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Academic research paper on topic "Weighted Estimates for Maximal Commutators of Multilinear Singular Integrals"

Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2012, Article ID 128520,20 pages doi:10.1155/2012/128520

Research Article

Weighted Estimates for Maximal Commutators of Multilinear Singular Integrals

Dongxiang Chen and Suzhen Mao

Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China Correspondence should be addressed to Dongxiang Chen, chendx020@yahoo.com.cn Received 11 July 2012; Accepted 1 September 2012 Academic Editor: Ti- Xiao

Copyright © 2012 D. Chen and S. Mao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper is concerned with the pointwise estimates for the sharp function of the maximal multilinear commutators and maximal iterated commutator 7^, generalized by m-linear operator T and a weighted Lipschitz function b. The (Lp1 (p) x ■■■ x Lpm (p),Lr (p1-r)) boundedness and the (LP1 (p) x ■■■ x LPm (p),Lr (p1-mr)) boundedness are obtained for maximal multilinear commutator T^ and maximal iterated commutator T^, respectively.

1. Introduction and Notation

The theory of multilinear Calderon-Zygmund singular integral operators,originated from the work of Coifman and Meyers', has an important role in harmonic analysis. Its study has been attracting a lot of attention in the last few decades. So far, a number of properties for multilinear operators are parallel to those of the classical linear Calderon-Zygmund operators but new interesting phenomena have also been observed. A systematic analysis of many basic properties of such multilinear operators can be found in the articles by Coifman and Meyer [1], Grafakos and Torres [2-4], and Lerner et al. [5]. So we first recall the definition and results of multilinear Calderon-Zygmund operators as well as the corresponding maximal multilinear operators.

Definition 1.1. Let T be a multilinear operator initially defined on the m-fold product of Schwartz space and taking values into the space of tempered distributions:

T : S(Rn) x ■■■ x S(Rn) S'(Rn).

Following [2], we say that T is an m-linear Calderon-Zygmund operator if for some 1 < qj < go, it extends to a bounded multilinear operator from Lqi x ••• x Lqm to Lq, where 1/q = (1/q1) + - - + (1/qm), and if there exists a function K, defined off the diagonal x = y1 = ••• = ym in (M")m+1, satisfying

T(f1,...,fm) (x) = I K(x,y1,...,ym)f1(y1) •"fm(ym)dy1 ••• dym, (1.2)

J (R»)m

for all x / fXjhsuppfj.

lK(y,y1.....ym) I < --rmn, (1.3)

(sm/jyfc - yi o

1 / X1 A\yj -y'j\

K(y0,...,yj,...,ym) - ^y0,...,y'j ,...,ym)| < --' (L4)

(X^- yi1)

for some e > 0 and all 0 < j < m, where |yj - yj| < (1/2) max0<k<m|yj - yk|. The maximal multilinear singular integral operator was defined by

r(f) (x) = sup | Tg (f1.....fm) (x) |, (1.5)

where Tg is the smooth truncation of T given by

T6(f1,...,fm) (x) = K(x,y1,...,ym)f1(y1) ••• fm(ym) dy1 ••• dym.

J Ix-y |2+---+ x—2>62

As pointed in [4], T*(f) is pointwise well defined when fj e Lqj(Rn) with 1 < qj < g.

The study for the multilinear singular integral operator and its maximal operators attracts many authors' attention. For maximal multilinear operator T*, one can see [4] for details. We list some results for T* as follows.

Theorem A (see [4]). Let 1 < qj < g and q such that 1/q = (1/q1) + ••• + (1/qm) and w e Aq1 n ••• n Aqm. Let T be an m-linear Calderon-Zygmund operator. Then there exists a constant Cnq < g, such that for all f = (f 1,..., fm) satisfying

) < Cnq(A + W fj ||

Lq> (w)' (1.7)

where W is the norm of T in the mapping T: L1 x ••• L1 ^ L1/m,G°.

Theorem B (see [4]). Let T be an m -linear Calderon-Zygmund operator. Then, for all exponents p,p1,. ..,pm, satisfying (1/p1) + ••• + (1/pm) = 1/p, one has

T* : Lpi x ••• x LPm LP, (1.8)

when 1 <p1,...,pm <<x>, one also has

T* : Lpi x ••• x Lpm Lp,co, (1.9)

when at least one pj is equal one. In either cases the norm of T* is controlled by a constant multiple of A + W.

Definition 1.2 (see [5] (commutators in the jth entry)). Given a collection of locally integrable function b = (bi,...,bm), we define the commutators of the m-linear Calderon-Zygmund operator T to be

M/l.....fm) =£ jf), (1.10)

where each term is the commutator of bj and T in the j th entry of T, that is

Tl,(f) = j f.....fj.....fm) - T f.....bjfj.....fm)- (1-11)

In [6], the following more general iterated commutators of multilinear Calderon-Zygmund operators and pointwise multiplication with functions in BMO were defined and studied in products of Lebesgue spaces, including strong type and weak end-point estimates with multiple Ap weights. That is,

Tnb (/) (x) = [b1, [b2.....[bm-1, [bm, T]m]m-J J 1

= K(X,y1,...,ym)Yl(bj(x) - bj(Vj))f1(y1) ■■■fm{Vm)dy1 ••• dym.

J (R")m ,=1

(1.12)

For the operator [b,T], when T is the Calderon-Zygmund singular integral operator and b e Ap (Mn) (the homogeneous Lipschitz spaces), Paluszyrtski [7] established the (Lp,Lq) boundedness with 1 <p < 1/p and 1/q = 1/p + p/n. Hu and Gu [8] extended this results to the case: b e Lipp^ with ¡i e A\.

Now we present the definitions of two classes of maximal commutators of multilinear singular integral operators. One is

Tib(f) (x)

2 K(X'V1' ym)f\{y\) ••• (bj (x) - bj(yj))fj(yj) •••fm(ym)dy

j=\J 25=1 |x-yi|2>62

(1.13)

the other is

Tnb (/) (x) = s5up| ib1' [b2, [bm-1, [bm, TS ]m]m-U ■■■] 2] 1 (/) (x) |

^X,yi,...,ym^Yl(b) (X) - bj (Vj)) fl( y 1) fm{Vm)dy

■1rm=i\x-yi\2>62 j=1

(1.14)

where dy = dyi ■ ■ ■ dym. It is obvious to see that

T^bf (x) <X (x). (1.15)

The main purpose of this paper is to extend the results in [8] to the maximal commutators generated by multilinear singular integrals T and Lip^ ^ functions b. We can formulate our result as following.

Theorem 1.3. Assume that the kernel K satisfies (1.3) and (1.4). Let 1 <q1,..., qm, q < to be given numbers satisfying 1/q = (1/q1) + ••• + (1/qm). And assume that T maps Lq1 (Rn) x ••• x Lqm (Rn) to Lq (Rn). For j e {1,...,m} and let 1/r = (1/p) - (fi/n), 1 <p <r < to, 0 < fi < 1, and 1/p = 1/p1 + ••• + 1/pm with 1 <pi < to, i = 1,...,m. Given ¡i such that ¡i e A1(Rn) and bj e Lip^(Rn) (j = 1,..., m), then one has

IIw)IL(¡1-r) * C\\biK>Uo, j=1.....m (L16)

From (1.15) and (1.16), one can get

IML (¡1-r) * j Ibj11 Lip Pi ^ w. (1.17)

If i = 1, one can get the following.

Theorem 1.4. Assume that the kernel K satisfies (1.3) and (1.4). Let 1 <q1,..., qm, q < to be given numbers satisfying 1/q = (1/q1) + ••• + (1/qm). And assume that T maps Lq1 (Rn) x ••• x Lqm (Rn) to Lq(Rn). For j e {1,...,m} and let 1/r = (1/p) - (fi/n), 1 <p < r < to, 0 < fi < 1 and 1/p = (1/p1) + ••• + (1/pm) with 1 <pi < to, i = 1,...,m. Set bj e Lipp (Rn)(j = 1,...,m), then one has

Kf) II Lr (Rn) * cibj kii IMIlpi (R^ j = 1.....m. (1.18)

Journal of Function Spaces and Applications From (1.15), one can get

< IN j=1

\uPn m

\Lpi (

(1.19)

The following theorem states the weighted estimates with two different weights for maximal iterated commutator of multilinear singular integrals.

Theorem 1.5. Assume that the kernel K satisfies (1.3) and (1.4). Let 1 <q1,..., qm, q < to be given numbers satisfying 1/q = (1/qi) + ■ ■ ■ + (1/qm). And assume that T maps Lqi (Rn) x ■ ■ ■ x Lqm (Rn) to Lq(Rn). Let 1/ri = (1/p) - (pi/n), 1 < pt < n < to, 0 < pi < 1, i = 1,...,m with 1/p = (1/p1) + ■ ■ ■ + (1 /pm), 1/r = (1/r1) + ■ ■ ■ + (1/rm), and p = p1 + ■ ■ ■ + pm, 0 < p < 1. Given ¡i such that ¡i e A1(R") and hi e Lip„, ¡(Rn)(i = 1,.. .,m), then one has

Tnb(/)ILi-m0 < CYI\Ml^\\/¿\\L,(,y (1-.20)

^ ' ¿=1

Similarly as Theorem 1.4, one also obtains the unweighted estimates of maximal iterated commutators.

Theorem 1.6. Assume that the kernel K satisfies (1.3) and (1.4). Let 1 <q1,..., qm, q < to be given numbers satisfying 1/q = (1/q1) + - ■ ■ + (1/q m). And assume that T maps Lqi (Rn) x ■ ■ ■ x Lqm (Rn) to Lq(R"). Let 1/ri = (1/pi)-(pi/n), 1 <pi<n< to, 0 < pi < 1, i = 1,...,mwith 1/p = (1/p1 ) + ■ ■ ■ +(1/pm), 1/r = (1/r1) + ■ ■ ■ + (1/rm), and p = p1 + ■ ■ ■ + pm, 0 < p < 1. Set hi e Lipp,(Rn)(i = 1,...,m), then one has

< ^\MlíP(,

(1.21)

The rest of this paper is organized as follows. In Section 2, we recall some standard definitions and lemmas. Section 3 is devoted to the proof of our theorems. Throughout this paper, we use the letter C to denote a positive constant that varies line to line, but it is independent of the essential variable. For any 1 < p <to, the p' is always used to denote the dual index such that (1/p) + (1/p') = 1.

2. Preliminaries

A nonnegative function fi defined on Rn is called weight if it is locally integrable. A weight ^ is said to belong to the Muckenhoupt class Ap (Rn), 1 <p < to, if there exists a constant C such that

sup^T1 í u(x)dx}(T1 Í u(x)-1/(p-1) dx\V < C< to, (2.1)

b \IBI jb / MBI jb /

for every ball B c R". A weight i is said to belong to class A1 (R") if

i i(x)dx^ < C inf i(x), almost all x e R", (2.2)

\|B| Jb / xeB

for every ball B 3 x. The class Ato(R") can be characterized as Ato = UKp<TO Ap.

Many properties of weights can be found in the book [9], we only collect some of them in the following lemma which will be used bellow.

Lemma 2.1. (i) Ap c Aq for 1 < p < q <to;

(ii) if i e Ai, then i0 e Ai for 0 < 0 < 1;

(iii) for 1 <p < to, i e Ap if and only if e Ap.

A locally integrable function f belongs to the weighted Lipschitz space Lip/^R") for 1 < p < to, 0 < p < 1 and i e Ato if

1 / i f , \ 1/P

sup—--(- If (x) - fB\pi(x)l-pdx) < C < to. (2.3)

B3x l(B)" \B J B /

The smallest bound C satisfying (1.19) is then taken to be the norm of f denoted by ||f ||Lipp .

Put Lip = Lipp,i.

If i e A1r b e Lipp f(0 < p < 1), from the definition of ||f 11 Lipp , it is obvious to see

|bB - b2k+1 BI < Cki(x)i(B(x, 2fc+1K))/7"||b||LipAi, (2.4)

where bB = (17|B|) jBb(y) dy.

The important properties of the weights are the weighted estimates for the maximal function, the sharp maximal function and their variants. One first recalls the maximal function defined by

M(f )(x) = sup -If |f (y)|dy, (2.5)

B3x |B| Jb

It is well known that for 1 <p < to, M maps Lp(i) into itself if and only if i e Ap, see [10]. The sharp maximal function is defined by

M#(f)(x) = sup-1 f |f (x) - fB\dy « sup infj1 f |f (x) - c\dy. (2.6)

B3x |B| Jb B3x c |B| Jb

One also recalls the variants M5(f)(x) = (M(|f |5)(x))175, and M#s(f )(x) = (M#(|f |5)(x))175. We denote the weighted fractional maximal operators by

Maissf (x) = sup

^fcipw í. wriM«)"'- (27)

Recall that Ma := Ma/1/1 is the weighted fractional maximal operators, that is

Ma(f)(x) = sup( l(a/n}\B\f ШуШу)- (2-8)

The following lemmas are all from [ 1 1 ].

Lemma 2.2 (Kolmogorov's inequality). Let (X, ¡) be a probability measure space and let 0 <p < q < to then there exists a constant C = Cpq such that for any measurable function f

< C\\f II

Lq^(y)-

Lemma 2.3. Let 0 < p, 6 < to and ¡i e Ato(R"), there exists C > 0 depending on the Ato(R") constant of ¡i such that

\\mf \\l%u) < c\\mflw' (2-10)

for any function f for which the left side of the above inequality is finite.

Lemma 2.4. Suppose that 0 < a <n, 0 < s <p < a/n, 1/q = (1/p) - (a/n). If y € AOT(Mn), then there exists a constant C = Cpq such that for any measurable function f

\Ma,y,s(f) \L(u) < C\\f \\

Ilp(U)•

(2.11)

Lemma 2.5. Suppose that 0 < a <n, 0 < s < p < a/n, 1/q = (1/p) - (a/n). If y € A1+(q/p<) (Rn), then there exists a constant C = Cpq such that for any measurable function f

maf \\l,(u) < C\\f \\lp yp/qy (2.12)

3. Two Estimates for Maximal Multilinear Commutators

We will prove our theorems in this section. To begin, we prepare another two iterated operators to control the commutators.

Let e Cto([0, +to)) such that |<p'(t)j < C/t, \y'(t) \ < C/t and satisfying

№»)(*) < V(t) < XnMV'XimV) < V(t) < X[i/2,3](t)- (3-1)

We define the maximal operators

f (x) = sup

v ' n>0

J (R»)m

W f (x) = sup

K(x,yi,...,ym)y\

J (R»)m

f Vlx-y11 + ■■■ + 1 x ym 1

Vlx - y11 + ■■■ + 1 x ym 1

\Y\fi(Vi)dy 1=1

¡YlMvi)®

For simplicity, we denote Kv,rn(x,y1,. ..,ym) = K(x,y1r.. .,ym)y(\J |x - y1| + ... + |x - ym|/n), KfrH (x,y1,...,ym) = K(x,y1,...,ym)f(^J ^ - y11 + ••• + |x - ym|/rf) and

®n(f)(x) H Krn(x'y1'...'ym)Tlfi(yi)dy'

X 7 J( Rn)m i=1

Wjf) (x) = K„(x , y1,. .., ym

W J (R»)m ,=1

The kernels of ®n and ¥n satisfy conditions (1.3) and (1.4) uniformly in n, respectively. And by the same argument in [4], both and ¥* have the same weighted estimates to T* that appeared in Theorems A and B.

It is easy to see that T*(f) < (f) + ¥*(/). Moreover,

Tlb(/) < f) + ^b(/)' Tro(/) < ®m(f) + (f) (3.4)

(x) = sup

v 7 n>0

E Ky,n(x,y1,...,ym)f1( y1)

j=1J (Rn)m

j=1 n>0

(bj(x) - bj(yj))fj(yj) ■ ■■f^y^dy

Kf,n(x,y1'... ym)f1{ y1)

(bj (x) - bj(yj))fj(yj) ■ ■■frntym)^

E<(f) (x)'

^Jf) (x) = sup

E Ky,n(X'yi'---'Vm)fl{ yi)

j=1J (R»)m

< £ sup

j=1 n>0

• (bj (x) - bj(yj))fj(yj) ■■ ■ fm(ym)dy

yi,---,ym)fi{ yi)

J (R»)m

(bj (x) - bj(yj))fj(yj) • • • fm(ym)dit

IXjf) (X),

f (X) = s^p [b1 , [b2.....\pm/ J m-1 • • • ] J f (X)

Km(x, yi,..., ym)Y\ (bj (X) - bj jYlMy^dy

J (R»)m j=i ¿=i

^Ib (/) (X) = sup [bi, [b2.....[^ J m-i ..] 2]^f) (X)

yi,...,ym^Y\(bj (X) - tyjUMyi)*®

J (Rn)m j=i ¿=1

For simplicity, we will only prove for the case m = 2. The arguments for the case m > 2 are similar. For the similarity to the two commutators ®^b(/) and ^b(/), we might as well consider the former. We only consider the former. And we establish the following crucial lemma.

Lemma 3.1. Let y e Ai(M") and bj e Lipp, with 0 < p < 1, j = 1,2. Let 0 <6 < 1/2 < 1 < s < n/p. Then one has

Mt Kfi^)] (X) < Cy(X)Ubj IL^^Cf!^)) (X) + C^(X)\\bj\\Upy x (Mß,y,s (f i ) (x) M f) (x) + M (f i ) (x) Mß,y,s f) (x) ), Mt f f2) (X) < Cy(x) II bj II L ¡Pßy Mß,y,s (f 1, f2))(x) + Cy(x)\IbjI\Upy

x ((^^-ßfyfs (fi ) (x) M (f2) (X) + M (f 1 ) (X) (f2) (x) ) .

Proof. Without loss of generality, we only consider the case j = 1 and denote b\ by b for convenience. Fix x e M" and let B = B(x,R), X = bB* be the average of b on B*, where B* = B(x, 2R). To proceed, we decompose fi = f0 + f", where f0 = fiX&, i = 1,2. Let c be a constant to be fixed along the proof. Since 0 <6 < 1, we have

Vfi / 1 C . , 6 \ ^

flf (v) - cl

i^{J|®b'1(/l,/2)(y)|6 -|c|6|dy) - cfdy

MjBijjCKy) -A)®*(/1,/2)(y)|6dy

jliBK( (b - a)/0'/20) (y)fdy)

1 /* 6 \ 1/6 1 |BK (b - /'

i r 6 \ 1/6

(b - a)/?/00) y| dy)

1 C \ 1/6

(b - I)/!*/0) y6|dyj

B f sup|®,((b - A)//)(y) - c|6dy

Bl Jb n>0

'B n>0

:= I + II + III + IV + V.

For I, since 0 < 6 < 1, ^ e A\ and b e Lip^, by Holder' inequality, we have 1

I < iui

^|j(b(y) - A)®*(/1,/2)(y)|dy

< ^L,|b(y) - bB*|^(y)-1/s®*(/1,/2)(y)^(y)1/sdy

< CL* (|b(y) - bB*|S'^(y)1-8' d^ ^'(J^* ®*(/1,/2 )(y)SKy)dy

/ ll(B*)(ß/n)+(1/s')

1 / 1 r , 1 / \ 1/S M(B*)(ß/n)T(1/S)

< C-|b(y) - bB* |S|(y)1-Sdy) l(B ,--(3.9)

" |(B«)ß/A|(B«) Jb*1 y B I ^ |B*|

--f ®*(/1,/2)(y)Si(y)dy) |B)(1/s)-(ß/n)

l(B*)1-(sß/n) J b

< ^MIli^M^/1,/)) (x)^

< C|(x) ||b||Lip Mtes(/1,/2)) (X).

To estimate the second term II. Since 0 < 6 < 1/2, using Kolmogorov's inequality with p = 6, q = 1/2, X = B, w = dx/jBj and the (L1(Rn) x L1(R"),L1/2-TO(R"))-boundedness of we derive that

II < ||®'((h - A)f10,f2

L1/2-TO(dy/|B|)

< KIB L 1 (h(y1} - hB*}f1 |B, If2 (^2)|dy^ ^

< C( ^ |BJ(h(y1) - hB' )f1(y1)ld^) jBJf2(y2)|dy2

< Cy(x) yhyLipp,dMp/d,4f0 (x)Mf^ (x),

where we have used the analogous technique in I to get the last inequality.

For the term III, using the fact jy - y2| ~ |y2 - xj for any y2 e (B*)c, y e B, and note that KVrH satisfies (1.3) uniformly in r[, we obtain

III < |B|{bK((h - ^)f?,f2TO) (y) |dy

1 ( ( A|KyO-l

- y2D2"

|f2(y2)|

< B -71-i—i-¡r~2n-dy1 dy2 dy

jBj J B J B*x(R"\B*) (|y - y1| + | y - y2|)

< f |h(y1) - hB. j f1(y1)|dyj M y |2"dy2

J B' ■/Rn\B' |y2 - x|

< c([ |h(y1) - hB-11 f 1 (y 1 )|dy^ (£f fy2ldy2

\JB' / \k=1 J 2kB'\2k-1B' |y2 - X|

(3.11)

<C w\( L IKy°- hB-| |f^ y° |dy0 (|rfcn PFi L1/2 ( y^ |dy2

< Cy (x) 11 h 11 Lipp,d Mp,d/S (f 1) (x) M (f2 ) (x).

For the term IV, using the fact jy - y1j~ jy1 - xj for any y1 e (B')c, y e B, and note that KOT satisfies (1.3) uniformly in n, and using (2.4), we obtain

IV < hIH (h - f f (y)|dy

< 1 f| A|hW - A||f. W ||f2(f)| dy, dy2 dy

jBj JBJ (R"\B')xB' (|y - y1| + |y - y21) "

< ^ |Kyl) - hB' l|f (*) | dy f |/-(y2)|dy2

J R"\B' |x - y1| •'B'

|b(y1) - bB*||/1(y1 )|

K~J2k+1 B*\2kB* ^ - y1|

<C Jf IKy. -M!/■ (y.)ldyVf |/2(y2)|dy2)

\k-0J2k+1 B*\2kB* \X-vA / \Jb* /

< q X^^f |b(y1) - bB*||/1 (y1)|dy^i | /2 (y2) |dy:

\fc-0 |2kB*|2 J2k+1B* J\Jb*

< ^f*"*/2..,,.^ " bB*||/. <»> (Bi Jb. |/2 <»> |d»)

k-0 |2k+1B

< CM(/2) (x)^2-kn—+1— ( f ^(y1) - b2k.1 b* | |/1 (y1)|dy1

k-0 2 B \ J 2k+iB*

+|b,* - b2k+1 ,*| |/1(y1) |dy 1 )

J 2k+i B* /

< C^(/2)(x)£2-kn( |(x)|byLjpß,|Mß,|,^(/0(X)

u(B(x,2k+1R))ß/n f , , sl

.wm^ J2k+1B*IMyO|dy1

< Cu(x)Hb^M/(x)Mß,u,s(/1)(x).

(3.12)

For V, fix the value of c by taking c = ®*((b - X)f100/f200)(x), recall that KV/H satisfies (1.4) uniformly in n, then we can obtain

1 . , , W X

. I / ,, . s X _

V < , fg |((b - A)/r,/2°°)(y) - ((b - A)/f,/2°) (x) |dy < Bf sup|®n((b - A)//)(y) - ®n((b - A)/r,/2°°)(x)|dy

|B| J B n>0

< B if 2sup|km (y,y1,y2) - Km (x,Уl,У2)||b(Уl - A)||/1(y1)|

|B| J BJ (R™\B*)2 n>0

x | /2 (y2) | dy 1 dy2 dy C C C | x - y|

< B -:—:-772n+-e KKyO - MMyO/?^) |dy1 dy2 dy

|B| J BJ (Rn\B*)2 (|y - y1| + | y - y2|)

C * C |x — y|£

< 1-^ KKyO - M .MyO ||/^y^ |dy1 dy2 dy

^ tT0J2k+1B*\2kB* |y1 - x\

< 4 X, B fl/n\ |(b(y1) - bB* )/1(y1)|dy^f | /2 (y2) | dy2 ^

Vk-0 2kB* 2+£/M2k+1 B* J\J2k+1B* /

* C(h-keJz^Lb I <*<*> - )f ^y) (wBF\ .L» f

< Cy(x)UbUUpe Mß^f)(x)M(f2)(x),

" (3.13)

where in the last inequality, we use the same computation in the IV term.

Consequently, combining the estimates of I, II, III, and V, we conclude the proof of Lemma 3.1. □

Now we are ready to return to prove Theorem 1.3.

Proof. First, by Lemma 2.1, we have that y e A1 c Ar, and hence y1-r e Ar c A^. Then by Lemma 2.3, we obtain

IK f f2)(x)||Lr (y1-r) < ||m (< f1, f2))(x)||Lr (y1-r)

/ • x „ (3.14)

< ||MK OfO) (x)||Lr (y1-r ) •

For j = 1,2, by Lemma 3.1, we reduce to bound the || ■ ||Lr(y1-r) norm of the right-hand side of (3.6). For the first term, since 1/r = (1/p) - (ß/n) and taking s such that 1 < s < p < n/ß,by Lemma 2.4, and Theorem B(ii), we have

||M,,s(®*(f1,f2))||ir (y1-r) = ||M^,s(®*(f1,f2))||ir y

< C||®*(f1,f2)||LP(y) (3.15)

< C|fl y LP1 (y) ||f2 ||LP2(y).

For the second term, we let 1 /r = 1 /p2 + 1/l, and 1/l = 1 /pi - ß/n. Then by Lemma 2.4 again, together with Holder's inequality, we obtain

||yMtes(f1)M(f2)||ir (y1-r) = HMß,y,s(f1)M(f2 )||Lr (y)

<||Mß,y,s(f1)||ii(y)||M(f2)||ip2 (y) (3.16)

< f LP1 y 11 f2 | LP2

We can obtain that

Similarly, we have

I®;-1-

fif2)L (y- ) < ^Up» IIfiI LPi (y) Wf2\\lP2(y) . (3.17)

^ fi,fOII^ y-, ) < ^^p* IIfiI LPi (y) IIf2\\lp2(y). (3.18)

Consequently, by the above arguments, we conclude the proof of Theorem 1.3.

Similarly as the proof of Lemma 3.1 and that of Theorem 1.3, we only consider the case m = 2 and establish the following sharp maximal function for ®nb •

Lemma 3.2. Let ¡i e Ai(M") and bi e Lipp , i = 1,2; p = p1 + p2, and 0 < p < 1. And let 0 <6 < 1/3 < 1 < si< n/pi, i = 1,2. Then one has

M#6^rn(f1,f2))(x) < ¡(x)2nllbjIk

j=1 (3.19)

X (Mp^s (®* (f1,f2)) (x) + MpAs(f1)(x)MpAs(f2 )(x)), 2

M#( ^Ibf f^) (x) < ¡(x)2U llbj 11 Lip p

j=1 pj'A (3.20)

X (Mp,A,s(®*(f1, f2))(x) + MMp ¡,s (f 1) (x) MMp ¡,s (f2 )(x)).

Proof. Fix x e Mn and let B = B(x, R) with n > 0. Taking Xi = (bi)B,, the average of b on B*, i = 1,2, where B* = B(x, 2R). Let c be a constant to be fixed along the proof. We split Q*nb(f1, f2)(y) in the following way:

®hb (f1,f2 )(y) = sup|(b1(y) - X1)(b2(y) - X2)On (f1,f2)(y)

- (b1(y) - X1)On(f1, (b2 - X2)f2)(y) - (b2(y) - X2)®n (3.21) X ((b1 - X1)f1,f2)(y) + ®n ((b1 - X1)f1, (b2 - X2)f2)(y) l.

Since 0 < 6 < 1/3, then we have

, \ 1/6

6 i„I6

l^nb f ,J2) wi

jBJjl^Cf! ,f2 )(y) l6 -Icl

iUBl®nb(f1,f2)(y) - c\°dy

< ' jliBl(b1(y) - X1)(b2(y) - X2)0*(f1 ,f2)(y)|6dy

^ (supl(b1(y) - X1)°n (f1, (b2 - X2)f2)(y)|^ dy)

. .6 \ /

1 f (sup|(b2(y) -X2)On((b1

' B \ n>0

jl L (supK^ - XO(b1 - X1)f1,f2) (y) | J dy

-1 | sup^n ((b1 - ^f^ (b2 - X2)f2) - c|6dy |B| Jb n>0 /

>B n>0

:= U + U2 + U3 + U4.

(3.22)

For the term U1, since 0 <6 < 1/3, and p = p1 + p2, then by Holder's inequality, we have

U1 < (B £ |h1(y) - J.|36 dy)"'"(B £ |h2(y) - 12|3V 1/36

( iB £ i®*(/1,/2)(y)|36rfy

< C( Bl L |h1(y) - (h1)B-|d^ (Bi L |h2(y) - (h2)B- |dy)

( 1 x ( -

(f1,f2)(y)|dy)

< C(Bi{B, |h1(y) - (h1 )B-|S'y(y)1-s(y(y)d^

x |B, |h2(y) - (h2)B-y(y)1-s' dy) (B| | y(y)d^ (3.23)

|®'(f1,f2)(y)|Sy(y)d^^y(y)-S'/SdyX 1/8

\jB*j JB' / \IB I ./ B'

1 ( 1 f IW N ^ , V/S' y(B')(p1/n)+1

|h1 (y) - (h1)B'|sy(y)1 'dy

y(B* )p,/ny(B') J*1 ^ V jB'j

¡B^ (yB) L |h2(y) - (h2)B' |sy(y)1-s'dy)1/s^ '

1 f ^ \1/sy(B')(1/8)-(p/n)y(B')(1/8')-1

■[ ®'(f1,f2)(y)sy(y)dy

vy(B*)1-(sp/n) J B' jB'j

< Cy(x)2\\h1\\Upp1y \\h2\\UpeiMpyS(&(f1,f2))(x).

For the term U2, noting that 0 < 6 < 1/3, we use the facts 1 = 6 + (1 - 6) and 0 < 6/(1 - 6) < 1/2, then by Holder's inequality and Komolgorov's inequality (Lemma 2.2) and Theorem B, we have

(1-6)/6

1 f........ \/ 1 ' x (

U2 < c( B| JBJh1 (y) - (hl)в.|dy^ B| sup^ f (h2 - l2)f2)(y) |6/(1-6) dy

faiB' lh1(y) - ^ |d^ L (h2 - ^2)f2)(y)|6/(1-6)

^^ |BJh1 (y) - (h1)B'|d^ 11°' (fl, (h2 - W) lU~(dy/|B'|)

< (Bi J^|h (y) - ^B^y) (Bi L |f1 (y1) |dy0

16 Journal of Function Spaces and Applications

K^iB*|(b2(y2) - (b2)B*)/2(y2)|dy2) <(]Bf\\b, |b1(y) - (b1)B*|s' u(y)1-s' dy) ({ u(y)d^

X (Bl Ll/1(y1)ru(y1 )dy1) (^iB* u(y1)-s'/sdy1/

X ^ (iß* ^2(y2) - (b2)B* U(y2)1-s'(J |/2(y2) |sU(y2)dy^

1 / 1 C |b( ) b ( )1-sd \1/s'l(B*)(ß1/n)+1

u^KmJ,*-bB*|u(y) dy) |b*|

( 1 C |/( )|s ( )d V/su(B*)(1/s)-(ß/n)u(BT1/s

Au(B*)1-(sß/n) Jb* 1/1 (y01 u(y1)dy7 -B*-

1 / 1 C . () ^ ( )1-s^ \1/s' U(B*)(ß2/n)+(1/s,)

X IBM!,) Jb* ^ - (b2)B*|l(y) dyV -B*-

¡^^y^uy^) u(B*)(1/s)-(ß/n)

l(B*)1-(sß/n JB*■—'■' -J < C|(x)2|b1|Lipß ||b2|Lipß Mftu,s(/1)(x)Mft|,s(/2)(x)

(3.24)

Similarly, for the term U3, we have

U3 < CA(x)2||b1||Lip yb2yLipp Mp/A/s(f1)(x)Mp,A/s(f2)(x). (3.25)

Now we turn to estimate the last term U4. To proceed, we denote that fi = f + f00, where f = fi^B*, i = 1,2. Let c = c1 + c2 + c3, where

c1 = ®n((b1 - X1)f0, (b2 - X2)f2°)(x),

c2 = ®n ((b1 - X1)f0, (b2 - X2)(x), (3.26)

c3 = ®n((b1 - X1) f0, (b2 - X2)f2°) (x).

We split IV in the following way:

U4 < U41 + U42 + U43 + U44,

(3.27)

U41 = ^B\jB sup|®^((b1 - if (b2 - W20) (y) |d

U42 ={sup|°^(b1 -A1)f0, (b2-Wz") (y)

6 *1/6

( Ii- \ - ) (00 \ ( <v) 1 / 1

((b1 -I1)f0, (b2 -A2)f?) (x) | dy

U43 = ^|B £ «up|0^(b1 -l1)f0, (b2-l2)fz0) (y) (3.28)

((b -11) f0, b-12) f20) (x) |6d^ 1/6

U44 = (B f sup|®n((b1 -If00,b-l2)f20)(y) \\B\ Jb n>0

((b1 -l1)f0, b-l2)f20) (x) |6dy)

For the term U41, we choose 1 < p0 < 1/26 and use Kolmogorov's inequality and Theorem B, then we use the same computation as U2 to deduce that

U41 < (iBJj^(h1 - if (h2 - !2)f20) (y)|p°V) 1/p0S

<||S>*( (b1 - If (b2 - l2)f20)||

L1/2-°(dy/|B|)

< (M U h (y1) - A1)fl(y1)|dy^ {J (h2(y2) - X2)f0(y2

< c(^ \B, | (h1(y1) - (h1 )B')f1(y1) |dy^ | (h2(y2) - (h2)B')f2(y2) |dy2)

< Cy(x)2\\h1 WLip^ WMLip^Mpys (f1) (x)Mpys (f2) (x).

(3.29)

For U42, by the fact jy - y2| ~ jy2 - xj, for any y2 e (B*)c, y e B, and note that KOT satisfies (1.4) uniformly in n, then we get that

U42 fg sup|®^(b1 - If0, (b2 - 12)f0) (y)

-®„( (b1 - I1)f0, (b2 -12) fO

4J(i I(b1(y1)-i1)f1(V1 )ldy1 f ly-1''^y2J'1 dyAdy

\B\ Jb\Jb' J(Br (|y-y11 + |y-y20 /

18 Journal of Function Spaces and Applications

<(Jj(b1(y1) - (b1)B* )/1(y1)|dy1)

£ f |y - x| My2) - (b2 + B* UMy2) | y

k-0 J 2k+1B*\2kB* | x - y2|2n+"

^L*|(b1(y1) - (b1)B*)/1(y1)|dy1)

* IB* l,/n C

k-0 ^^J2k+1B* ^ rt - (b2)B* ^ y2) ^

^L*|(b1(y1) - (b1)B*)/1(y1)|dy1)

)B*J jn

x 1 i2-k£^kr f |b2(y2) - bh-Ufry^y

k-0 12 B | J 2k+1 B*

-k, _J_

k-1 |2k+1B*|

< Cu(x)|b1 Uup^M?*/)^2-k£mb*|b2(y2) - ^WH^)^

+ |(b2)B«-(b2)2k+1B-M \ J2 (y2) \dy2 j

J2k+1B* /

< Ca(x)2 ||b1 ||LipA,A ||b2HLip^MPAs (f1)(x)MpA,s(f2) (x) ,

(3.30)

where we have used the same computation of IV to gain the last inequality. Similarly as U42, we can get the estimates for U43,

U43 < CA(x)2||b1||Lip ^ Hb2HLrpfMpA,s(f1)(x)MpA,s(f2 )(x). (3.31)

Now we turn to U44, by the fact |y-y1| ~ |y1 -x|, |y-y2| ~ |y2 -x| for any y1, y2 e (B*)c, y e B, and recalling that satisfies (1.4) uniformly in n, then we can obtain

|O ((b1 - X1)f0, (b2 - X2)f0)(y) - On ((b1 - X1)f0, (b2 - Xf00) (x) l

< Cx 1 1 y

k-0 J (2k+1B*\2kB*)2 (|y - y1| + |y - y2\) 2n+"

| b1 y1 - A1 /1 y1 b2 y - 2 - A2 /2 y2 | dy1 dy2

* C I x _ y I£

< C£ 2 r-y2n+7 Kb1(yO - AOMyO (k (y - 2) - A2)Myi) |dy1 dy2

k-0-J (2k+1B*\2kB*)2 | y - y1|

< CX, ^n f |(b1(y1) - A1)/1(y1)| f |(b2(y2) - A2) /2 (y2) | dy2

k-0 |2kB* 12+£/n J 2k+1B* J 2k+1 B*

k=0 \I2~B* 1

< CX2-k£( f l(b1 (y1) -11 )f1(y1)ldy1

k=0 \ 2 B J2k+1B'

I 1 (b2 (y2) - 12)f2(y2) |dy2

B* J2k+1B*

|2k+1B^ J2k+1B* < Cy(x)2||b1 ||LipAy Mu^Mßys(f1)(x)Mßryrs(f2)(x).

(3.32)

Therefore, 1

U44 < iB i sup|®n((b1 - l1)f0, (b2 - l2)f0)(y) - ®n((b1 - l1)f0, (b2 - l2)f20)(x)|dy

\B\ J B n>0

< Cy(x)2\\h1 \\Lippi y\\h2 H^Mp^ (ft) (x)Mpys(f2) (x).

(3.33)

Consequently, the estimate for U1, together with those of U2, U3, and U4, can conclude the proof of Lemma 3.2. □

Now we return to prove Theorem 1.5.

Proof. Similarly as the proof of Theorem 1.3,

ll°!h(f1 ,f2 ) 11 Lr (y-r) < llM6 (°nh (f 1 ,f2 )) 11 Lr (y1-2r) < ||M# (°?!h (f1,f2))||Lr yl_lr) . (3.34)

We reduce to bound the \\ ■ \\Lr¡1-r) norm of the right-hand side of (3.19). We estimate each term as follows. For the first term, since 1/r = (1/p) - (p/n) and choosing s such that 1 < s < p <n/p, by Lemma 2.4 and Theorem A and observe that y e A1, we obtain

||y2 Mp,y,s(0'(f1,f2 ))||Lr (y1-2r) = ||Mp,y,s(®'(f1,f2))||ir (y)

< C||°' (f1 ,f2) 11lp(y) (3.35)

< C|f1 (y)11f2 11Lp2(y).

For the second term, since 1/r = (1/r1) + (1/r2), by Holder's inequality and Lemma 2.4, we get

|y2My,s(f1) Mß,y,s(f2) ||Lr (y1-2r) = UMßMf1) Mß,y,s(f2) || Lr (y)

<HMß,y,s(f1)HLn {JMßys(f2)HLr2 (y) (3.36)

< C11 f 111LP1 (y)||f:z||LP2(y).

Similarly, we also have

Il^bf f2)IILr (¡1-2r) < C11 f 111 (¡) 11 f2 11LP2 (¡). (3.37)

This estimate together with that for (f1, f2) finishes the proof of Theorem 1.5. □

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (1096015, 10871173, and 11261023) and the Natural Science Foundation of Jiangxiang Teacher's Division (GJJ10397).

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