10,1tC
f
(the Schwartz function class on ) (see [11]).
1n
R

2
Remark 1. The boundedness of
p
LR
M
is
known for many

t
’s. A few prominent examples are
as follows:
1) If

t
is a real-valued polynomial, then
M
is
bounded on for , see [16, p. 477] or [15].
2p
LR

tt
1p
2) If
with
0,1
, then
M
is
bounded on for , see [13].
2p
LR

1
tC
1p
3) If such that
0,1

00

0

0t
and is a convex increasing function for ,

t
M
is bounded on for , see [13], see [7,
Corollary 5.3].
2p
LR
1p
4) Let If

.btt tt



20,1tC
,

t
is convex on
with , and
there exists an
0,
0
 
000



so that for each 0,t

,bt
bt
t then
M
is bounded on
2p
LR for
, see [4, Theorem 1.5]. In particular, if
1p
t
is
ether even or odd and there exists a so that
for each
0C
0,t

2t
,Ct
then
M
is bounded
on for , see [3] or [4].

2p
LR 1p
In order to obtain Theorem 1, we let ,
S
be the op-
erator defined by




1
,1
1/ 2
2
1
0
,
d
:',d
n
n
n
S
Sfxx
r
yfxryxyy r







 .
Clearly, if
21
,hLRrdr

, then



21
,1 ,
,
,,
hn n
LRr dr
TfxxhS fxx



1
.
Therefore Theorem 1 can be deduced immediately
from the next theorem.
Theorem 2. Let
and be as in Theorem 1.
Then
1)
 
21
21
,.
n
nLR
LR
Sf Cf
2)
 
1
1
,,
pn
pn LR
LR
Sf Cf
2p
, provided that the maximal operator
M
in (1.3) is bounded on
2p
LR for 1. p
Remark 2. By the similar arguments as in [1], we re-
mark that the condition is op-
timal. Precisely, there exists an lies in
for all

1/2 1
log n
LLS


1
log n
LLS
 12
and satisfies (1.1)
such that ,
S
is unbounded on . And it is
worth pointing out that the size condition is much weaker
than that for the classical Calderón-Zygmund singular
integral operators.
1pn
LR
This paper is organize as follows. In Section 2 we will
give the proofs of our theorems. An extension of our
main results will be given in Section 3. We would like to
remark that the main ideas in the proofs of our results are
taken from [7,9,17].
Throughout this paper, we always use letter to
denote positive constants that may vary at each occur-
rence but are independent of the essential variables.
C
2. Proofs of Main Results
Let us begin with a lemma, which will play a key role in
the proofs of our main results.
Lemma 2.1. Let and satisfy (1.1). If
the maximal operator
11n
LS

M
in (1.3) is bounded on
2p
LR for, then the following maximal operator ,
M
defined by



,1
1
0/2 | |
,
:sup, d
n
n
n
RRyR
Mfxx
y
f
xyxyy
y



(2.1)
is bounded on
1pn
LR
with bound

11
,1 .Cp
n
LS

Copyright © 2011 SciRes. APM
M. L. CAO ET AL.
336
Proof. Since


 


 


 


1
1
1
,1
1
/2
0
1
/2
0
,' 1
,
d
sup, d
d
|sup ',d
,d
n
n
n
n
R
n
RS
R
R
n
SR
R
yn
S
Mfxx
r
yfxryx r y
r
r
yfxryxr
r
yMf xxy



 

y  




where




2
,' 11
0
1
,:sup, d
R
yn n
R
R
.
M
fxxfxryxrr
R


Thus by Minkowski's inequality, we have


 

1
11
,,'
d.
n
pn pn
y
S
LR LR
M
fyMf



y
It remains to show that

 
1
1
,pn
pn
yLR
LR
Mf Cf
with independent of .
C
1y
1.
n
Let For each fixed
1, 0,,0S1
'n
y
S
,
choose a rotation
such that y
1. Denote the in-
verse of
by 1
and define the function
f
by
. Then

11
,,
nn
xf xx

fx



1
11
,,
nn
f
xryxr fxrxr



1
.
This together with the
p
L-bondedness of
M
, and
change of variables, show that

 
1
1
,, 1,
pn
pn
yLR
LR
Mf Cfp

where is independent of. Lemma 2.1 is proved.
C
Next we introduce some notations. Assume that
and satisfies (1.1). For any

1/2 1
log n
LLS

l, let


11
: 22
nl l
l
EyS y


 Also,
we let Set


1
0:| |2
n
EySy

 


4
:l
l
Dl E
 2
and for 1,l


 
1
1
1
:
d,
l
nl
n
lE
E
S
yyyS
yyy



 

and
 
0:.
l
lD
y
y


y
0,
Then we have the
following:
 
1d0,
nl
Syyl


(2.2)
11
2:2,
l
lEl
,
A
lD
 (2.3)
00
12
,CC (2.4)
{0} ,
l
lD
y


y

(2.5)


1/ 21
1/2
{0} log
1,
n
l
lD LLS
lAC 
 
(2.6)
where 1
l
lE
A
 for lD
and
01.A
Now we give the proofs of our theorems as follows.
Proof of Theorem 2. For each
0lD , we let




1
,1
12
2
1
0
,:
d
,d
l
n
n
ln
S
Sfxx
r
yfx ryxyyr



 



 .
By (2.5) and Minkowski's inequality, we have

,1 ,
{0}
,,
l
nn
lD
SfxxS fxx

 

1
.
(2.7)
For any
0lD , let’s argue as in the proof of
Theorem 2.1 in [1], choose a collection of C
function
,lj
j
on
0,
with the following properties:
supp

11 11
,2,2
jl jl
lj
  
,
(2.8)
,
01
lj t
,
and (2.9)

2
,1,
lj
jt


,
d,
d
lj tC
tt
(2.10)
where
0,t,
and C
is a constant independ-
ent of l.
For each j
and
0lD , denote by
the multiplier
,lj
S




,1,
,,
ljn ljn
Sf f


1
,
and by
2
,
lj
S



2
,1,,
,,
ljnljljn
Sfxx SSfxx

1
.
Then by (2.9) and Minkowski’s inequality,



 






11
11
1
1
,1
2
2
12
2
2
,1
2
1
1/2
2
2
,1
,1
,
d
,d
d
(',)d
:,,
l
jl
jl n
l
n
n
l
jk
S
lj kn
l
kj S
lj kn
lk n
k
Sfxx
y
r
Sfxryxr y
r
y
r
Sfxryxr y
r
Ifxx



















where
Copyright © 2011 SciRes. APM
M. L. CAO ET AL.337



 


1
1
2
,11
1/2
2
2
,1
,
d
',d.
l
n
lk nl
jS
lj kn
Ifxxy
r
Sfxryxry
r





This together with (2.7) implies





1
1
,,
0.
pn
pn lk
lD kLR
LR
Sf If


(2.11)
Now we estimate

1
,pn
lk LR
If
in the following
cases:
Case 1. For , we claim that there exists 2p0
such that for ,
lD
0




21
21
1/2
||
,21 n
n
k
lkl LR
LR
IfC lAf
 , (2.12)
where C is independent of l and k.
Indeed, by Plancheral’s theorem and Fubini’s theorem,



,
2
2
,1,1
2,dd,
ljk
lknl jn
j
Iff J



(2.13)
where
 

R
111
,::22,
jklljkl
n
lj kR

  
 
 


11
1
2
22'
,1
d
ed
ljl
n
iry
lj l
S
r
Jy r



 y
In order to prove (2.12), we first estimate
,lj
J
.
Obviously, we have
 
22
,1
11
ljll .
J
lClA
  (2.14)
By (2.2), a straightforward computing shows that
 



2
211
,1
2
11
2
12
12,
jl
lj l
jl
l
Jl
Cl A


 

(2.15)
Using interpolation between (2.14) and (2.15), we get
 
 
2/ 1
11
2
,12
l
jl
ljl
JClA


 . (2.16)
On the other hand, we have
 


 
 


 
1
11
1
11
11
2
,1
2
22
1
d
edd
d
edd
l
nn
jl
nn
ljl
ljll
SS
iryu
ll
SS
iryu
Jyu
r
yu
r
yu
ryu
r














.
And by integration by parts,





11
22
1
1
1
d
e
12
ljl
iryu
jl
r
r
Cly u


 
,
with the easy fact


11
22
1
d
e1
ljl
iryurl
r


,
we obtain





11
22
1
1/4
1
d
e
12
ljl
iryu
jl
r
r
Cly u


 
.
Thus, by Holder’s inequality
 

  



11
1/4 2
1
,2
1/2
1/2
1/4 2
1
2
12
12 .
nn
jl
lj l
SS
jl
l
JCl
yudydu
Cl




 

 

Note that 00
2,CCA and for , lD
3
2
ll
ll
ACE C
2, we have


1/2 21
1
222
l
l
ll
CEC
.
l
A 
Consequently,
 

1/4
41 1
2
,12 2
ljl
ljl
JClA


 .
This together with (2.14) and an interpolation implies
 
 
14 1
1
2
,12
l
jl
ljl
JClA

 .
(2.17)
Then by the fact that ,,'lj kljk
  whenever
1,,1 ,jjjj
 (2.12) follows from (2.13), (2.16)
and (2.17).
Case 2. For we shall show that there exists 2,p
0
such that for ,

0lD




1
1
1/2
||
,21 pn
pn
k
lkl LR
LR
IfC lAf
 .
(2.18)
where C is independent of l and k.
Indeed, choose such that

/2 1
pn
gL R


/2 11
pn
LR
g
and




 




1
2l
111
2
,1
2
11
2
,1
11
2',2 d
d,dd.
pn nn
lk l
j
LR RS
jl jl
lj kn
nn
If y
Sfxryxr y
rgxx xx
r






 

By Holder’s inequality, we get
Copyright © 2011 SciRes. APM
M. L. CAO ET AL.
338
 
 





1
11
2
2
,1
11
1
2
2
,11
1
sup
d
2,2 d
,dd.
l
nn
lkj l
pRS
jl jl
n
lj knnl
j
If y
r
gxryxryr
Sfxx xx





 

Note that





 

1
1
2
1
1(1)
1
,1
sup
d
2,2 d
1,.
l
n
l
jl
S
jl jl
n
n
y
r
gxryxryr
lM gxx


 


Applying Holder’s inequality again, it follows from
Lemma 2.1 and the Littlewood-Paley theory (see [18,
Chapter 4]) that
 


 
 
22
,/2
1
1/2
2
2
,
1/2
2
22
,
1
22 2
2
1
1
1
11
lkl p
p
lj k
j
p
lljk
j
.
p
ll
p
p
If Clg
Sf
ClSf
ClfClA f



This together with (2.12) and an interpolation implies
(2.18).
Therefore, by (2.11), (2.12) and (2.6), we get








21
21
21
1/2 121
1/2
,{0}
||
1/ 2
0
log
1
2
1
.
n
n
n
nn
lDk
LR
k
lLR
l
lD LR
LLSLR
Sf Cl
Af
ClAf
Cf
 
 




This prove 1) of Theorem 2.
If
M
is bounded on
2p
LR for 1, then
by (2.11), (2.18) and (2.6), we obtain that for ,
p
2p

  
1/2 11
1
,log ,
npn
pn LLSLR
LR
Sf Cf
 

which completes the proof of Theorem 2.
Proof of Theorem 1. By the fact





21
,1 ,
,
,,
1hn n
LRr dr
TfxxhS fxx



,
and Theorem 2, it follows that ,h
T
is bounded on
for . On the other hand, by duality
we can obtain that ,h
T
1pn
LR
3. Further Results
In this section, we will extend the definition of T
,h to
higher dimensional cases. Let Φ(t)=(
1(t), ···,
m(t)) be a
curve on Rm, , where each
i(t) is a real-valued
continuous function. For x, yRn, and x*Rm, we define
2m





,,* ..
,* d.
n
n
hR
Tfxx pvhyyy
fx yxyy

 
(4.1)
Also, we define the operator S,Φ by




1
,0
1/2
2
,*
d
,* d,
n
S
Sfxx y
r
fx ryxryr



 

(4.2)
and the lower dimensional maximal function MΦ by


1*
1
01*
/2
,
sup,d .
r
rr
Mfxx
rfxtxt

t
(4.3)
In [10], Fan and Zheng extended the result of Theorem
B to the operator TΦ,h for hγ(R+) for γ > 1. Here, we
will obtain the following results.
Theorem 3. Suppose that hL2(R+, r
1dr) and 
L(log+L)1/2(Sn1). Then
1)
 
2
2
,.
nm
nm
hLR
LR
TfCf
2)
 
,
p
nm
pnm
hLR
LR
TfCf
, 1,p
provided that the maximal operator
M
defined in (4.3)
is bounded on
1pm
LR
for all
1.p
Theorem 4. Let Φ and be as in Theorem 3. Then
1)
 
2
2
,.
nm
nm LR
LR
SfCf

2)
 
,,
pnm
pnm LR
LR
SfCf
 , 2p
provided that the maximal operator
M
defined in (4.3)
is bounded on
1pm
LR
for all 1.p
Clearly, if m = 1 then Theorem 3 and 4 are reduced to
Theorem 1 and 2. For m 2, by the same arguments as in
the proof of Lemma 2.1, we can obtain the following
lemma.
Lemma 4.1. Let L1(Sn1) and satisfy (1.1). If the
maximal operator MΦ in (4.3) is bounded on Lp(Rm+1) for
p > 1, then the following maximal operator M,Φ defined
by



,*
0*
/2 | |
()(, )
:sup ,
rn
ryr
Mfxx
yfx yxydy
y



(4.4)
2p
is bounded on for
. Theorem 1 is proved.
1pn
LR
21p is bounded on
p
nm
LR
with bound

11n
LS
C
,
Copyright © 2011 SciRes. APM
M. L. CAO ET AL.
Copyright © 2011 SciRes. APM
339
[8] D. Fan and Y. Pan, “Singular Integral Operators with
Rough Kernels Supported by Subvarieties,” American
Journal of Mathematics, Vol. 119, No. 4, 1997, pp. 799-
839. doi:10.1353/ajm.1997.0024
1p
.
Then Theorem 3 and 4 follow from this lemma with
the arguments and the estimates similar to those in the
proofs of our theorems in Section 2. The details are
omitted. [9] D. Fan and Q. Zheng, “Maximal Singular Integral Op-
erators along Surfaces,” Journal of Mathematical Analy-
sis and Applications, Vol. 267, No. 2, 2002, pp. 746-759.
doi:10.1006/jmaa.2001.7812
4. Acknowledgements
[10] W. Kim, S. Wainger, J. Wright and S. Ziesler, “Singular
Integrals and Maximal Functions Associated to Surfaces
of Revolution,” Bulletin London Mathematical Society,
Vol. 28, No. 3, 1996, pp. 291-296.
doi:10.1112/blms/28.3.291
The authors are grateful to the referee for his/her helpful
comments and suggestions.
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