A Class of Singular Integral Operators Associated to Surfaces of Revolution

doi:10.4236/apm.2011.16060 Published Online November 2011 (http://www.SciRP.org/journal/apm)

A Class of Singular Integral Operators Associated to

Surfaces of Revolution

*

Muliang Cao1, Huoxiong Wu2#

1_{Guangzhou Branch of Institute of Electronic Technology, PLA Information Engineering University,}

Guangzhou, China

2_{School of Mathematical Sciences, Xiamen University, Fujian, China}

E-mail: ml_cao@163.com,#_{[email protected]}

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 

2

n

1 _Rn

d d

 

 .

 







, ; n

y y y R



  

n

 be the surface of revolution generated by a suitable function . For

nonzero points





: 0, R x R , we denote x'x x. Let be a homogeneous function of degree zero on



and satisfy

n1



 _{L S}1

n

R

   

1 d

n

S  y  y



0, (1.1) Suppose that is a radial measurable function. De-fine the singular integral operator

h

,h

T_ in _Rn1 _along 

 by

 





 

 

_

_{ }

, 1

1

,

. . , d

| | n

h n

n n

R

T f x x

h y y

p v f x y x y y

y









 



_

  



(1.2)

for all _fn1 (the Schwartz function class on _Rn1_),

where



, n1



1

n n

x x R  R R .

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







2 _0,

C

  



 

0 0

  . Suppose that _{ C}

 

_Sn1 n

R

is a homogeneous function of degree zero on and sat-isfies (1.1). Then T,1 is bounded on L Rp

 

n1 for

1  p .

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



1

n C S 

 



 

1

q n L S 

  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

   

t 0 Ct

   for some  and small , where is a constant independent of . Suppose that

t

C t

 

1

1 n

H S 

  and h L_ 





0,_





_{. Then}

,h T_ is bounded on _{L R}p

 

n1 _{for 1 , provided that the}

maximal operator

p

  



 defined by

  

2 1

2 1 p

2

j j

j

, s

j

u



,

 



d

f r s

 f r







_

t s t t

 

is bounded on _{L R}p

 

2 _{for p1.}



h L 

Actually, the condition can be

weak-ened to the case:









0,

 

1

 

0 0 { ;

s

h sup s

h  R s h 

 t dt }, 1

  

_

  

with 1 p1 2 min



1 2,1



(see [9]), and the size condition on  in Theorem B is the best one, so far,



even if 

 

t 0 (see [8]).

 

t

 

On the other hand, for , it is known in [17], if , that



2 _, 1

h L R r dr   ,h

T

0 _ is bounded on _{L R}p

 

n1

 

_S 1



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



1 n

H S

p

  



1





2 _{R r dr}_, 1 n

R





1/ 2

log

L L

 





1/ 2

log

L L

 



 

1/ 2 ₁

log n

L L S

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 

1

n





 

_Sn1

 

1

n



R

h L  

1) T,h f 2

_{ }

n1 _{L R}2

_{ }

n1.

L R C f  2) T,h

 

f L Rp

_{ }

n1 C f L Rp

_{ }

n1,

for provided that the lower dimensional maximal operator

1  p ,

M defined by

  

1



 



/ 2 0

, sup R , d

R R

M_ f r s R f r t s  t t





_

  (1.3)

is bounded on _{L R}p



2



for 1_{  p .}

In Theorem 1, if 

 

t is continuously differentiable on



0,



and satisfy 

   

t  0 Ct 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

1

n







1 _0,1

t C





f  

(the Schwartz function class on _Rn1) (see [11]).

 

2

Remark 1. The L Rp boundedness of 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 _{L R}p



2 for , see [16, p. 477] or [15].

 

t t



1 p

2) If _{ }  _{with  }



_0,1



_{, then M}  is bounded on _{L R}p



2 for , see [13].

 

_{t C}1



 





p1

3) If



0,1



such that 

 

0 

 

0 0 0 t

and 

 

t  is a convex increasing function for , M_ is bounded on for , see [13], see [7, Corollary 5.3].



2

p

L R



p1

4) Let b t

 

t

   

t  t . If _

 

_{t C}2





_0,1





_,

 

t

 is convex on



with , and

there exists an



0,

0

 

 

0 0 0

  

 

 

so that for each t0,

 

,

b t b t t then M_ is bounded on _{L R}p

 

2 _for

, see [4, Theorem 1.5]. In particular, if 1

p 

 

t is

ether even or odd and there exists a so that for each

0  C

0,

t 

 

Ct 2

 

t , then M_ is bounded on _{L R}p

 

2 for _p1, see [3] or [4].

In order to obtain Theorem 1, we let S, be the op-erator defined by

 



 



 



 

1

, 1

1/ 2 2 1

0 ,

d

: n ', d

n

n S

S f x x

r

y f x ry x y y

r



 



 





 _{ } 

_    _



 

 .

Clearly, if _{h L R r dr} 2



_, 1



_{, then}

 



2

_

1

_

 





,h , n1 L R r dr, , , n T f x x_  h   S_ f x x_1 . Therefore Theorem 1 can be deduced immediately from the next theorem.

Theorem 2. Let  and be as in Theorem 1. Then



1) S,

 

f _{L R}2

_{ }

n1 C f _{L R}2

_{ }

n1.

2) S,

 

f _{L R}p

_{ }

n1 C f _{L R}p

_{ }

n1,

2  p , provided that the maximal operator M

in (1.3) is bounded on _{L R for} p

 

2 _{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

L L S

 













_log

 

n1

L L  S

   1 2 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.



1

p n L R 





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



1 n1

L S 

 

M in (1.3) is bounded on

 

2

p

L R for, then the following maximal operator M,

defined by

 





 

_

_{ }

_

, 1

1 0 _{/ 2 | |}

,

: sup , d

n

n n

R _{R y R}

M f x x y

f x y x y y

y





 

  _{ }

 



_

   (2.1)

is bounded on _{L R}p

 

n1 _{with bound}

 

1 1 , 1 .

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 R S

R

R

n

S _R R

y n

S

M f x x

r

y f x ry x r y

r r

y f x ry x r

r

y M f x x y





 

 









 

 

 



  

   

 y

   

 

 

 









where

 



2



 



, ' 1 1

0

1

, : sup R , d

y n _R n

R .

M f x x f x ry x r r

R

   







_

 

Thus by Minkowski's inequality, we have

 

_{ }

1 1

 

 

_{ }

1

 

, _{L R}p n _Sn , 'y _{L R}p n d .

M_ f  



  y M_ f   y

It remains to show that

 

_{ }

1

_{ }

1

,y L Rp n L Rp n

M_  f  C f 

with C independent of .

1

y



_ n1_.

Let 



1,0, ,0 S For each fixed _y'Sn1_,

choose a rotation such that   y 1. Denote the

in-verse of  by _1 _{and define the function f}  by . Then



,xn1



 f



x x, n1



f_ x

 



, n 1



1



, n 1

 



f x ry x    r  f_ x r x 1   r . This together with the _Lp_{-bondedness of M}

, and change of variables, show that

 

_{ }

1

_{ }

1

,y L Rp n L Rp n , 1 ,

M_  f  C f    p





where C is independent of. Lemma 2.1 is proved. Next we introduce some notations. Assume that and satisfies (1.1). For any



 

1/ 2 ₁

log n

L L S

 

l, let



n1_{: 2}l

 

₂l 1



l

E  yS    y   Also,

we let



1

 



Set

0 : | | 2

n

E _ y_S  _ y_{ }

 



_: 4l



l D l   E 2



and for l1,

 

 

 

 

     

1

1 1 :

d ,

l

n l

n

l E

E S

y y y S

y y y

 

 



 

  

   

 





and ₀

 

y : 

 

y 



_{l D} _l

 

y. 0,



Then we have the following:

   

1 d 0,

n l

S  y  y  l



(2.2)

1 2 l 1: 2 ,

l E A l Dl ,

     (2.3)

0 ₁ C 0 ₂ C ,

      (2.4)

 

y l D {0} l

 

y ,

 



 (2.5)





1/ 2

_

_{  }

1/2 ₁

{0} 1 l log n ,

l D  l A C L L S



(2.6)

where

1

l

l E

A   for l D and A₀1.

Now we give the proofs of our theorems as follows. Proof of Theorem 2. For each l D 

 

0 , we let

 



 



 



 

1

, 1

1 2 2 1

0

, :

d

, d

l

n n

l n

S

S f x x

r

y f x ry x y y

r



 



 







 _{     } 

 



 

 .

By (2.5) and Minkowski's inequality, we have

 



 

, , n1 l D {0} l, , n

S_ f x x_ 



_{ } S_{ } f x x_1



. (2.7) For any l D 

 

0 , let’s argue as in the proof of Theorem 2.1 in [1], choose a collection of C function

 

l j, _j on



0,



with the following properties:

supp

 

 1 1  1 1

, 2 , 2

j l j l l j

 _       _,

   (2.8)

 

,

0l j t 1, and

 

(2.9)

2

, 1,

l j j  t  





 

, d

, d

l j t C

t t



 



 _ (2.10)

where t0,  , and C_ is a constant independ-ent of l.

For each j and l D 

 

0 , denote by

the multiplier l j,

S



_{ }

_

_

_{ }



_

_

, , 1 , ,

l j n l j n

S f   _   f   _1 , and 2 by

,

l j

S

 







 







2

, , 1 , , ,

l j n l j l j n

S f x x S S f x x1 .

Then by (2.9) and Minkowski’s inequality,

 



 

    

 



 



 

 

 

 

 

 



1 1

1 1

1 1

, 1

2 2

1 2 2 2

, 1

2 1

1/ 2 2 2

, 1

, 1

,

d

, d

d

( ', )d

: , ,

l

j l

j l n

l n n

l

j k S

l j k n

l

k j S

l j k n

l k n

k

S f x x

y

r

S f x ry x r y

r y

r

S f x ry x r y

r

I f x x



 

 

 

 

 

 

 

 

 

 

 

 _

_ 





 

   _



 _

 _ 



 

   _

 











  



 

 



 



 

 



 



 

1 1

2

, 1 ₁

1/ 2 2 2

, 1

,

d

', d .

l n

l k n j S l

l j k n

I f x x y

r

S f x ry x r y

r

 

 

 

 

 _

_ 



 

   _



  



This together with (2.7) implies

 

_{ }

1 _{ }

 

_{ }

1

, L Rp n l D 0 k l k, L Rp n .

S_ f  



 



 I f  (2.11)

Now we estimate Il k,

 

f _{L R}p

_

n1



in the following

cases:

Case 1. For p2, we claim that there exists  0

such that for l D 



0



,

 

2

_{ }

1





2

_{ }

1

1/ 2 | |

, n 2 1 n

k

l k L R l L R

I f  C  l A f  , (2.12)

where C is independent of l and k.

Indeed, by Plancheral’s theorem and Fubini’s theorem,

 



_

_

_{ }

,

2 2

, ₂ , 1 , d d 1,

l j k

l k j n l j n

I f f   J   

  

 



 

_ 

(2.13) where

     



1 1 1



R

, : : 2 2 ,

j k l l j k l n

l j k  R 

        

     

 

 

 

 

1 ₁

1

2

2 _{2 '}

, ₁

d

e d

l _{j l}

n

i ry

l j _S l

r

J y

r



   

 

 



_{ }

  y

In order to prove (2.12), we first estimate Jl j,

 

 .

Obviously, we have

  



2





2

, 1 1 1

l j l l .

J    l C l A (2.14)

By (2.2), a straightforward computing shows that

  



  





  

2 2 1 1

, 1

2 1 1 2

1 2

1 2 ,

j l

l j l

j l

l

J l

C l A

 



   

  

 

(2.15)

Using interpolation between (2.14) and (2.15), we get

 





₂  ₁ ₁ 2/ 1

, 1 2

l j l

l j l

J  C l A     . (2.16)

On the other hand, we have

 

   

  _{ }

   

   

  _{ }

   

1

1 1 1

1 1 1 ₁

2

, ₁

2

2 ₂ 1

d e d d

d

e d d

l

n n

j l

n n l _{j l}

l j _{S S} l l

i r y u

l l

S S

i r y u

J y u

r

y u

r

y u

r _{y u}

r







 

 



  

 

 _

    



   

 

  

 



 

  

 



 





.

And by integration by parts,

  _{ }





 

_

_

1 ₁

2 ₂ 1

1 1

d e

1 2

l _{j l}

i r y u

j l

r r

C l y u





 _  _{ }

  _{ }

   



,

with the easy fact

  _{ }

1 1

2 ₂ 1

d

e 1

l _i j l _{r y u} r

l r



 _  _{ }

 



,

we obtain

  _{ }





 

_

_

1 1

2 ₂ 1

1/ 4 1

d e

1 2

l j l

i r y u

j l

r r

C l y u





 _

   

  _{ }

   



.

Thus, by Holder’s inequality

 





 





   









 

1 1

1/ 4 ₂

1

, 2

1/ 2 1/ 2

1/ 4 ₂

1

2

1 2

1 2 .

n n

j l

l j l

S S j l

l

J C l

y u d y d u

C l

 

  



 

 

 

 

  

    

  

  



Note that _{0 2} C CA₀, and for l D ,

 

3

2l l

l l

A _C  E _C_{2 ,} _{we have}

 

1/ 2 2 1 1

2 2 2

l l

l C  El C

 _.

l A

   

Consequently,

 





₄ ₁  ₁ 1/ 4 2

, 1 2 2

l j l

l j l

J  _C l_ A    _.

This together with (2.14) and an interpolation implies

 





₂  ₁ 1 4 1

, 1 2

l j l

l j l

J  C l A     . (2.17)

Then by the fact that l j k,   l j k, '   whenever



1, , 1 ,



j j j j (2.12) follows from (2.13), (2.16) and (2.17).

Case 2. For p2, we shall show that there exists

0

  such that for l D 

 

0 ,

 

_{ }

1





_{ }

1

1/ 2 | |

, p n 2 1 p n

k

l k L R l L R

I f  C  l A f 



  .



(2.18)

where C is independent of l and k.

Indeed, choose _{g L} p/ 2



_Rn1 such that p/2

_{ }

n1 1

L R

g   and

 

_{ }

 

 



 



 





_{ }





1

2l

1 1 1

2

, ₁

2

1 1

2

, 1

1 1

2 ', 2 d

d

, d d .

p n n n

l k L R R j S l

j l j l

l j k n

n n

I f y

S f x ry x r y

r

g x x x x r

 

  _ 

 

 

 



 

  









 



 

 

 



 







 

 



1

1 1

2 2

, ₁

1 1

1 2 2

, 1 1 1

sup

d

2 , 2 d

, d d .

l

n n

l k _{p R} j _S l

j l j l

n

l j k n n l

j

I f y

r

g x ry x r y

r

S f x x x x

 



  

 



  





 



  





 











Note that

 

 









 





 





1 1

2 1

1 ( 1)

1

, 1

sup

d

2 , 2 d

1 , .

l n

l

j _S l

j l j l

n

n

y

r

g x ry x r y

r l M  g x x

 

  

 



 

 



  

 

 





Applying Holder’s inequality again, it follows from Lemma 2.1 and the Littlewood-Paley theory (see [18, Chapter 4]) that

 





_{ }

 











 











2 2

, 1 / 2

1/ 2 2 2

,

1/ 2 2

2 ₂

, 1

2 2 ₂ 2

1

1

1

1 1

l k p l p

l j k j

p l j l j k

.

p

l p l p

I f C l g

S f

C l S f

C l f C l A f



 

 

  



  

    









This together with (2.12) and an interpolation implies (2.18).

Therefore, by (2.11), (2.12) and (2.6), we get

 

_{ }





 





_{ }

 



  

 

2 1 2 1

2 1 1/2 _{1 2 1}

1/ 2

, {0}

| |

1/ 2 0

log

1

2

1

.

n

n

n

n n

l D k L R

k

l L R

l

l D L R

L L S L R

S f C l

A f

C l A f

C f











  

   



 





 

 







 

This prove 1) of Theorem 2.

If M_ is bounded on _{L R}p

 

2 _{for 1 , then}

by (2.11), (2.18) and (2.6), we obtain that for ,

p

  

2

p

 

_{ }

_

_{  }

1/2 ₁

_{ }

₁

1

, _{L R}p n _Llog _{L S}n _{L R}p n ,

S_ f  C    f 

which completes the proof of Theorem 2. Proof of Theorem 1. By the fact

 



2

_

1

_

 



,h , n1 L R r dr, , , n1

T_ f x x  h   S_ f x x , and Theorem 2, it follows that T,h is bounded on

for . On the other hand, by duality we can obtain that T,h



1

p n L R 

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 m2 x, yRn_{, and x*R}m_{, we define}

 



 

 

 





, , * . .

, * d .

n

n

h _R

T f x x p v h y y y

f x y x y y



   

  



_(4.1)

Also, we define the operator S,Φ by

 





 

 





 

1

, ₀

1/ 2 2

, *

d

, * d ,

n S

S f x x y

r

f x ry x r y

r







    



 

   _



 



(4.2)

and the lower dimensional maximal function MΦ by

 



 





1 *

1

0 _{/ 2} 1 *

,

supr _rr , d .

M f x x

r f x t x t

  



_

  t (4.3)

In [10], Fan and Zheng extended the result of Theorem B to the operator TΦ,hfor h∆γ(R+) for γ > 1. Here, we

will obtain the following results.

Theorem 3. Suppose that hL2(R+, r−1dr) and  L(log+_L)1/2_(Sn−1_{). Then}

1) T,h

 

f L R2

_

n m

_

C f L R2

_

n m

_

.

2) T,h

 

f L Rp

_

n m

_

C f L Rp

_

n m

_

, 1  p , provided that the maximal operator M defined in (4.3)

is bounded on _{L R}p



m1



_{for all p1.}

Theorem 4. Let Φ and  be as in Theorem 3. Then

1) S ,

 

f _{L R}2

_

n m

_

C f _{L R}2

_

n m

_

.

2) S ,

 

f _{L R}p

_

n m

_

C f _{L R}p

_

n m

_

, 2  p ,

provided that the maximal operator M defined in (4.3)

is bounded on _{L R}p



m1



_{for all p1.}

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(Sn−1) 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 | |

( )( , )

: supr n ,

r y r

M f x x y

f x y x y dy y

 

  

 



_

    (4.4)



2  p

 is bounded on for . Theorem 1 is proved.



1

p n L R 



2

1 p is bounded on _{L R}p



n m



_{with bound}

 

1 n1

L S

[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

1  p .

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.

5. References

[11] S. Lu, Y. Pan and D. Yang, “Rough Singular Integrals Associated to Surfaces of Revolution,” Proceedings of

the AMS—American Mathematical Society, Vol. 129,

2001, pp. 2931-2940.

doi:10.1090/S0002-9939-01-05893-2

[1] A. Al-Salman and Y. Pan, “Singular Integrals with Rough Kernels in L(log+_L)1/2_(Sn–1_{),” Journal of the London}

Mathematical Society, Vol. 66, No. 2, 2002, pp. 153-174.

doi:10.1112/S0024610702003241

[12] Y. Pan, L. Tang and D. Yang, “Boundedness of Rough Singular Integrals Associated with Surfaces of Revolu-tion,” Advances in Mathematics (China),Vol. 32, No. 2, 2003, pp. 677-682.

[2] H. Al-Qassem and Y. Pan, “Singular Integrals along Sur-faces of Revolution with Rough Kernels,” SUT Journal of

Mathematics, Vol. 39, No. 1, 2003, pp. 55-70.

[3] H. Carlsson, M. Christ, A. Cordoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger and D. Weinberg, “Lp _{Estimates for Maximal Functions and}

Hil-bert Transforms along Flat Convex Plane Curves in R2_,”

Bulletin of the American Mathematical Society, Vol. 14,

No. 2, 1986, pp. 263-267.

doi:10.1090/S0273-0979-1986-15433-9

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