A note on the multilinear oscillatory singular integral operators

A note on the multilinear oscillatory singular integral operators

Zefu Chu, Guoen Hu and Zhibo Lu

(Received December 17, 1999) (Revised November 13, 2000)

Abstract. In this paper, we consider theL^p…Rⁿ† boundedness for a class of multi- linear oscillatory singular integral operators with polynomial phases. We show that if the polynomial phases are non-trivial and the homogeneous kernels satisfy a certain minimum size condition, then the L^p…Rⁿ† boundedness for the multilinear oscillatory singular integral operators can be deduced from the L^p…Rⁿ† boundedness for the corresponding local multilinear singular integral operators.

1. Introduction

We will work on Rⁿ…nb2†. Let P…x;y† be a real-valued polynomial on RⁿRⁿ,W…x†be homogeneous of degree zero which has a mean value zero on the unit sphere S^nÿ1. De®ne the oscillatory singular integral operator

Tf…x† ˆ …

Rⁿe^iP…x;y†W…xÿy†

jxÿyjⁿ f…y†dy:

…1†

It is well-known that the operators of this type have arisen in the study of Hilbert transforms along curves, singular integrals supported on lower- dimensional varieties and singular Radon transforms, etc. A celebrated result of Ricci and Stein [9] says that ifWALip₁…S^nÿ1†, thenTis bounded onL^p…Rⁿ† for 1<p<y, with a bound depending only on n, p and degP (the total degree of P), not on the coe½cients of the polynomial. Chanillo and Christ [2] showed thatWALip₁…S^nÿ1†is also su½cient forTto be a bounded mapping from L¹ to weak L¹, and the bound depends only on n and degP. Lu and Zhang [7] improved the result of Ricci and Stein, and proved that if WA 6_q>1L^q…S^nÿ1†, then T is bounded on L^p…Rⁿ† with a bound C…n;p;degP† for 1<p<y.

In this paper, we will study the multilinear operators de®ned by T_A1;...;Akf…x† ˆ

Rⁿe^iP…x;y† W…xÿy†

jxÿyj^n‡m Y^k

jˆ1

R_mj‡1…A_j;x;y†f…y†dy;

…2†

2000Mathematics Subject Classi®cation. 42B20

Key words and phrases. multilinear operator, oscillatory singular integral, BMO.

The research was supported by the NSF of China (19701039) and the NSF of Henan Province.

wherekandmj …jˆ1;. . .;k†are positive integers,mˆP_k

jˆ1mj,Aj …jˆ1;. . .;k†

has derivatives of orderm_j in BMO…Rⁿ†,R_mj‡1…A_j;x;y†denotes the …m_j‡1†-th Taylor series remainder of Aj at x about y, that is,

Rm‡1…Aj;x;y† ˆAj…x† ÿ X

jajamj

1

a!D^aAj…y†…xÿy†^a:

Operators of this type have been studied in [3], [4], [6] and many other works.

It is easy to see that the operator TA1;...;Ak is closely related to the oscillatory singular integral operator de®ned by (1) and the multilinear singular integral operator de®ned by

T~_A1;...;Akf…x† ˆ …

Rⁿ

W…xÿy†

jxÿyj^n‡m Y^k

jˆ1

R_mj‡1…A_j;;x;y†f…y†dy:

…3†

Using good-l-inequality techniques, Cohen and Gosselin [5] showed that if W satis®es a certain vanishing moment and WALip₁…S^nÿ1†, then for 1<p<y,

kT~A1;A2fkpaY²

jˆ1

X

jajˆmj

kD^aAjk_BMO…Rⁿ_† 0

@

1 Akfkp:

In [3], Chen, Hu and Lu considered the L^p…Rⁿ† boundedness for the operator TA1;A2 and proved that if WA6_q>1L^q…S^nÿ1†, and the polynomial P…x;y† is non-trivial, then the L^p…Rⁿ† boundedness for T_A1;A2 can be obtained from the L^p…Rⁿ† boundedness for the local multilinear singular integral operator

SA1;A2f…x† ˆ …

jxÿyja1

W…xÿy†

jxÿyj^n‡m1‡m2 Y²

jˆ1

Rmj‡1…Aj;;x;y†f…y†dy;

(see [2, Theorem 2]). The purpose of this paper is to show that if WA L…logL†^k‡1…S^nÿ1†, and P is non-trivial, then the L^p…Rⁿ† boundedness for

TA1;...;Ak can be obtained from the L^p…Rⁿ† boundedness for the local version of

the operator T~_A1;...;Ak. Our main result in this paper can be stated as follows.

Theorem1. Let 1<p<y, k and mj …jˆ1;2;. . .;k†be positive integers, mˆP_k

jˆ1mj, Aj …jˆ1;2;. . .;k†be functions on Rⁿ whose derivatives of order mj

are inBMO…Rⁿ†. Suppose thatWis homogeneous of degree zero and belongs to the space L…logL†^k‡1…S^nÿ1†, that is,

S^nÿ1jW…x⁰†jlog^k‡1…2‡ jW…x⁰†j†dx⁰<y;

and the operator

SA1;...;Akf…x† ˆ …

jxÿyja1

W…xÿy†

jxÿyj^n‡m Y^k

jˆ1

Rmj‡1…Aj;x;y†f…y†dy …4†

is bounded on L^p…Rⁿ†. Then for any real-valued non-trivial polynomial P…x;y†, the operator T_Ade®ned by(2)is also bounded on L^p…Rⁿ†, with a bound depending on n, p, mj …jˆ1;. . .;k†, Q_k

jˆ1…P

jajjˆmjkD^ajAjk_BMO…Rⁿ_†† and degP, not on the coe½cients of P.

2. Proof of Theorem 1

We begin with some preliminary lemmas.

Lemma1 (see [5]). Let b…x†be a function onRⁿ with derivatives of order m in L^q…Rⁿ† for some n<qay. Then

jRm…b;x;y†jaCm;njxÿyj^m X

jajˆm

1 jI…x;~ y†j

I…x;~ y†jD^ab…z†j^qdz

!_1=q

; where I…x;~ y† is the cube centered at x with diameter 5 

pn

jxÿyj.

Lemma 2. Let 1<p<y, k and m_j …jˆ1;2;. . .;k† be positive integers, mˆP_k

jˆ1mj, Aj …jˆ1;2;. . .;k†be functions on Rⁿ whose derivatives of order mj

are inBMO…Rⁿ†. Suppose thatW~ is homogeneous of degree zero and belongs to the space L^y…S^nÿ1†. Set

lW;k~ ˆinf l>0:kWk~ 1

l log^k 2‡kWk~ y

l

! a1

( )

:

Then for any r>0, the operator

UA1;...;Ak;rf…x† ˆr^ÿnÿm

r=2<jxÿyjarjW…x~ ÿy†jY^k

jˆ1

jRmj‡1…Aj;x;y†jjf…y†jdy …5†

is bounded on L^p…Rⁿ†with a bound C…n;m;p†lW;k~

Q_k

jˆ1…P

jajjˆmkD^ajAjk_BMO…Rⁿ_††.

Proof. Note that for each t>0, l_tW;~k ˆinf l>0:ktWk~ ₁

l log^k 2‡ktWk~ _y l

! a1

( )

ˆinf tl~: ~l>0;ktWk~ 1

tl~ log^k 2‡ktWk~ y

tl~

! a1

( )

ˆtlW;k~ :

Thus we may assume that lW;k~ ˆ1=2. Therefore, kWk~ 1log^k…2‡ kWk~ y†a1:

De®ne the operator E by Eh…x† ˆ

jxÿyja1jW…x~ ÿy†jh…y†dy:

Denote by E the adjoint operator of E, that is, Eh…x† ˆ

jxÿyja1jW…y~ ÿx†jh…y†dy:

Let b₁;b₂;. . .;b_kABMO…Rⁿ†and Q be a cube with side length 1. Denote by mQ…bj† the mean value of bj on Q. We claim that for 1<p<y, supph H10nQ and non-negative integer lak,

QjEh…x†j^pY^l

jˆ1

jbj…x† ÿmQ…bj†j^pdx …6†

aClog^ÿk‡l†p…2‡ kWk~ _y†Y^k

jˆ1

kb_jk_BMO…R^p n†khk_p^p; with the interpretation that when lˆ0, Q_l

jˆ1jbj…x† ÿmQ…bj†j11. To prove (6), we can assume that khk_p ˆ1. Choose 1<r_j <y such that P_k

jˆ11=r_j

ˆ1. By the well-known John-Nirenberg inequality, there is a positive constant Cj ˆC…p;rj;n† such that

Qjbj…x† ÿmQ…bj†j^2prjdx

_1=…2rj†

aCjkbjk_BMO…R^p n†:

We may also assume thatkb_jk_BMO…R^p n†ˆ1=C_j for all 1ajak. We shall carry out our argument by induction onl. If lˆ0, the Young inequality gives that

QjEh…y†j^pdyaCkWk~ ₁^pkhk_p^paClog^ÿkp…2‡ kWk~ _y†:

Now let dakÿ1 be a non-negative integer and assume that the estimate (6) holds forlˆd. We will show that (6) holds forlˆd‡1. Observe thatF…t†

ˆtlog^p…2‡t† is a Young function and its complementary Young function is C…t†Aexpt^1=p. By the general HoÈlder inequality, it follows that

QjEh…x†j^pY^d‡1

jˆ1

jb_j…x† ÿm_Q…b_j†j^pdx

aCinf l>0: …

Q

jEh…x†j^p

l log^p 2‡jEh…x†j^p l

Y^d

jˆ1

jbj…x† ÿmQ…bj†j^pdxa1

( )

inf l>0: …

Qexp jb_l‡1…x† ÿm_Q…b_l‡1†j l^1=p

Y^d

jˆ1

jb_j…x† ÿm_Q…b_j†j^pdxa2

( )

;

(see [1] or [8]). Applying the Young inequality again, we have kEhkyakWk~ ykhk1aCkWk~ ykhkpaCkWk~ y. Our induction assumption now gives that

QjEh…x†j^plog^p 2‡jEh…x†j^p l

Y^d

jˆ1

jb_j…x† ÿm_Q…b_j†j^pdx

aClog^p 2‡CkWk~ _y^p l

!

log^ÿk‡d†p…2‡ kWk~ y†:

Set l₀ˆlog^ÿk‡d‡1†p…2‡ kWk~ _y†. An easy computation then leads to that …

QjEh…x†j^plog^p 2‡jEh…x†j^p l₀

Y^d

jˆ1

jbj…x† ÿmQ…bj†j^pdxaCl0:

On the other hand, by the HoÈlder inequality, …

Qexp jbl‡1…x† ÿmQ…bl‡1†j l^1=p

Y^d

jˆ1

jbj…x† ÿmQ…bj†j^pdx

a …

Qexp 2jb_l‡1…x†ÿm_Q…b_l‡1†j l^1=p

dx

₁₌₂Y^d

jˆ1

Qjb_j…x†ÿm_Q…b_j†j^2prjdx

_1=…2rj†

a …

Qexp 2jb_l‡1…x† ÿm_Q…b_l‡1†j l^1=p

dx

₁₌₂

;

which together with the John-Nirenberg inequality implies that

inf l>0: …

Qexp jb_l‡1…x† ÿm_Q…b_l‡1†j l^1=p

Y^d

jˆ1

jbj…x† ÿmQ…bj†j^pdxa2

( )

aC;

Therefore, …

QjEh…x†j^pY^d‡1

jˆ1

jb_j…x† ÿm_Q…b_j†j^pdxaClog^ÿk‡d‡1†p…2‡ kWk~ _y†:

We can now prove our Lemma 2. By dilation-invariance, it su½ces to consider the case rˆ1. Write Rⁿˆ6_jIj, where each Ij is a cube having side length 1 and the cubes have disjoint interiors. Let w_j be the characteristic function of Ij. Set f_jˆ fw_j. Then

f…x† ˆX

j

f_j…x†; a:e: xARⁿ:

Since the support of UA1;...;Ak;1f_j is contained in a ®xed multiple of Ij, the supports of various terms fU_A1;...;Ak;1f_jg have bounded overlaps, and so we have

kUA1;...;Ak;1fk_p^paCX

j

kUA1;...;Ak;1f_jk_p^p:

Thus we may assume that supp fHI for some cube Iwith side length 1. Set Aej…y† ˆAj…y† ÿ X

jajjˆmj

1

a_j!mI…D^ajAj†y^a: A straightforward computation shows that for x;yARⁿ,

R_mj‡1…A_j;x;y† ˆR_mj‡1…Ae_j;x;y†.

Choose n<q<y Lemma 1 now tells us that jR_mj…Ae_j;x;y†j

aCjxÿyj^mj X

jajjˆmj

1 jI~…x;y†j

I…x;~ y†jD^ajAj…z† ÿmI…D^ajAj†j^qdz

!_1=q

aCjxÿyj^mj X

jajjˆmj

1 jI~…x;y†j

I…x;~ y†jD^ajAj…z† ÿmI…x;~ y†…D^ajAj†j^qdz

!_1=q

‡Cjxÿyj^mj X

jajjˆmj

jm_I…D^ajA_j† ÿmI…x;y†~ …D^ajA_j†j

aCjxÿyj^mj X

jajjˆmj

…kD^ajA_jk_BMO…Rⁿ_†‡ jm_I…D^ajA_j† ÿmI…x;y†~ …D^ajA_j†j†:

Note that if yAI and jxÿyja1, then I~…x;y†H100nI. This in turn implies that for yAI and 1=2ajxÿyja1,

jm_I…D^ajA_j† ÿmI…x;y†~ …D^ajA_j†jaCkD^ajA_jk_BMO…Rⁿ_†. Thus in this case, we have

jR_mj…Ae_j;x;y†jaCjxÿyj^mj X

jajjˆmj

kD^ajA_jk_BMO…Rⁿ_†aC X

jajjˆmj

kD^ajA_jk_BMO…Rⁿ_†: Let

f…y† ˆY^k

jˆ1

X

jajjˆmj

…kD^ajA_jk_BMO…Rⁿ_†‡ jD^ajA_j…y† ÿm_I…D^ajA_j†j†

0

@

1 A: We can write

U_A1;...;Ak;1f…x†aE…jffj†…x†:

A standard duality arguement and the HoÈlder inequality then show that

kU_A1;...;Ak;1fk_pa sup

supphH10nI;khkp0a1

E…jffj†…x†h…x†dx

ˆ sup

supphH10nI;khkp0a1

jEh…y†f…y†f…y†jdy aCkfk_p sup

supphH10nI;khkp0a1kfEhk_p⁰;

where p⁰ is the dual exponent of p, i.e. p⁰ˆp=…pÿ1†. Invoking the estimate (6) for 0alak, we ®nally obtain

kUA1;...;Ak;1fkpaCY^k

jˆ1

X

jajjˆmj

kD^ajAjk_BMO…Rⁿ_† 0

@

1 Akfkp:

This completes the proof of Lemma 2.

Proof ofTheorem1. Without loss of generality, we may assume that for

1ajak, X

jajjˆmj

kD^ajAjk_BMO…Rⁿ_†ˆ1:

Letk₀ be a positive integer and P…x;y†be a real-valued non-trivial polynomial having degree k0 in x and degree l0 in y. Write

P…x;y† ˆ X

jmjˆk0;jnjˆl0

am;nx^myⁿ‡R…x;y†;

where R…x;y† is a real-valued polynomial which has degree less k0 in x. By dilation-invariance, we may assume thatP

jmjˆk0;jnjˆl0ja_mnjˆ1. SpliteT_A1;...;Ak as TA1;...;Akf…x† ˆ

jxÿyja1e^iP…x;y† W…xÿy†

jxÿyj^n‡m Y^k

uˆ1

Rmu‡1…Au;x;y†f…y†dy

‡X^y

jˆ1

2^jÿ1<jxÿyja2^je^iP…x;y† W…xÿy†

jxÿyj^n‡m Y^k

uˆ1

R_mu‡1…A_u;x;y†f…y†dy

ˆT_A⁰_1;...;Akf…x† ‡X^y

jˆ1

T_A^j_1;...;Akf…x†:

We ®rst consider the operator T_A^j_1;A2;...;Ak for jb1. Let E0ˆ fx⁰AS^nÿ1, jW…x⁰†ja2g and E_l ˆ fx⁰AS^nÿ1;2^l<jW…x⁰†ja2^l‡1g for positive integer l.

Let W_l be the restriction of W on E_l. De®ne the operator T_A^j_1;...;Ak;l by

T_A^j_1;...;Ak;lf…x† ˆ …

2^jÿ1<jxÿyja2^je^iP…x;y† W_l…xÿy†

jxÿyj^n‡m Y^k

uˆ1

R_mu‡1…A_u;x;y†f…y†dy:

To estimate the L^p…Rⁿ† boundedness for T_A^j_1;...;Ak;l, we will use the following lemma.

Lemma 3. Let the polynomial P…x;y†, k, mu and Au …uˆ1;. . .;k† be the same as above, W~ be homogeneous of degree zero and belong to the space L^y…S^nÿ1†. De®ne the operator

Vjf…x† ˆ …

1<jxÿyja2e^{iP…2jx;2jy†} W…x~ ÿy†

jxÿyj^n‡m Y^k

uˆ1

Rmu‡1…Au;x;y†f…y†dy:

Then for1< p<y, there exists positive constants C anddwhich are depending only on n, p and degP such that

kVjfkpaCkWk~ y2^ÿdjkfkp.

For the case of kˆ1, this lemma was proved essentially in [3, page 43±46]. For general positive integer k, Lemma 3 can be proved by induction on k. We omit the details.

We now estimateT_A^j_1;...;Ak;l. Note that forbABMO…Rⁿ† and t>0, b_t…x†

ˆb…tx† also belongs to the space BMO…Rⁿ† and kbtk_BMO…Rⁿ_†ˆ kbk_BMO…Rⁿ_†. Thus by dilation-invariance and Lemma 3,

kT_A^j_1;...;Ak;lfkpaC2^ÿdj2^lkfkp: …7†

On the other hand, Lemma 2 states that

kT_A^j_1;...;Ak;lfk_paCl_Wl;kkfk_p: …8†

Set l_l^k ˆl^kkWlk1‡2^ÿl. A trivial computation gives that kWlk1

l_l^k log^k 2‡kWlky

l_l^k

!

a kWlk1

l^kkW_lk₁ log^k 2‡kWlky

2^ÿl

aC;

which in turn implies

l_Wl;kaC…l^kkW_lk₁‡2^ÿl†:

…9†

Our hypothesis on Wnow says thatP

l>0l^k‡1kWlk1<y. Let Nbe a positive integer such that N>2d^ÿ1. Combining the inequalities (7) and (8) yields that

X

jb1

X

lb0

T_A^j_1;...;Ak;lf

_paX

jb1

kT_A^j_1;...;Ak;0fkp‡X

l>0

X

j>Nl

kT_A^j_1;...;Ak;lfkp

‡X

l>0

X

1ajaNl

kT_A^j_1;...;Ak;lfkp

aCX

jb1

2^ÿdjkfkp‡CX

l>0

2^l X

jbNl

2^ÿdjkfkp

‡CX

l>0

llWl;kkfkpaCkfkp:

Now we turn our attention to the operatorT_A⁰_1;...;Ak. The estimate for this term follows from the following lemma directly.

Lemma 4. Let 1<p<y, and S_A1;...;Ak be de®ned by (4) with WA L…logL†^k…S^nÿ1†. Suppose that S_A1;...;Ak is bounded on L^p…Rⁿ†. Then for any real-valued polynomial P…x;~ y†, the operator

W_A1;...;Akf…x† ˆ …

jxÿyja1e^iP…x;y†~ W…xÿy†

jxÿyj^n‡m Y^k

uˆ1

R_mu‡1…A_u;x;y†f…y†dy;

is bounded on L^p…Rⁿ† with a bound C…n;m;p;degP†.~

Proof. We follow along the same line as in the proof of Lemma 6 in [3].

We shall carry out the arguement by a double induction on the degree in x and yof the polynomial. Obviously, Lemma 4 holds if the polynomialP…x;~ y†

depends only onx or only on y. Let u andvbe two positive integers and the

polynomialP…x;~ y†have degree u in x andv in y. We assume that Lemma 4 is known for all polynomials which are sums of monomials of degree less than u in x times monomials of any degree in y, together with monomials which are of degree u in x times monomials which are of degree less than v in y.

We can now write

P…x;~ y† ˆ X

jmjˆu;jnjˆv

bmnx^myⁿ‡P0…x;y†;

where P0…x;y† satis®es the inductive assumption. We consider the following two cases.

Case I. P

jmjˆu;jnjˆvjbmnja1. As in the proof of Lemma 2, we may assume that supp fHI for some cube I centered at x₀ and having side length 1. By translation-invariance (note that our result is independent of the coe½cients of the polynomial), we may assume that supp fHI₀, the cube centered at the origin and having side length 1. Set

P…x;y† ˆP0…x;y† ‡ X

jmjˆu;jnjˆv

bmny^m‡n:

Observe that if jxÿyja1 and yAI0, then

je^{iP…x;~ y†}ÿe^iP…x;y†jaCjxÿyj.

Thus,

jW_A1;...;Akf…x†ja …

jxÿyja1e^iP…x;y† W…xÿy†

jxÿyj^n‡m Y^k

jˆ1

R_mj‡1…A_j;x;y†f…y†dy

‡C …

jxÿyja1

jW…xÿy†j jxÿyj^n‡mÿ1

Y^k

jˆ1

jRmj‡1…Aj;x;y†jjf…y†jdy

a …

jxÿyja1e^iP…x;y† W…xÿy†

jxÿyj^n‡m Y^k

jˆ1

Rmj‡1…Aj;x;y†f…y†dy

‡CX^y

jˆ0

2^ÿjU_A1;...;Ak;2^ÿjf…x†;

where U_A1;...;Ak;2^ÿj is de®ned by (5). Set U_A^l_1;...;Ak;2ÿjf…x† ˆ2^ÿj…n‡m†

2^ÿjÿ1<jxÿyja2^ÿjjW_l…xÿy†jY^k

uˆ1

jR_mu‡1…A_u;x;y†jjf…y†jdy:

It follows from Lemma 2 and the inequality (9) that X^y

jˆ0

2^ÿjkU_A1;...;Ak;2^ÿjfkpaCX^y

jˆ0

2^ÿjX

lb0

kU_A^l_1;...;Ak;2ÿjfkp

aCkfk_p‡X^y

jˆ0

2^ÿjX

lb1

l_Wl;kkfk_paCkfk_p:

This via the induction hypothesis tells us that

kWA1;...;AkfkpaC…n;m;p;degP†k~ fkp.

Case II. P

jmjˆu;jnjˆvjb_mnj>1. Set J ˆ …P

jmjˆu;jnjˆvjb_mnj†^1=…u‡v†. Let Q…x;y† ˆ X

jmjˆu;jnjˆv

bmn

J^u‡vx^myⁿ‡P0…x=J;y=J†:

Then P…x;~ y† ˆQ…Jx;J y†. De®ne the operator W~_A1;...;Akf…x† ˆ

jxÿyjaJe^iQ…x;y† W…xÿy†

jxÿyj^n‡m Y^k

uˆ1

R_mu‡1…A_u;x;y†f…y†dy:

By dilation-invariance, it su½ces to prove that

kW~_A1;...;Akfk_paC…n;m;p;degP†k~ fk_p: …10†

We splite the operator W~_A1;...;Ak as W~A1;...;Akf…x† ˆ

jxÿyja1e^iQ…x;y† W…xÿy†

jxÿyj^n‡m Y^k

uˆ1

Rmu‡1…Au;x;y†f…y†dy

‡X^j0

jˆ1

2^jÿ1<jxÿyja2^je^iQ…x;y† W…xÿy†

jxÿyj^n‡m Y^k

uˆ1

Rmu‡1…Au;x;y†f…y†dy

‡ …

2^j0<jxÿyjaJe^iQ…x;y† W…xÿy†

jxÿyj^n‡m Y^k

uˆ1

R_mu‡1…A_u;x;y†f…y†dy

ˆW~^If…x† ‡W~^IIf…x† ‡W~^IIIf…x†;

where j₀ is the positive integer such that 2^j0 <Ja2^j0‡1. The conclusion of Case I applies to W~^I, so

kW~^IfkaC…n;m;p;degP†k~ fk_p.

By the inequalities (7), (8) and (9) as in the estimate for P

jb1T_A^j_1;A2;...;Ak, we can obtain that

kW~^IIfk_paC…n;m;p;degP†k~ fk_p.

On the other hand, it follows from Lemma 2 and the estimate (9) that kW~^IIIfk_paC…n;m;p;degP†k~ fk_p.

This leads to the estimate (10), and completes the proof of Lemma 4.

Acknowledgement

The authors would like to thank the referee for some valuable suggestions and corrections.

References

[ 1 ] R. A. Adams, Sobolev Space, Academic Press, New York, 1975.

[ 2 ] S. Chanillo and M. Christ, Weak …1;1† bounds for oscillatory singular integrals, Duke Math. J. 55 (1987), 141±155.

[ 3 ] W. Chen, G. Hu and S. Lu, Criterion of…L^p;L^r†boundedness for a class of multilinear oscillatory singular integrals, Nagoya Math. J. 149 (1998), 33±51.

[ 4 ] W. Chen, G. Hu and S. Lu, On a multilinear oscillatory singular integral (II), Chin Ann.

of Math.18:A (1997), 73±82.

[ 5 ] J. Cohen and J. Gosselin, A BMO estimate for multilinear oscillatory singular integral, Illinois J. of Math. 30 (1986), 445±464.

[ 6 ] G. Hu, S. Lu and D. Yang, On a class of multilinear oscillatory singular integral operators, J. Math. Soc. Japan 48 (1998), 623±637.

[ 7 ] S. Lu and Y. Zhang, Criterion ofL^pboundedness for a class of oscillatory singular integrals with rough kernels, Rev. Mat. Iberoamericana 8 (1992), 201±219.

[ 8 ] C. PeÂrez, Endpoint estimates for commutators of singular integral operators, J. Funct.

Anal. 128 (1995), 163±185.

[ 9 ] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals (I), oscillatory integrals, J. of Funct. Anal. 73 (1987), 179±194.

Zefu Chu, Guoen Hu and Zhibo Lu Department of Applied Mathematics University of Information Engineering P.O. Box 1001-747, Zhengzhou 450002

People's Republic of China Z. Chu: [email protected] G. Hu: [email protected] Z. Lu: [email protected]