On multilinear square function and its applications to multilinear Littlewood–Paley operators with non-convolution type kernels

Accepted Manuscript On multilinear square function and its applications to multilinear Littlewood–Paley operators with non-convolution type kernels Qingying Xue, Qingying Xue

PII: DOI: Reference:

S0022-247X(14)00863-4 10.1016/j.jmaa.2014.09.039 YJMAA 18855

To appear in:

Journal of Mathematical Analysis and Applications

Received date: 20 June 2014

Please cite this article in press as: Q. Xue, Q. Xue, On multilinear square function and its applications to multilinear Littlewood–Paley operators with non-convolution type kernels, J. Math. Anal. Appl. (2015), http://dx.doi.org/10.1016/j.jmaa.2014.09.039

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ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS TO MULTILINEAR LITTLEWOOD-PALEY OPERATORS WITH NON-CONVOLUTION TYPE KERNELS QINGYING XUE AND JINGQUAN YAN

Abstract. Let m ≥ 2, n ≥ 1 and x ∈ Rn , define the multilinear square function T 1/2 ∞  m , where by T (f)(x) = 0 | (Rn )m Kt (x, y1 , · · · , ym ) j=1 fj (yj )dy1 . . . dym |2 dt t the kernel K satisfies a class of integral smooth conditions which is much weaker than the standard Calder´ on-Zygmund kernel conditions. In this paper, we first established the Lp1 (w1 ) × · · · × Lpm (wm ) → Lp (νω ) estimate of T when each pi > 1 and weak type Lp1 (w1 ) × · · · × Lpm (wm ) → Lp,∞ (νω ) estimate of T when there is m p/p a pi = 1, where νω = i=1 ωi i and each wi is a nonnegative function on Rn . As applications of the the above results, we obtained the boundedness of multilinear Littlewood-Paley operators with non-convolution type kernels, including mulitinear g-function, Marcinkiewicz integral and gλ∗ -function.

1. Introduction The multilinear Calder´on-Zygmund theory was originated in the works of Coifman and Meyer on the Calder´on-Zygmund commutator [4] [5] in the 70s. Coifman and Meyer ([5] and [6]) first considered the strong (Lp1 × · · · × Lpm , Lp ) type estimate for the following convolution-type multilinear Calder´on-Zygmund singular integral operator,  m   K(y1 , · · · , ym ) fi (x − yi ) dyi . (1.1) T (f )(x) = p.v. (Rn )m

i=1

Later, Kenig and Stein [18] showed that T was also bounded from L1 × · · · × L1 to 1 on-Zygmund operators with L m ,∞ . In recent years, the theory on multilinear Calder´ more general kernels has attracted much attentions as a rapid developing field in harmonic analysis, see for examples, [7], [16], [17], [19] and the references therein. In order to see how the kernels become more general, let us consider the bilinear Fourier multiplier.   B(f, g)(x) =

m(ξ, η)fˆ(ξ)ˆ g (η)eix·(ξ+η) dξdη. Rn

Rn

Date: Jun 18, 2014. Key words and phrases. Multilinear square function; integral smooth conditions; multilinear Littlewood-Paley operators. The first author was supported partly by NSFC (Grant No. 11471041), the Fundamental Research Funds for the Central Universities (No. 2012CXQT09) and NCET-13-0065. Corresponding author: Qingying Xue Email: [email protected]. 1

2

QINGYING XUE AND JINGQUAN YAN

where m = m(ξ, η) ∈ S  (Rn × Rn ), there are several important examples, such as Example 1. Suppose B satisfies the translation invariance property: B(τh f, τh g)(x) = τh B(f, g)(x) and let K(x, y) = F −1 (m(ξ, η))(x, y), then it is easy to see that this kernel coincides with the convolution type kernel in (1.1) for m = 2 and   K(x − y1 , x − y2 )f (y1 )g(y2 )dy1 dy2 . B(f, g)(x) = Rn

Rn

Example 2. Suppose B(f, g) is an operator of non-translation invariance, and let K(x, y1 , y2 ) = F −1 (m(x, ξ, η))(y1 , y2 ). Then  K(x, x − y1 , x − y2 )f (y1 )g(y2 )dy1 dy2 . B(f, g)(x) = R2n

Moreover, instead of considering the kernel with the form K(x, x − y1 , x − y2 ), one may consider the variable coefficient operators of the more general form K(x, y1 , y2 ), that is,  B(f, g)(x) =

R2n

K(x, y1 , y2 )f (y1 )g(y2 )dy1 dy2 ,

which is called bilinear operators with non-convolution type kernels. In [16], Grafakos and Torres defined and studied the multilinear (m-linear) singular integral operators with non-convolution type kernels as follows:  (1.2) T f(x) = K(x, y1 , · · · , ym )f1 (y1 ) · · · fm (ym )dy1 · · · dym . (Rn )m

As a multilinearization of Littlewood-Paley’s g-function, Coifman and Meyer [6] also introduced the following multilinear operator (bilinear, one dimendional).  ∞ m(t) (f ∗ φt )(a ∗ Φt ) (1.3) B(a, f ) = dt. t 0 They studied the L2 -estimate of this operator by using the notion of Carleson meaˆ have compact support, with 0 ∈ ˆ sures (with m(t) ∈ L∞ , a ∈ BM O, φˆ and Φ / supp Φ). In 1982, Yabuta [26] obtained the Lp (p ≥ 1) boundedness and BM O type estimates of B(a, f ) by weakening the assumptions in [6]. In 2001, Sato and Yabuta [20] studied the Lp1 × · · · × Lpm to Lp (m ≥ 2, p ≥ 1/m) boundedness of the following multilinear Littlewood-Paley g-function,  ∞ m dt ((ϕi )t ∗ fi )(x) . (1.4) Tg (f)(x) = t 0 i=1 The importance of the multilinear Littlwood-Paley g-function and related multilinear Littlewood-Paley type estimates were shown in PDE and other fields by Coifman, McIntosh and Meyer [3], Coifman, Deng and Meyer [2], David and Journe [8] and also by Fabes, Jerison and Kenig [9], [10], [11]. In [11], the authors studied a class of multilinear square functions and applied it to Kato’s problem. In (1.3) and (1.4), the kernels are restricted to separable variable kernels, and it is only a special case of the kernel in (1.1). Therefore, in [1], [21], [25] the authors considered the multilinear Littlewood-Paley g-function, gλ∗ function and Marcinkiewicz

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

3

integral with the same kind of convolution-type homogeneous kernel as in (1.1). It seems that the methods used in [1], [21], [25] doesn’t work for the above LittlewoodPaley operators with more general non-convolution type kernels as in (1.2), by the reason that the estimates there rely heavily on the convolution-type kernels and the well known Marcinkiewicz function studied in [14]. In this paper, we take a further step. We consider the multilinear Littewood-Paley operators with non-convolution type kernels as in (1.2). This is done by establishing some pretty much elementary inequalities which provide a foundation for our analysis, and considering a class of multilinear square function associated with the following more general non-convolution type kernels: Definition 1.1 (Integral smooth condition of C-Z type I). For any t ∈ (0, ∞), let Kt (x, y1 , . . . , ym ) be a locally integrable function defined away from the diagonal x = y1 = · · · = ym in (Rn )m+1 and denote (x, y ) = (x, y1 , . . . , ym ). We say Kt satisfies the integral condition of C-Z type I, if for some positive constants γ, A, and B > 1, the following inequalities hold:  1  ∞ A |Kt (x, y )|2 dt 2 ≤ m , (1.5) ( j=1 |x − yj |)mn 0 (1.6)



 0

∞

|Kt (z, y ) − Kt (x, y )|2 dt

 12

A|z − x|γ ≤ m , ( j=1 |x − yj |)mn+γ

whenever |z − x| ≤ B1 maxm j=1 {|x − yj |}; and  ∞  12  A|yi − yi |γ  2  |K (x,  y ) − K (x, y , . . . , y , . . . , y )| dt ≤ (1.7) t t 1 m i mn+γ ( m 0 j=1 |x − yj |) i| for any i ∈ {1, . . . , m}, whenever |yi − yi | ≤ |x−y . B The multilinear square function T is defined by   m   ∞ dt 1/2  (1.8) T (f )(x) = | Kt (x, y1 , · · · , ym ) fj (yj )dy1 . . . dym |2 , t (Rn )m 0 j=1

/ m for any f = (f1 , · · · , fm ) ∈ S(Rn ) × S(Rn ) × · · · × S(Rn ) and all x ∈ j=1 suppfj . Throughout this paper, we assume that T can be extended to be a bounded operator for some 1 ≤ q1 , · · · , qm ≤ ∞ with 1q = q11 + · · · + q1m , that is (1.9)

Lq1 × · · · × Lqm → Lq .

In fact, in the convolution type kernel case, by using the multilinearization method, we know that there is some certain relationship between the square function T and multilinear Fourier multiplier. By this fact and together with the results of GrafakosMiyachi-Tomita [15], we know that if the kernel is sufficient smooth, then T does satisfy the boundedness in (1.9), which shows that our assumption (1.9) is reasonable. More details can be found in [25]. Part of the main results of this paper for T are:

4

QINGYING XUE AND JINGQUAN YAN

Theorem 1.1. Let T be the operator defined in (1.8) with the kernel satisfying the integral condition of C-Z type I. Then T can be extended to a bounded operator from 1 L1 × · · · × L1 to L m ,∞ . Theorem 1.2. Let T be the operator defined in (1.8) with the kernel satisfying the integral condition of C-Z type I. Let p1 = p11 + · · · + p1m with 1 ≤ p1 , p2 , · · · , pm < ∞, and assume that ω  satisfies the Ap condition, then the following results hold: (i) If there is no pi = 1, then (1.10)



T (f)

Lp (νω )

m 



fi

≤C

Lpi (ωi )

.

i=1

(ii) If there is a pi = 1, then (1.11)



T (f) p,∞ L (ν

ω )

≤C

m 



fi

Lpi (ωi )

.

i=1

See Section 2 for the definition of multiple weights class Ap Now, we give some direct applications of Theorems 1.1-1.2. Two more definitions will be needed first. Definition 1.2.  Let K be a function defined on Rn × Rmn with supp K ⊆ B := 2 {(x, y1 , . . . , ym ) : m j=1 |x − yj | ≤ 1}. K is called a multilinear Marcinkiewicz kernel if for some 0 < δ < mn and some positive constants A, γ0 , and B1 > 2, (1.12)

(1.13)

A ; |K(x, y )| ≤ m ( j=1 |x − yj |)mn−δ A|yi − yi |γ0 , |K(x, y ) − K(x, y1 , . . . , yi , . . . , ym )| ≤ m ( j=1 |x − yj |)mn−δ+γ0

whenever (x, y1 , . . . , ym ) ∈ B and |yi − yi | ≤ (1.14)

1 |x B1

− yi | for all 0 ≤ i ≤ m, and

A|x − x |γ0 , |K(x, y ) − K(x , y1 , . . . , ym )| ≤ m ( j=1 |x − yj |)mn−δ+γ0

whenever (x, y1 , . . . , ym ) ∈ B and |x − x | ≤ B11 max1≤j≤m |x − yj |. 1 K( xt , yt1 , . . . , ytm ). We will always Given a kernel K, denote Kt (x, y1 , . . . , ym ) = tmn use this notation throughout this paper. The operator T associated with a multilinear Marcinkiewicz type kernel Kt is called the multilinear Marcinkiewicz integral of nonconvolution type, we denote it by μK . Remark 1.1. Assume that Ω is homogeneous of degree zero and Lipschitz continuous on (Sn−1 )m , Ω vanishes on the product of unit sphere (Sn−1 )m . Take the kernel Ω(x−y1 ,...,x−ym ) K(x, y ) = |(x−y m(n−1) χ(B(0,1))m (x − y1 , . . . , x − ym ), then it is easy to see that 1 ,...,x−ym )| K satisfies (1.12)-(1.14) and thus it is a multilinear Marcinkiewicz kernel. The operator T associated with this K coincides with the following multilinear Marcinkiewicz

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

5

integral μΩ with convolution-type kernel which has been studied in [1]:  μΩ (f)(x) =

∞ 0

 1 tm

Ω(x − y1 , . . . , x − ym ) (B(x,t))m

|(x − y1 , . . . , x − ym )|m(n−1)

2 12 dt fi (yi ) dy . t i=1

m 

Moreover, if m = 1 and n ≥ 2, then μΩ is just the well known Marcinkiewicz integral of higher dimension which was defined and studied by Stein [22] in 1958. Definition 1.3. Let K(x, y1 , . . . , ym ) be a locally integrable function defined away from the diagonal x = y1 = · · · = ym in (Rn )m+1 . K is called a multilinear LittlewoodPaley kernel if for some positive constants A, γ0 , δ, and B1 > 1, |K(x, y )| ≤

(1.15)

(1 +

A , mn+δ j=1 |x − yj |)

m

and (1.16)

|K(x, y ) − K(x, y1 , . . . , yi , . . . , ym )| ≤

whenever |yi − yi | ≤ (1.17)

1 |x B1

− yi |, for all 1 ≤ i ≤ m, and

|K(x, y ) − K(x , y1 , . . . , ym )| ≤

whenever |x − x | ≤

1 B1

A|y − yi |γ0 m i , (1 + j=1 |x − yj |)mn+δ+γ0

A|x − x |γ0 m , (1 + j=1 |x − yj |)mn+δ+γ0

max1≤j≤m |x − yj |.

The operator T associated with a multilinear Littlewood-Paley kernel is called a multilinear Littlewood-Paley g-function, we denote it by gK . If the kernel K is in the form of convolution type, which is K(x − y1 , . . . , x − ym ), then gK coincides with the operator defined and studied by Xue, Peng, Yabuta in [25]. For m = 1, n ≥ 2, gK is just the well known classical Littlewood-Paley g-function studied by Wang in [24]. For multilinear Marcinkiewicz kernel and Littlewood-Paley kernel, we obtain: Theorem 1.3. Let K be a multilinear Marcinkiewicz kernel. Then Kt satisfies the integral smooth condition of C-Z type I with 0 < γ ≤ min{δ, γ0 }. Theorem 1.4. Let K be a multilinear Littlewood-Paley kernel. Then Kt satisfies the integral smooth condition of C-Z type I with 0 < γ ≤ min{δ, γ0 }. Corollary 1.5. The results in Theorem 1.1 and Theorem 1.2 still hold for the multilinear Marcinkiewicz integral μK with 0 < γ ≤ min{δ, γ0 } and Littlewood-Paley g-function gK with 0 < γ ≤ γ0 . The above results can be extended to multilinear Littlewood-Paley gλ∗ -function, but with much more difficulty to overcome. We begin with the definition below. Definition 1.4 (Integral smooth condition of C-Z type II). For any t ∈ (0, ∞), let Kt (x, y1 , . . . , ym ) be a locally integrable function defined away from the diagonal

6

QINGYING XUE AND JINGQUAN YAN

x = y1 = · · · = ym in (Rn )m+1 . We say Kt satisfies the integral condition of C-Z type II, if for some positive constants γ, A, and B > 1,  nλ   dzdt  1 A t |Kt (z, y )|2 n+1 2 ≤ m ; (1.18) mn |x − z| + t t ( |x − y |) j Rn+1 j=1 + (1.19)  

 Rn+1 +

dzdt  1 A|x − x |γ t nλ |Kt (x − z, y )−Kt (x − z, y )|2 n+1 2 ≤ m , |z| + t t ( j=1 |x − yj |)mn+γ

whenever |x − x | ≤ B1 maxm j=1 {|x − yj |}; and  nλ   dzdt  1 t |Kt (z, y )−Kt (z, y1 , . . . , yi , . . . , ym )|2 n+1 2 |x − z| + t t Rn+1 + (1.20)

A|yi − yi |γ ≤  , m mn+γ ( |x − yj |) j=1

i| . for i ∈ {1, . . . , m}, whenever |yi − yi | ≤ |x−y B The multilinear square function associated with the above kernel K is defined by  

12  m   nλ t 2 dzdt  | Kt (z, y ) fj (yj )dy | n+1 , (1.21) Tλ (f )(x) = |x − z| + t t Rn+1 Rnm + j=1 / m whenever f = (f1 , · · · , fm ) ∈ S(Rn ) × S(Rn ) × · · · × S(Rn ) and x ∈ j=1 supp fj . We have the following results:

Theorem 1.6. Let K be a kernel satisfying integral smooth condition of C-Z type II. Then Theorem 1.1 and Theorem 1.2 still hold for Tλ with λ > 2m. Theorem 1.7. Let λ > 2m. Suppose K is a multilinear Littlewood-Paley kernel. Then Kt satisfies the integral smooth condition of C-Z type II with 0 < γ ≤ min{(λn − 2mn)/2, γ0 , n/2}. If the kernel K in (1.21) is a multilinear Littlewood-Paley kernel, we denote Tλ by gλ∗ . Then we obtain the corollary as below, Corollary 1.8. Let λ > 2m and 0 < γ ≤ min{(λn − 2mn)/2, γ0 , n/2}. Then, the results in Theorem 1.1 and Theorem 1.2 still hold for the multilinear gλ∗ -function with non-convolution type kernel. Remark 1.2. If the kernel K is in the form of convolution type, K(x−y1 , . . . , x−ym ), then gλ∗ -function coincides with the operator defined and studied by Shi, Xue, Yabuta in [21]. Moreover, if m = 1, n ≥ 2, and assume the kernel is the Poisson kernel which is a pretty much smooth function, then the operator is just the well known classical gλ∗ function of higher dimension which was first defined and studied by Stein [23] in 1961 and later studied by Fefferman [12] in 1970. The article is organized as follows. In Section 2, we give some definition and notations which will be used later. Section 3 contains the proof of Theorem 1.1. We

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

7

give the proof of Theorem 1.2 in Section 4 inspired by some methods of [19]. Proof of Theorems 1.3-1.4 will be shown in Section 5. The remainder of this paper will be devoted to prove Theorems 1.6-1.7, the results of multilinear gλ∗ function, which relies heavily on some pretty much elementary inequalities. 2. Definitions and notations We first give the definition of multiple weights Ap . Definition 2.1. ([19] Multiple weights) Let 1 ≤ p1 , · · · , pm < ∞, p satisfies p1 =  p/pi 1 + · · · + p1m . Given ω  = (ω1 , · · · , ωm ), set νω = m . We say that ω  satisfies i=1 ωi p1 the Ap condition if   p1 m  1   m  1  1−p pi 1 p/pi i ω ω < ∞, sup |B| B i=1 i |B| B i B i=1  when pi = 1,

1 |B|





1−p ω i B i

1/pi

is understood as (inf ωi )−1 . B

We will use the boundedness of the new multilinear maximal operator M which was introduced in [19] in the following way  m  1  |fi (y)|dy, M(f )(x) = sup |Q| Q x∈Q i=1 where the supremum is taken over all cubes Q containing x. We will need the sharp maximal function via Lemma 2.1. Definition 2.2. (Sharp maximal functions) (see [13] and [19]) For δ > 0, Mδ is the maximal function 

1/δ  1 δ 1/δ δ Mδ f (x) = M (|f | ) (x) = sup |f (y)| dy . Qx |Q| Q In addition, M is the Sharp maximal function of Fefferman and stein [13],   1 1

|f − c|dy ≈ sup |f − fQ |dy. M f (x) = sup inf Qx c |Q| Q Qx |Q| Q and Mδ f (x) = M (|f |δ )1/δ (x). Lemma 2.1. ([13]) Let 0 < p, δ < ∞ and ω be any Mackenhoupt A∞ weight. Then there exists a constant C independent of f such that the inequality   p (Mδ f (x)) ω(x)dx ≤ C (Mδ f (x))p ω(x)dx, (2.1) Rn

Rn

holds for any function f for which the left-hand side is finite.

8

QINGYING XUE AND JINGQUAN YAN

Moreover, if ϕ : (0, ∞) → (0, ∞) is doubling, then there is a C depending on the A∞ constant of ω and the doubling condition of ϕ, such that (2.2) sup ϕ(λ)ω({x ∈ Rn : Mδ f (x) < λ}) ≤ C sup ϕ(λ)ω({x ∈ Rn : Mδ f (x) < λ}), λ>0

λ>0

again for any f such that the left hand side is finite. 3. Proof of Theorem 1.1 Proof. For any α > 0, set Eα = {x ∈ Rn : |T (f1 , . . . , fm )(x)| > α}. We are going to show that there exists a constant C such that |Eα | ≤ Cα

1 −m

m 

1

||fi ||Lm1 .

i=1

The proof is via the Calder´on-Zygmund decomposition which has become standard for proving the weak type estimate upon some known strong estimate. By the linearity 1 of T , we can assume ||fi ||L1 = 1, i = 1, . . . , m. Apply the decomposition at level α m for fi , i = 1, . . . , m. We then  obtain gi , fi and families of disjoint cubes Qi,j such that fi = gi + bi and bi = k bi,k , where 1

||gi ||L∞ ≤ 2n α m ,

(3.1)

||gi ||L1 ≤ 1,

 (3.2)

1

suppbi,j ⊂ Qi,j , Qi,j

bi,j = 0, ||bi,ki ||L1 ≤ 2n+1 α m |Qi,ki |, |



1

Qi,ki | ≤ α− m .

ki

For i = 1, . . . , m, β = (β1 , . . . , βm ) with βi = 0 or 1, set h0i = gi , h1i = bi , and     hβ = (hβ1 1 , . . . , hβmm ). Also set Eαβ = {x ∈ / i,ki Q∗i,ki : |T (hβ )(x)| > α}, where Q∗i,ki denote the cube with five times the length and the same center of Qi,ki . Then we have |Eα | ≤



 |Eαβ |

+|

 i,ki

 j ∈{0,1} β,β

Q∗i,ki |

≤

  β

 |Eαβ |

n

+5 α

1 −m

m 

1

||fi ||Lm1 .

i=1

Let B = ||T ||qLq1 ×···×Lqm →Lq . By the Lq1 × · · · × Lqm → Lq bounededness of T , |Eα0,...,0 | ≤

m m 1 B  B  qqm q −m i ||g || ≤ α = Bα . q i i L αq i=1 αq i=1

  So it suffice to estimate |Eαβ | with β =  (0, . . . , 0). Denote E ∗ = i,ki Q∗i,ki , we will 

fuscous on the estimate T (hβ )(x) with x ∈ (E ∗ )c . Without loss of generality assume

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

9

that βi = 1, for i = 1, . . . , l, and βi = 0, for i = l + 1, . . . , m, 1 ≤ l ≤ m. We have  ∞ dt 1   β T (h )(x) = ( |Ft (hβ )(x)|2 ) 2 t 0  ∞  l m   dt 1 =( | Kt (x, y1 , . . . , ym ) bi (yi ) gi (yi )dy |2 ) 2 t Rnm 0 i=1 i=l+1 (3.3) l m   ∞    dt 1 ≤ ( | Kt (x, y1 , . . . , ym ) bi,ki (yi ) gi (yi )dy |2 ) 2 t Rnm i=1 k1 ,...,kl 0 i=l+1  =: Ik . k1 ,...,kl

Assume that Qi0 ,ki0 has the smallest length in {Qi,ki }li=1 . Then by (3.2), the cancelation condition of bi0 ,ki0 and the Minkowski’s inequality, (3.4)   l m    ∞ dt 1 | Kt (x, y1 , . . . , ym ) bi,ki (yi ) gi (yi )dy |2 ) 2 Ik = t Rnm 0 i=1 i=l+1  ∞   = | (Kt (x, y1 , . . . , yi0 , . . . , ym ) − Kt (x, y1 , . . . , ci0 ,ki0 , . . . , ym )) ×

Rnm

bi,ki (yi )

i=1

 ≤

0 l 

 Rnm l 

×

m 

gi (yi )dy |2

i=l+1



∞ 0

dt 1 )2 t

|(Kt (x, y1 , . . . , yi0 , . . . , ym ) − Kt (x, y1 , . . . , ci0 ,ki0 , . . . , ym )|2

|bi,ki (yi )|

i=1

m 

dt 1 )2 t

|gi (yi )|dy .

i=l+1

By (1.7), one obtains  Ik ≤ (3.5)

Rnm

m l   |yi0 − ci0 ,ki0 |γ  |bi,ki (yi )| |gi (yi )|dy . mn+γ ( m i=1 |x − yi |) i=1 i=l+1

Noting that if x ∈ (E ∗ )c , yi ∈ E for i ∈ {1, . . . , l}, we have |x − yi | ∼ |x − ci,ki | and l m  |x − y | ∼ |x − c | + then m i i,k i i=1 i=1 i=l+1 |x − yi |. So    |yi0 − ci0 ,ki0 |γ li=1 |bi,ki (yi )| m i=l+1 |gi (yi )| Ik ≤ C dy1 . . . dym l m mn+γ Rnm ( i=1 |x − ci,ki | + i=l+1 |x − yi |)  l m   |yi0 − ci0 ,ki0 |γ ≤C ||gi ||L∞ |bi,ki (yi )|dy1 . . . dym l (3.6) ln+γ Rnm ( i=1 |x − ci,ki |) i=1 i=l+1 ≤C

l  i=1

||bi,ki ||L1

m 

l(Qi0 ,ki0 )γ ||gi ||L∞ l . ln+γ ( |x − c |) i,k i i=1 i=l+1

10

QINGYING XUE AND JINGQUAN YAN

Therefore, by (3.2), we have γ l  |Qi,ki |1+ ln Ik ≤ Cα γ . |x − ci,ki |n+ l i=1



We are now in a position to estimate |Eαβ |.  |Eαβ |

≤



1 1

|T (h )(x)| dx ≤ C

l   

1+

γ

|Qi,kiln

 1l

dx |x − ci,ki |  γ l  l   |Qi,k |1+ lnγ  1l   |Qi,ki |1+ ln  1l i dx ≤ C γ γ dx n+ l n+ l (E∗)c i=1 k |x − ci,ki | (E∗)c k |x − ci,ki | i=1 i i

αl  =C

≤C



1 l

 β

(E∗)c

l   i=1

|Qi,ki |

 1l

(E∗)c

k1 ,...,kl i=1

n+ γl

1

≤ Cα− m .

ki



Theorem 1.1 is thus proved.

4. Proof of Theorem 1.2 To prove Theorem 1.2, we need the pointwise estimate of Mδ# (T f)(x): Lemma 4.1. Let 0 < δ < m1 , then there exists constant C such that for any bounded and compact supported functions fi , i = 1, . . . , m. (4.1)

Mδ# (T f)(x) ≤ CM(f)(x).

Proof. Fix a point x ∈ Rn and a cube Q containing x, let 0 < δ < show that there exists a constant cQ such that   1 1 |T (f)(z) − cQ |δ dx m ≤ CM(f)(x). |Q| Q

1 . m

we need to

∞ 1 Take cQ = ( 0 |cQ,t |2 dtt ) 2 . Then   1 1 |T (f)(z) − cQ |δ dz δ |Q| Q 

1δ  ∞   ∞  m  1 1 δ 1 dt dt 2 2 ( = | Kt (z, y ) fj (yj )dy | ) 2 − ( |cQ,t | ) 2 dz |Q| Q 0 t t Rnm 0 j=1 

1δ   ∞  m  δ  1 2 dt 2 ≤ | Kt (z, y ) fj (yj )dy − cQ,t | dz =: I. |Q| Q 0 t Rnm j=1

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

11

Splitting each fj as fj = fj χQ + fj χQc = fj0 + fj∞ . Write m 

m 

fj (yj ) =

j=1

(fj0

fj∞ )

+

=

=

m 

α

fj j (yj )

α1 ,...,αm ∈{0,∞} j=1

j=1 m 



fj0 (yj )

j=1

0

=f +



m 



+

α

fj j (yj )

α1 ,...,αm ∈{0,∞},∃αj ,αj =∞ j=1

fα ,

α  , α =0

 αj α where α  = (α1 , . . . , αm ), αi = 0 or ∞, fα = m j (yi ). Denote T (f )(x) = j=1 f   αm T (f1α1 , . . . , fm )(x) and let cQ,t = α ,α =0 Rnm Kt (x, y ) m y . We have j=1 fj (yj )d (4.2)

1δ    ∞  m  δ  1 0 2 dt 2 | Kt (z, y ) fj (yj )dy | dz I≤C |Q| Q 0 t Rnm j=1

1δ m   1   ∞   δ αj 2 dt 2 +C | (Kt (z, y ) − Kt (x, y )) fj (yj )dy | dz |Q| Q 0 t nm R j=1 α  , α =0  1   1  =: CI0 + C Iα ,Q (z)δ dz δ =: CI0 + C Iα . |Q| Q α  =0

α  =0

By the boundedness of T , 1 I0 = ( |Q| (4.3)

  (

∞ 0

Q

 |

Rnm

Kt (z, y )

m 

fj0 (yj )dy |2

j=1

1 dt δ ) 2 dz) δ t

≤ C T (f1 , . . . , fm L m1 ,∞ (Q, dx ) |Q|

 m  1 ≤ C T L1 ×···×L1 →L1/m,∞ |fj |dyj |Q| Q j=1 ≤ C T L1 ×···×L1 →L1/m,∞ M(f)(x).

We now estimate Iα with α  = 0. Assume without loss of generality that αj1 = / {j1 , . . . , jl }, 0 ≤ l < m. By Minkowski’s inequality · · · = αjl = 0 and αj = ∞, if j ∈ and (1.6), we have  Iα ,Q (z) ≤





Rnm



≤ Aγ

Rnm

∞ 0

(

|Kt (z, y ) − Kt (x, y )| |z − x| mn+γ j=1 |z − yj |)

m

γ

2 dt

m  j=1

m  12 

t

α

|fj j (yj )|dy

j=1 α

|fj j (yj )|dy .

12

QINGYING XUE AND JINGQUAN YAN

Thus

(4.4)

 

δ m  1 |x − z|γ αj m |fj (yj )| dy dz) δ mn+γ Q Rnm ( i=1 |x − yi |) j=1   γ ∞    |Q| n ≤ Aγ |fj (yj )|dyj |fj (yj )|dyj 1 k |Q n |)nm+γ kQ (3 Q 3 k=1 j∈{j1 ,...,jl } j ∈{j / 1 ,...,jl }  m ∞  1  1 ≤ Aγ |fj (yj )|dyj 3kγ j=1 |3k Q| 3k Q k=1

1 Iα ≤ Aγ ( |Q|

≤C

Aγ M(f)(x). γ

Combine (4.3) and (4.4). This finishes the proof of this lemma.



Lemma 4.2. Suppose sup fi ⊂ B(0, R). Then there is a constant C such that for |x| > 2R, T (f )(x) ≤ CM(f)(x). Proof. This estimate is direct by using the Minkowski’s inequalty and the size condition of the kernel. In fact,     m m   ∞  ∞ dt 1  2 dt 12 | Kt (z, y ) fj (yj )dy | ) ≤ |Kt (z, y )|2 ) 2 |fj (yj )|dy t t nm Rnm R 0 0 j=1 j=1  m  1 m ≤ |fj (yj )|dy mn Rnm ( j=1 |x − yj |) j=1 ≤ CM(f)(x).  Proof of Theorem 1.2 Proof. Now we are in a position to prove Theorem 1.2. Based on Lemma 4.1 and the above estimate, the proof of Theorem 1.2 is similar as the proof of Corollary 3.9 in [19] and will only be given for the reader’s convenience. Without loss of generality, we may suppose fi > 0 and f ∈ (Cc∞ (Rn ))m . Similar arguments as in [19] also allow p   us to assume that Rn T (f) υω dx < ∞. By Lemma 2.1, we only need to prove that

1/p  p (Mρ (T (f))) νw < ∞. (3.2) Rn

Since w ∈ A∞ , there exists p0 > 1, such that w ∈ Ap0 . We can take ρ > 0 small enough and p/ρ > p0 such that w ∈ Ap/ρ . The Lp/ρ bounds of M and (4.8) yield

1/p

1/p   p p (Mρ (T (f))) νw ≤C (T (f)) νw < ∞. Rn

Rn

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

13

Thus, we obtain the desired estimates by applying Lemma 2.2, 

p1   

p1    p p1   p   p (T (f)) νω ≤ ≤ Mρ T (f) νω Mρ T (f) νω Rn

Rn

 ≤C

Rn



p M(f) νω

p1

≤C

Rn  m   i=1

Rn

p i fi ωi

p1

i

.

The proof of Theorem 1.2 (ii) can be treated as that of 1.2 (i) with only a slight modifications, we omit its proof here.  5. Proof of Theorems 1.3-1.4 Proof of Theorem 1.3. Proof. Let K be a multilinear Marcinkiewicz kernel. To prove (1.5), recalling the condition (1.12),    ∞ 1  ∞ x y dt  1 2 dt 2 |Kt (x, y )| = |K( , )|2 χ(x,y)∈tB (t) 2mn+1 2 t t t t 0 0  ∞  1 1 1  t−2δ−1 dt 2 m ∼ m . mn−δ 1 m ( j=1 |x − yj |) ( j=1 |x − yj |)mn ( j=1 |x−yj |2 ) 2 To prove Kt satisfies (1.6), write   ∞ dt  12 |Kt (z, y ) − Kt (x, y )|2 t 0   ∞ x y z y dt  1 = |K( , ) − K( , )|2 χ(z,y)∈tB,(x,y)∈tB (t) 2mn+1 2 t t t t t 0  1  ∞  z y dt 2 + |K( , )|2 χ(z,y)∈tB,(x,y)∈tB / (t) 2mn+1 t t t 0   ∞ x y dt  12 + |K( , )|2 χ(z,y)∈tB,(x, . / y )∈tB (t) 2mn+1 t t t 0  By (1.12) and (1.13), and assuming without loss of generality that ( m j=1 |x −  1 1 m yj |2 ) 2 ≤ ( j=1 |z − yj |2 ) 2 ,   ∞ dt  12 |Kt (z, y ) − Kt (x, y )|2 t 0  2γ0  ∞ |z − x| t2δ dt  12 m ≤A χ (t) (z, y )∈tB,(x, y )∈tB ( j=1 |x − yj |)2mn+2γ0 −2δ t 0  ∞  1 t2δ dt  12 m +A χ (t) (z, y ) ∈tB,(x, / y )∈tB ( j=1 |x − yj |)2mn−2δ t 0 =: I + II.

14

QINGYING XUE AND JINGQUAN YAN

As |z − x| ≤ 12 maxm j=1 {|x − yj |}, (

m 

2

1 2

|x − yj | ) ∼

j=1

m 

|x − yj | ∼

j=1

m 

|z − yj | ∼ (

j=1

m 

1

|z − yj |2 ) 2 .

j=1

We can estimate part I by   12  ∞ |z − x|γ0 CA|z − x|γ0 −2δ−1   t dt ≤ . I≤A  mn+γ0 −δ mn+γ0 1 ( m ( m 2 2 ( m j=1 |x − yj |) j=1 |x − yj |) j=1 |z−yj | )  m 2 δ 2 δ δ Using the fact that ( m j=1 |x − yj | ) − ( j=1 |z − yj | ) ≤ C|x − z| , it follows 

II ≤ A



m

(

m

(

1

j=1

j=1

|z−yj |2 ) 2

t−2δ−1 dt

1

 12

|x−yj |2 ) 2

(

1 CA|z − x|δ  ≤ . mn−δ mn+δ ( m j=1 |x − yj |) j=1 |x − yj |)

m

Taking γ ≤ min{δ, γ0 }, we get   ∞ 1 |z − x|γ |Kt (z, y ) − Kt (x, y )|2 dt 2 ≤ CA m . ( j=1 |z − yj |)mn+γ 0 The proof of Kt satisfying (1.7) is similar. By (1.12) and (1.13),   ∞ dt 1 |(Kt (x, y1 , . . . , yi , . . . , ym )(x) − Kt (x, y1 , . . . , yi , . . . , ym )(x)|2 ) 2 t 0  ∞  γ  1 |yi − yi | 0 ≤A t−2δ−1 χ(x,y)∈tB χ(x,yi )∈tB dt 2 m ( j=1 |x − yj |)mn+γ0 −δ 0  ∞   12 1 m +A t−2δ−1 χ(x,y)∈tB χ(x,yi )∈tB / dt ( j=1 |x − yj |)mn−δ 0   ∞ −2δ−1  12 1 m +A t χ(x,y)∈tB . / χ(x, yi )∈tB dt ( j=1,j =i |x − yj |)mn−δ 0 =: I  + II  + III  . As |yi − yi | ≤ |x − yi |/B1 , we have |x − yj | ∼ |x − yi | and then (

m 

2

1 2

|x − yj | ) ∼

j=1

m 

|x − yj | ∼

j=1

∼(

m 

m 

|x − yj | + |x − yi |

j=1,j =i 1

|x − yj |2 + |x − yi |2 ) 2 .

j=1,j =i

So  A|yi − yi |γ0 I = m mn+γ −δ 0 ( j=1 |x − yj |) 

CA|yi − yi |γ0 ≤ m . ( j=1 |x − yj |)mn+γ0



∞ 1 m 1  2 2 2 2 2 max{( m j=1 |x−yj | ) ,( j=1,j=i |x−yj | +|x−yi | ) }

t−2δ  12 dt t

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

15

For part II  and III  , noticing that |(

m 

2

2 δ

|x − yj | + |x − yi | ) − (

j=1,j =i

we have II  + III  ≤ C|yi − yi |δ /( (1.7) with γ ≤ min{δ, γ0 }.

m j=1

m 

|x − yj |2 )δ | ≤ |yi − yi |δ ,

j=1

|x − yj |)mn+δ . Thus we finish the proof of 

Proof of Theorem 1.4. Proof. To prove (1.5), recalling the condition (1.15),    ∞ 1  ∞ x y dt  1 2 dt 2 |Kt (x, y )| = |K( , )|2 2mn+1 2 t t t t 0 0   ∞  12 1 2δ−1 m  t dt (t + j=1 |x − yj |)2mn+2δ 0   12  ∞ t2δ−1 1 1  =A dt ∼ m . m mn 2mn+2δ+2γ 0 ( j=1 |x − yj |) (t + 1) ( j=1 |x − yj |)mn 0 By (1.16), for i ∈ {0, 1, . . . , m},   ∞ dt  12 |Kt (y0 , y1 , . . . , yi , . . . , ym ) − Kt (y0 , y1 , . . . , yi , . . . , ym )|2 t 0    ∞ y0 yi ym y0 y ym 2 dt  12 = |K( , . . . , , . . . , ) − K( , . . . , i , . . . , )| 2nm+1 t t t t t t t 0   2γ0 1  ∞  |y − yi | m i ≤A t2δ−1 dt 2 (t + j=1 |y0 − yj |)2mn+2δ+2γ0 0   12  ∞ t2δ−1 |yi − yi |γ0  =A dt m ( j=1 |y0 − yj |)mn+γ0 0 (t + 1)2mn+2δ+2γ0 |yi − yi |γ0 ≤ CA m . ( j=1 |y0 − yj |)mn+γ0 This proves that Kt satisfies the integral condition of C-Z type I with γ ≤ γ0 .



6. Proof of Theorems 1.6-1.7 First, we give some elementary inequalities which provide a foundation for our analysis in the proof of our Theorems; we list them in the following lemma for simplicity. Lemma 6.1. (i). Let θ > 0, αi > 0 if i ∈ {1, . . . , N }, α0 = 0, 0 < A1 ≤ A2 · · · ≤ AN . Assume for some k ∈ {1, . . . , N }, α1 + · · · + αk−1 < 1 + θ < α1 + · · · + αk , then 

∞

(6.1) 0

 1+θ− k−1 α

i=1 i sθ Ak ds ∼ .  N αi (s + A1 )α1 . . . (s + AN )αN i=k Ai

16

QINGYING XUE AND JINGQUAN YAN

(ii). For θ > n, A > 0, C > 0,  (6.2) |z|≤A

1 AC n 1 ) . dz ∼ ( (|z| + C)θ Cθ A + C

(iii). Let θ1 , θ2 > 0, A ≤ B. Denote θ0 = min{θ1 , θ2 }, θ3 = max{θ1 , θ2 } > n, then  dy  An−θ3 (|x| + B)−θ0 . (6.3) θ θ 1 2 Rn (A + |y|) (B + |x − y|) Proof. (i). Since the left side of (6.1) can be decomposed as follows: 



A1

A2

+

( 0

A1

 +··· +

∞ AN

) N

sθ

αi i=1 (s + Ai )

ds =:

N 

Ii .

i=1

Note  that   −αi A1 θ −αi θ+1 s ds ∼ N A1 . I0 ∼ N i=1 Ai i=1 Ai 0  N  N −αi A2 θ−α1 i 1 1 I1 ∼ i=2 Ai s ds ∼ i=2 A−α (Aθ+1−α − Aθ+1−α ). 2 1 i A1 .. .  k−1    N θ+1− k−1 θ+1− i=1 αi −αi Ak −αi θ− k−1 i=1 αi i=1 αi ds ∼ Ik−1 ∼ N A s A (A − A ). k k−1 i=k i i=k i Ak−1 k k   N  θ+1− i=1 αi θ+1− α Ak+1 θ− k αi −αi i i=1 Ik ∼ i=k+1 A−α s ds ∼ N (Ak − Ak+1 i=1 i ). i i=k+1 Ai Ak

.. .  −1  −1  AN θ−N −1 α θ+1− N θ+1− N −αN i=1 αi i=1 αi N i i=1 s ds ∼ A (A − A ). IN −1 ∼ A−α N N N −1 N AN −1 N  ∞ θ−N α θ+1− i=1 αi i=1 i ds ∼ A IN ∼ A N s . N Thus, (6.1) follows by summing up the above estimates from I0 to IN . (ii). The proof of (6.2) is quite direct.





A

rn−1 dr = C n−θ θ (r + C)



A C

rn−1 1 AC n dr ∼ θ ( ) , θ (r + 1) C A+C |z|≤A 0 0  D rθ1 where we use the easy estimate that if θ1 > −1, and 1 + θ1 < θ2 , then 0 (r+1) θ2 dr  A θ1 +1 ( A+1 ) . (iii). Note that θ3 = max{θ1 , θ2 }, θ0 = min{θ1 , θ2 }. (6.3) can be based on the much simpler case when n = 1 and A = B = 1, which means, if θ3 > 1, then  dy  (|x| + 1)−θ0 . (6.4) θ θ 1 2 R (1 + |y|) (1 + |x − y|) 1 dz = (|z| + C)θ

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

17

Assuming (6.4) holds, it follows   dy dy ≤ θ θ θ θ2 1 2 1 R (A + |y|) (B + |x − y|) R (A + |y|) (A + |x + B − A − y|)  dy = A1−θ1 −θ2 (6.5) x+B−A θ 1 − y|)θ2 R (1 + |y|) (1 + | A |x| + B −θ0 ∼ A1−θ1 −θ2 ( = A1−θ3 (|x| + B)−θ0 . ) A For the general dimension case, by Rotation,  dy θ1 θ2 Rn (A + |y|) (B + |x − y|)  dy1 . . . dym−1 dym ∼ . θ1 θ2 Rn (A + |y1 | + · · · + |ym−1 | + |ym |) (B + |y1 | + · · · + |ym−1 | + ||x| − ym |) By (6.5), the term in the right side of the above inequality can be control by a constant multiplication of  (A + |y1 | + · · · + |ym−1 |)1−θ3 (B + |y1 | + · · · + |ym−1 | + |x|)−θ0 dy1 . . . dym−1 . Rn−1

A repeated application of (6.1) will then finish the proof, recalling θ3 > n. It remains to establish inequality (6.4), and it suffices to assume that |x| is much bigger than one and θ1 ≤ θ2 . Therefore, (6.6) dy θ θ2 1 R (1 + |y|) (1 + |x − y|)  ∞  ∞ dy dy  + θ θ θ 1 2 1 (1 + |y|) (1 + |x − y|) (1 + |y|) (1 + |x + y|)θ2 0 0  |x|  3|x| 2 2 dy 1 1 −θ2 −θ1  |x| + |x| ( + )dy θ θ θ2 2 |x| (1 + |y|) 1 (1 + |x − y|) (1 + |x + y|) 0 2  ∞ dy + 3|x| |y|θ1 +θ2 2  |x| 2 dy −θ2  |x| + |x|−θ1 (|x|1−θ2 + 1) + |x|1−θ1 −θ2 . θ1 (1 + |y|) 0  |x| dy In cases θ1 < 1, θ1 = 1 and θ1 > 1, 0 2 (1+|y|) θ1 can be controlled respectively 1−θ1 , ln |x| and one. In any case, it follows that by a constant multiplication of |x|  |x| dy 2  |x|θ2 −θ1 . Therefore, the last inequality in (6.6) is controlled by |x|−θ1 . 0 (1+|y|)θ1 (6.4) is thus proved.   ∞  t α  dzdt tδ Denote IA,D,E = 0 D+t . |z|
18

QINGYING XUE AND JINGQUAN YAN

Corollary 6.2. Assume there exits a constant 0 < ε < 12 such that 0 < A ≤ εD, and β < α, δ < n, then 1 A )min{α−β,n} . IA,D,E  ( β (D + E) D + E Proof. By (6.2),



IA,D,E ∼ A

∞

n 0

tα+δ−n−1 dt. (D + t)α (t + E)β+δ−n (A + E + t)n

By (6.1), if D < E, , IA,D,E ∼

An ; E β (A + E)n

IA,D,E ∼

An ; Dβ (A + E)n

if E ≤ D ≤ A + E, if E ≤ A + E ≤ D, and α < β + n,

An (A + E)α−β 1 A + E α−β A n = ( ( ) ) ; Dα (A + E)n Dβ D A+E if E ≤ A + E ≤ D, and α > β + n, An IA,D,E ∼ β+n . D As 0 < A ≤ εD, if E ≤ D ≤ A + E, we have D ∼ E. IA,D,E ∼



Proof of Theorem 1.6. Proof. The proof of Theorem 1.6 will be omitted by the reason that it is almost the same as the proof of Theorem 1.1-1.2 with few modification.  Proof of Theorem 1.7.  t nλ  12   |Kt (z, y )|2 tdzdt , and Proof. Denote I = n+1 |x−z|+t Rn+1 +    t nλ dzdt  1 |Kt (x − z, y ) − Kt (x − z, y )|2 n+1 2 , I0 = |z| + t t Rn+1 +    nλ dzdt  1 t |Kt (z, y ) − Kt (z, y1 , . . . , yi , . . . , ym )|2 n+1 2 , Ii = |x − z| + t t Rn+1 + for 1 ≤ i ≤ m. By (1.15),  nλ   t2δ dzdt  12 t m . I 2mn+2δ tn+1 |x − z| + t (t + |z − y |) j Rn+1 j=1 + m  |a − b | ∼ Using the simple geometric fact m j j=1 j=2 |b1 − bj | + |a − b1 |, which we will use several times again, we have  nλ  t2δ dzdt t 2 m I  . 2mn+2δ |x − z| + t (t + j=2 |y1 − yj | + |z − y1 |) tn+1 Rn+1 +

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

19

Assuming without loss of generality that nλ < 2mn + 2δ, by (6.3),  ∞ tn−2mn−2δ tnλ+2δ dt 2 m I  nλ n+1 (t + j=2 |y1 − yj | + |x − y1 |) t 0  ∞ tnλ−2mn−1 1 m ∼ dt ∼ m . nλ (t + j=1 |x − yj |) ( j=1 |x − yj |)2mn 0 We now estimate Ii . By (1.15) and (1.17),  ∞ nλ  t2δ dzdt t 2 m Ii  2mn+2δ |z−y | |x − z| + t (t + j=1 |z − yj |) tn+1 0 |yi −yi |≥ |B |i 1  ∞  nλ t2δ dzdt t m +  |z−y | |x − z| + t (t + j=1,j =i |z − yj | + |z − yi |)2mn+2δ tn+1 |yi −yi |≥ |B |i 0 1  ∞ nλ  t2δ dzdt t  2γ0 m + |yi − yi | 2mn+2δ+2γ 0 |z−y | |x − z| + t (t + j=1 |z − yj |) tn+1 |yi −yi |< |B |i 0 =

2 Ii,1

+

2 Ii,2

+

1

2 Ii,3 .

i| i| , if |yi − yi | ≥ |z−y , we have Assume that B1 /B ≤ 1/2. Since |yi − yi | < |x−y B |B1 | B1 1  |z − yi | < B |x − yi | ≤ 2 |x − yi |. Thus |x − z| ∼ |x − yi | ∼ |x − yi |. Then

2 Ii,1

 ∼

∞ 0



nλ t |x − yi | + t

 |yi −yi |≥

|z−yi | |B1 |

(t +

t2δ dzdt . 2mn+2δ tn+1 j=1 |z − yj |)

m

Use the geometric fact mentioned above, and then (6.2),   ∞ nλ  t2δ dzdt t 2 m Ii,1 ∼ . 2mn+2δ tn+1 |z−y | |x − yi | + t |yi −yi |≥ |B |i (t + 0 j=1,j =i |yi − yj | + |z − yi |) 1

By Corollary 6.2, and note that |yi − yi | < |x − yj |, 0 < γ ≤ n/2 min{(λ − 2m), 1}, it yields that 2 Ii,1

|yi − yi |min{λn−2mn,n}

|yi − yi |2γ

    . m m ( |x − yj |)2mn+min{λn−2mn,n} ( |x − yj |)2mn+2γ j=1

j=1

The estimate of Ii,2 is similar and will be omitted. Now we estimate Ii,3 . (6.7)  tnλ+2δ dzdt 2  2γ0  Ii,3  |yi − yi | m nλ 2mn+2δ+2γ 0 (|z| + t) (t + j=1 |z + x − yj |) tn+1 Rn+1 +  tnλ+2δ dzdt  2γ0  ∼ |yi − yi | . m nλ 2mn+2δ+2γ 0 (|z| + t) (t + j=2 |y1 − yj | + |z + x − y1 |) tn+1 Rn+1 +

20

QINGYING XUE AND JINGQUAN YAN

By (6.3), as nλ < 2mn + 2δ + 2γ0 ,  ∞ tnλ+2δ 1 dt 2  γ0 m Ii,3  |yi − yi | 2mn+2δ+2γ −n nλ n+1 0 t (t + j=2 |y1 − yj | + |x − y1 |) t 0   ∞ rnλ−2mn−2γ0 −1  12 |yi − yi |2γ0 ∼ m dr (6.8) ( j=1 |x − yj |)2mn+2γ0 0 (1 + r)nλ |yi − yi |2γ0 ∼ m . ( j=1 |x − yj |)2mn+2γ0 The estimate of I0 is similar. By (1.15) and (1.17),  ∞  t nλ t2δ dzdt 2  I0  m 2mn+2δ max{|x−yj −z|} |z| + t (t + j=1 |x − yj − z|) tn+1 |x−x |≥ 0 |B1 |  ∞  t nλ t2δ dzdt  + m  2mn+2δ max{|x−yj −z|} |z| + t (t + j=1 |x − yj − z|) tn+1 |x−x |≥ 0 |B1 |  ∞  t nλ |x − x |2γ0 t2δ dzdt  + m 2mn+2δ+2γ max{|x−yj −z|} 0 tn+1 |z| + t (t + j=1 |x − yj − z|) |x−x |< 0 |B | =

2 I0,1

+

2 I0,2

+

2 I0,3 .

1

max{|x−y −z|}

j As |x − x | < B1 max{|x − yj |}, if |x − x | ≥ , we have max{|x − yj − z|} < |B1 | B1 max{|x − yj |} ≤ 12 max{|x − yj |}. Now assume max{|x − yj |} = |x − yi |. Thus B |z| ∼ |x − yi |. Then  ∞   nλ t2δ dzdt t 2  I0,1 ∼ m 2mn+2δ max{|x−yj −z|} |x − yi | + t (t + j=1 |x − yj − z|) tn+1 |x−x |≥ 0 |B1 |   ∞ nλ  t2δ t−(n+1) dzdt t m  . 2mn+2δ |x−y −z| (t + |x − yi | + t |x−x |≥ |Bi | 0 j=1,j =i |yi − yj | + |x − yi − z|) 1

Again, by Corollary 6.2, it yields that |yi − yi |2γ 2   . I0,1 m 2mn+2γ ( |x − yj |) j=1

The estimate of I0,2 is similar with I0,2 and I0,3 similar with Ii,3 , both will be omitted. Thus, we finish the proof of Theorem 1.7.  Acknowledgements. The authors want to express their sincerely thanks to the referee for his or her valuable remarks and suggestions which made this paper more readable. References [1] X. Chen, Q. Xue and K. Yabuta, On Multilinear Littlewood-Paley Operators, submitted. [2] R. R. Coifman, D. Deng and Y. Meyer, Domains de la racine carr´ee de certains op´erateurs diff´erentiels accr´etifs, Ann. Inst. Fourier (Grenoble). 33 (1983), 123-134.

ON MULTILINEAR SQUARE FUNCTION AND ITS APPLICATIONS

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QINGYING XUE AND JINGQUAN YAN

Qingying Xue, School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China E-mail address: [email protected] Jingquan Yan, School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China E-mail address: [email protected]