Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 686017, 12 pages http://dx.doi.org/10.1155/2014/686017

Research Article

On the Multilinear Singular Integrals and Commutators in the Weighted Amalgam Spaces

Feng Liu, Huoxiong Wu, and Daiqing Zhang

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China Correspondence should be addressed to Daiqing Zhang; zhangdaiqing2011@163.com Received 28 September 2013; Revised 5 January 2014; Accepted 5 January 2014; Published 4 May 2014 Academic Editor: Yoshihiro Sawano

Copyright © 2014 Feng Liu et al. 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 norm estimates for the multilinear singular integral operators and their commutators formed by BMO functions on the weighted amalgam spaces (Lqv^,Lp) (r"). Some criterions of boundedness for such operators in (Lqv^, Lp) (r") are given. As applications, the norm inequalities for the multilinear Calderon-Zygmund operators and multilinear singular integrals with nonsmooth kernels as well as the corresponding commutators on (Lqv^, Lp ) (r") are obtained.

1. Introduction

Let R" (n > 2) be the n-dimensional Euclidean space equipped with the Euclidean norm | • | and the Lebesgue measure dx. For 1 < p, q < x; the amalgam spaces (Lq,Lp)(Rn) of Lp(Rn) and L^R") are denoted by the set of all measurable functions f : R" ^ C, which are locally in Lq(Rn) and satisfy

11(1«

\\fXB(y.

Jr" 11

i) f/y

where B(y, r) := {x e R" : lx - yl < r} for r > 0 and y e R" We remark that the amalgam spaces (Lq,Lp)(Rn) were introduced by Fofana in [1] in connection with the study of the continuity of the fractional maximal operator of Hardy-Littlewood and of the Fourier transformation in R". In [1], Fofana also considered the subspace (Lq,Lp)a(R") of (Lq, LP)(R"), which consists of measurable functions f such that for 1 < a < >x>,

ll(L«,LfT(r")

:= sup

[ (\B(y,r)\

Jr» v

\/u-\/q-\/p\

fx B(y,,

b(ys) iii«(r")

< x, 1 < p,q < x

By the definitions, it is clear (also see [1]) that (Lq, Lq)(Rn) = Lq(Rn), (Lq,L°°)a(R") = Lq'(nq/a)(Rn), where Lq'X(Rn), with 1 < q < x and 0 < X < n, is the classical Morrey space that consists of measurable functions f: R" ^ C such that

Hl«-a(r")

sup \B(y, r)\

y£R",r>0

1 I \f(x)\qdx)

JB(y,r) /

In this paper, we focus on the weighted version of (Lq,Lp)"(R"). Precisely, letting w be a weight on R" and 1 < q,p,a < x, we define the weighted amalgam spaces (Lqw,Lp)"(R") as the space of all measurable functions f satisfying

\\f\\(Ll„L?f(R")

sup (f (w(B(y,r))1/a-1/q-1/p

r>0 \ JR" V

II II \p \1/p

^fXB(y,r)\\Llm) dy]

and a suitable modification version for p = x or q = x.

< x, 1 < p,q < x

and a suitable modification version for p = <x> or q = <x>, where Lqw(R") is the weighted Lebesgue space.

It is easy to check that when k = 1 - qla and 1 < q < a < >x>, the space (Lqw, Lm)a(R") is nothing but the weighted Morrey space ^^(R"), which is the set of all measurable functions f such that (see [2])

II/IL^r")

:= sup(^ £ \f(X)\lW(X)dXy<™, (5)

1 <q< <m, 0 <k < 1.

As is well known, the boundedness of the classical operators in the harmonic analysis on the weighted Morrey spaces has extensively been studied (see [2-6] and references therein). In particular, Wang and Yi [6] recently showed that the m-linear commutators and the iterated commutators of the m-linear Calderon-Zygmund operators are bounded on weighted Morrey spaces.

Based on the above, we feel that it is natural and interesting to study the boundedness of the classical operators in harmonic analysis on the amalgam spaces and the weighted versions. Indeed, a lot of attention has recently been given to this topic (e.g., see [7-10]). Here, we will continue the investigation along this line. The main purpose of this paper is to study the boundedness of the multilinear operators on the weighted amalgam spaces (Lqv^, Lp)a(Rn).

Let K(x, y1,...,ym) be a locally integral function defined off the diagonal x = y1 = ■ ■ ■ = ym in (R")m and let T : S(R")x---x S(R") ^ S'(R") be an m-linear operator associated with the kernel K(x, y1,..., ym) in the following way:

(T( f1,...,fm),g)

\ K(x,yi,...,ym) n fi (y i) 3 (x) dyi ■■■ dymd X,

r" J(r")m ¡=f

where f i,..., fm, g in S(R") with H"m=l supp( fj) R supp(3) = 0.

ForS = (b1t..., bm) e (BMO(R" ))m, we define the m-linear commutator of T denoted by Tj.^ as follows:

Ttí (f,->fm) ■■=!T¡ (f)

where each term is the commutator of bj and T in the jth entry of T; that is

V (f) = bjT(f1,...,fm)-T(f1,...,bjfj,...,fm)

and f = (f i,...,fm), where fj is a smooth function with compact support on R". The iterated commutator is defined by

Tni (!)■■= [ b2 ,...,[b„-1,[bm ,T]m]m-1,...]2]i (f).

If is associated with a distribution kernel, which coincides with the above function K, then we have, at a formal level,

T{f)(x)

= \ K(x,y1,...,ym)f1 (yi) ■■■ fm (ym )dyi ■■■dym'>

J(R")m

Tl {f)W

= Lr (»'W- "> (>■))

XK(X,yi,...,ym )fi (y ¿■■■fm bm) fy1 ■■■ fym'

Tni(f)M

Rn,m n( h (x)-bj(yj))K(x> yi,...,ym)

Xfl (y 1)---fm (ym)dyi ■■■dy„

Also, we recall the definitions of the classical Mucken-houpt classes Ap weights and the multilinear Ap conditions for multiple weights.

Definition 1. A weighted w on R", that is, a positive locally integrable function on R", belongs to Ap(R") for 1 < p < >x> if there exists a constant C > 0 such that

sup i i

Q cube in r^ V IQI JQ

X ( — [ W

IQI Jq

— [ w(x)

QI Jq ( )

1/(p-1) (x)dx)P 1 < C.

The infimum of these constants C is called the Ap constant of w and denoted by [w]A .A weight w belongs to the class A ^R") if there exists a constant C > 0 such that

sup — J w (x) dx( infw (x)) < C, (12)

Q cube in r" IQI jq !

and the infimum of these constants C is called the A1 constant of w and is denoted by [w\A .

Definition 2. Let me N with m > 1 and 1 < p1,...,pm < >x>, 1/m < p < ra,and 1/p = J™=1 1/p¡. LetP = (p1,...,pm) and W = (w1,...,wm). Set

We say that w satisfies the Ap condition if

ik jV*(x)dx Q \\Q\ JQ

n gu ^

< œ>,

where p', = p;l{p; -1) for j = 1,..., m.

Obviously, for m = 1, Ap is the classical Muckenhoupt classes Ap condition. It is not difficult to check that for m > 1 (see [11]),

UAPj S AP-

which implies that something more general happens for the Ap classes. Also, the authors in [11] showed that the Ap conditions are the largest classes of weights because all m-linear Calderon-Zygmund operators are bounded on the weighted Lebesgue spaces.

To state our main results, we still need to recall and introduce some notations. For fixed y e R" and r > 0, we set B = B(y, r). For any A > 0, let XB = B(y, Xr) and \XB be the characteristic function of the set XB. Given any positive integer m and j e {1,..., m}, we denote by D™ the family of all finite subset a = {a(1),..., a(j)} of {1, ...,m} of j different elements. For any a e D™,we also denote the complementary sequence of a by x given by x = {1, ...,m} \ a. We remark that t=0 if and only if a e D™. Letting f = (f1,..., fm) for

a fixed j e {0,...,m} and a e DJ, we set f = (f1,..., fm)

and fi = /¡X(2B)C if i e a and f, = f^B if' e x. Now we can formulate our main results as follows.

Theorem 3. Let me N with m > 2 and T be an m-linear operator. Let 1 < qj < a j < pj < >x> (j = 1,...,m)

satisfy 1/q = J™=1 1/q,, 1/a = 1=1 1/at, 1/p = 1/p{, and p/pj = q/qj = a/aj (j = 1, ...,m). Assume that u> =

(w1,...,wm) e A^for Q= (ql,...,qm) with w1,...,wm e

A m and va = n^f'. IfT maps L* (Rn) x---x L^ (Rn) to LqV^ (R"), then the inequality

< cn\\f>\

j\\(,Vl ,Lpi ) j(r")

holds provided that for any ball B in R", any j e {1,...,m} and a e D™, there exist constants y > 0 and ft > 0 such that for a.e. x e B,

T(f)(x)

<c(nw№Tllq' WMibWlI

t€ak=1

2k+1B\\Lqi.

Theorem 4. Let me N with m > 2, 1 < qj < o-j <

pj < rn (j = 1,...,m), satisfy 1/q = 1==1 1/li, 1/a =

17=1 1/ai, 1/p = TiL\ 1/pi, and p/pj = q/qj = a/a} (j =

1, ...,m). Assume that w = (w1,...,wm) e A^for Q =

(q1,...,qm) with wi,...,wm e Am, = ^=1 w1q>, and b = (b1,...,bm) eBMOm.If

lv„ (r")

») j i iwîwl^(r")'

then the inequality

(Lq„a,Lf) (r")

(m \ m

ZI NI BUO(R") ) I! \\fi\\ (LqJ. ,L"i )"' (r»)

holds provided that for any ball B in R", any j e {1,..., m}, a e D™, ^ e a, and v ex, there exist constants y > 0 and ft > 0 such that for a.e. x e B,

T* (f)(x)

<c(n^,(2B)-1lq-\\ft X:

2B\\Lq>.

1 + Yk i^k+i^-11^

(2k+1B)

: Wtoi^

« î (\b„ (x)-(b„) )w„

l-, VI v-v ' \ P/2k+1B\ Il pWbmoJ p

(~,k+1T,\-1l%\\ r ||

«(2 B) \\fvX2^B\\Lq. >

Tl (f) (x) <c(n wi(2B)l1qt\\M2bIli)

ieak=1

1+yk [„k+iuY11?'

2kß -w>

¡¡foi**'

(2k+1Bj

BWti,'

«( I bv(x)-(bv)

2BI + \bMO) Wv « (2B)-1l^\\fvX2B\U .

Theorem 5. Let me N with m > 2, 1 < qj < aj <

pj < >x> (j = 1,...,m), satisfy 1/q = Ti=i 1/li, 1/a =

T=i 1/«i, 1/p = TZi 1/pi, and p/pj = q/qj = a/aj (j =

1, ..., m). Suppose that w = (wit..., wm) e A^ for Q =

(qi,..., qm) with u>i,...,u>m e A ^ = ^i^', and b = (bi,...,bm) eBMOm.If

\\Tni(f)\

lt (r")

±c(n\\b

,jllBMO(r") IL LP(r")' j=1 / j=i >

i nn^rl

then the inequality

\\Tnl {f)\\(Lla,L?f(Rn) f m

j llBMO(r") IL L\\J AVlH. ,Lpi)"' (r") j=l / j=l >

holds provided that for any ball B in R", any j e [1,...,m] and a e Dm, there exist constants y > 0 and ft > 0 such that for a.e. x e B,

Tni(f)M

<CI I (n\b>m-W

x ( nw.(2BT1,q'\\f.XiAlZ,

\ i£T

xnl Wf^

i€ak=1

I I Llt' (24)

where for any rj e {0,1,..., m], a0 e Dm and t0 = [1,...,m}\

Theorem 6. Let T be an m-linear operator with kernel K satisfying

\K (y0' yi'-..' ym)\ <

{ZZi=o\Vk -yi\)

Let 1 < qj < aj < pj < >x> (j = 1,...,m) satisfy 1/q =

T=i 1/%, 1/a = T=i 1/ai, 1/P = T=i 1/pi, and p/pj =

q/qj = a/aj (j = 1,..., m). Assume that u> = (wi,..., wm) e

nm=iAqi, v* = nm=^q', and b = (bi,...,bm) e BMOm. Then these inequalities (17), (20)-(21), and (24) hold.

Remark 7. We remark that for pi = ■■■ = pm = p = <x>, Theorems 3-6 are also true, just with the restrictive condition:

q/qj = a/aj (j = 1,..., m). Moreover, for v* = n'jl=lu'j'q' = w e Aq, that is, wi = ■■■ = wm = w e Aq, we can remove the restrictive condition p/pj = q/qj = a/aj (j = 1,...,m) in Theorems 3-6. See also [12, Theorem 3.5] for the unweighted case.

The rest of this paper is organized as follows. In Section 2, we will give the proofs of our main results. Some applications will be given in Section 3. Throughout this paper, the letter C, sometimes with additional parameters, will stand for positive constants, not necessarily the same one at each occurrence but independent of the essential variables. In what follows, we use the convention nje0aj = 1 and Tje0 aj = 0.

2. The Proofs of Main Results

Let us begin with a lemma, which will be used in the proofs of our main results.

Lemma 8 (cf. [6, Lemma 3.1]). Let m > 2, pi,...,pm e (0,<x>) and p e (0,<x>) with 1/p = T=i 1/Pi. Assume that

wi,..., wm e Am and v* = nrn=iw^Pi, then for any ball B, there exists a constant C > 0 such that

f\ wt (x) dx)'P' v* (x) dx. (26)

Proof of Theorem 3. For fixed x e B,we can write

T(f)(x) = T(fX2B)(x) + Z YT(f)(x). (27)

j=i oiDf

The boundedness of T from Lq^ x-^ x Lq™ to LqV^, (17) and Holder's inequality lead to

\\T(f) Xb\\l%

< Cn\\fiX2B\\L%.

+ CI I (n

j=1 ozD? \ ier

W, (B) w, (2B)

X \\fiX2B\\L%

xni1 + Yk( w,(B) Ii1 2kß U (2k+1 B)

WfiX2*+i

Note that for any k e N and w e Aq, there exists a constant ca > 0 that depends only on q, [w]A , n such that

w(B) >(2k+1 B)

< C2-(k+1)n°q

Hence, multiplying both sides of (28) by va(B)l/a-l/q-l/p, note that 1/a - 1/q - 1/p < 0 and p/pt = q/qt = a/at; by Lemma 8 and (29) we obtain

l/a-l/q-l/p\

< C fiwi(B)

(l/a-l/q-l/p) q/qi

\T(f) Xb\\

< cnw>(B)l/ai-l/qi-l/pi||^^(f) XbI

<cn^m)l/a'-l/i'-l/p' i=l '

cll( nWt(2B)l/a'-l/q'-l/p' UxiA^.

j=l o€Df \ ¡€t

k+l „^/"i-l/q—l/pi

^ 2knc (l/^-l/p+ß) ¡€ok=l2 '

x WfiXi^BWii>

■wi(2k+l B)

which combined with the fact that at < pt for all i e {1,..., m} leads to

\\T(f)

Theorem 3 is proved.

(Lqv, IF j\\(Ll>. ,Le> f' (R")'

Proof of Theorem 4. For fixed x e B, by linearity we can write

T^ (75\(x)

j=l 0€D"> \ ^€0

1 1(1 T ( f](x)

+1T (f )(*)]'

V€T )

Invoking (18), (20)-(21) and Holder's inequality, we have \\Ti;i (f) Xb\\lv

n\\f'X2.

'»IIBMO M IF 'a2b\\lch,. i=l ) i=l

<111 in(w^)

i=l o€Dm ß€o\ i€T^ w> (2n) '

i(B) \l/q'

x WfiX2B\\LV,.

4 n 11+ki w(B)

\i€a\{^}k=l 2

(2k+l B)

WfiX2k+\

x11-+k\\b,

£ 2kß 11 (2k+lB)J

X H/u^1 J

111(n(WW-2L)

j=l o€U" v€T \fe-r\M N W' (2B) '

X \\f>X2B\\Ll

1 + yk( w{ (B)

i€ok=

l 2kß \wt (2k+lB)

WfiX2k+i

* llb]lBMo( wiwj) Wf^Wi

Bya similar argument as in getting (31), we can conclude that

nat ,lp) (Rn)

<c(1INLo(r») )U\\fJ

•) i i if j\\(Lqi l?i)"' (r»)' j=l >

which completes the proof of Theorem 4. Proof of Theorem 5. For fixed x e B,we can write

Tni (f)M = Tni (h 2B)M

+ 1 1 Tni (f)M'

j=l o€Dm v 7 J )

Applying (22), (24) and Holder's inequality, we get that \\Tnb (f) Xb\\l^

im \ m

j=lo€Df n=0aa€D^ \ ¡€o(

c111 1 (niNlBMoHniNI

■Ay .

lYllu^B)\ 1/q'

, l£T X l

\+yk( wt (B)

2kß U (2k+1 B)

2k+1B\\L'1'. ■

By similar arguments as in getting (31) again, we can deduce that

\Ki(f)\\

(L%, ,Lff(R")

llBMO(r") / 1 L\\J S\\(Lq,i ,Lpi )"' (r"/ \j=l / j=l >

This completes the proof of Theorem 5. □

Proof of Theorem 6. For fixed x e B,it is easy to check that

2\z-y\<\z-x\<2\z-y\, if z e(2B)c. (38) Since wt e Aq , we have for any i e {1,...,m} and ¡3 e N,

\f (z)\dz < C\\fiX2Bh. \2B\ w,(2B)-llq', (39)

J2B w'

J(2B)C \y - Z\

to -if

<l\2kBI L * (z)\dz

kk={ 1 J2k+1B\2kB

TO -l/q-

[ \f (4

i(2B)c \y - ,

\(.ß+l)n

ïCÏWfiX*"

x \2k+lB\-Pw,(2k+1 B)liqt.

By Holder's inequality, (25) and (38), writing a = (cr(1), ..., a(j)}, we have

T(f)(x)

<C|M| \f (z)\dz

/ \ -mn

[ ' (l\x-y,\ ) U\f (y>)X(2By (y,)\dy,

\f (z)\dz

\U) (z)\

\f«(j) (4

il W jy-zjn JJ(2B)C \y - z\(m-j+1')n

-dz■

It follows from (39)-(42) that TÎf](x)

(37) <C(n\\f-X2B\\Ll. \2B\w, (2B)-1/q'

( j-1 '

x(ni-.(0(2k+1 B)-1/^\\fa{l)X2

\ i=1k=1

wct(')

TO . ,

\2 B\ wc(j)(2 B)

x HicrO'^+'BlLHi)

m<r(j)

<c(n\\f,X2B\\LlW,(2B)-1/q'

x(n TOw0(,)(2k+1 B)-1/q"') Wfornx* \ i=1k=1

xJj2-k(m-j)wa{j) (2k+1 B)-1/q°(i) X |fcr(j)X2fc+1B|L'îcr(j)

This implies (17) in the case of that ¡3 = 0 and y = 0 or ¡3 = m - j and y = 0.

For Tj.p a e D™ and i e a, we have from (25) and (38)

Tl(f)(x)

c[ \b, (x)-b, (y,)\

J( r")m

xU\fk (yk)\(!\y-yk\ ) dyx-dy,

„ Kb iüzh Mi, dz

2B ! i(2B)c \x - z\

[ V1 (ï \y-y\)

}(R"y \kia\li} J

-(m-1)n

x n \fk ( yk) X(2B)C (yk)\dyk■ keo\[i}

Since wV e Aq^,thus w1-qv e Aqi with q'V = qV/(qV -1). By the properties of functions in BMO(Rn) and Holder's inequality, we have for any ball Q and v e {1, ...,m},

jQ \bv (z)-(bv)Q\\fv (z)\dz <(jQ \fv (z)\q- w(z)dz) 'q

x(}Q \bv(z) - (bv)Qf w^-qV (z) dz)

<C\\fvXQlX INIbMO № Wv(Q.y1lqV■

It follows from (39) and (45) that for any v e {1,...,m} and 1 < qv < rn,

J(2B)C

\(b] (x)-bv (z))fv (Z)\

\x-z\n

\ (bv (x)-bv (z))fv (z)\ k=1 J2k+1B\2kB \y - z\n

<C^( I bv (x)-(bv)^B\ +k\\bv\\

xWv(2k+1B) l1qy\\fvX2k+iB\\L"vV

Let ^ = a(i0) for some i0 e {1,..., j}. We now consider two cases:

Case 1 (i0 = j). We have

Ti (f)(x)

<c( n\\f,X2B\\L% \2B\w,(2B)-1/q

n j \f<*0 (z)\

^iw-mw-W \y-z\

\foii) (4

(2B)C \y - z\(m-j+1)n

2k+1B\ II "IIBMO

l{2k+1B) ¡/"^bIU

<c(n\\f,X2B\\Ll M2B)-1/q

x( n Lw«(.)(2k+1 B)-1^ WUX*»b\\ \l6|1,...,j-1}\i l0}k=1

x^2-k(m-^)wa(j)(2k+1B)-1/q'w

| | fa(j)X2t+1B\\Tq^(i)

'•'„(j)

(45) xl( \ b^ (x)-(b^)

2fc+1 BI + "1 Ibmo

k+1Tyr1/qt

xwl^(2 B) \\Z^+IbII^ :

which satisfies (20) in the case of that ft = 0 and y = 0 or ft = m- j and y = 0.

Case 2 (i0 = j). We have

T.I (f)(x)

<C( fl 11 f,X2B\\Ll \2B\w,(2B)-1/q

n j I 0 (z) I „i j(

t.{1,...,j-2}i(2b) \y-A" \ Uj-1) (z)\

\(m-j+1)n

)(2B)C \y-Z

xl{\b, (x) - (b^

xw,(2k+1 b) llq" H/^+IbIU

<c(nUX2B\\Ll ^,(2B)-1/q-

x( n L"« 1)(2k+1B)-1/qa{l)\\fa{l)X2k+iB\\

\ie{1,...,j-2}k=1

,tv(i)

x L2 Wa(j-1) k=1

(2k+1B)

x \\fa(j-1)X2t+1B\\L^(i-i)

k+1 r,\-1lq"()-i)

xL( \ b, (x)-(bft):

'id-1)

2k+1B\ ' "IT"!IBMO

,(2k+1 B) 1/q"\\f"X2k+lB\L*X :

which satisfies (20) in the case of that ft = 0 and y = 0 or ¡3 = m- j.

For v e r,it follows from (25) and (38) that

cf \bv (x)-b] (yv)\

/ \ —mn

im \ m

*(!\y-yk\ ) U\fk M^-dy»

c( n l\f &\dz)n

\i£t\{v} J2B / i=1

\fo(0 (z)l )(2B)C \y-z\n

(2B) \y-,

\fpjj) (4

\(m—j+1)n

\(b] (x)- b] (z)) f] (z)\dz.

This together with (39)-(41), (45) and Holder's inequality leads to

<C( U WfiX2B\\L« \2B\w,(2B)—1/*

\ ier\{]}

j—1 m 1/ *ni^.)(2k+1 B) \\fa{l)X2k-i=1k=1

TO . ,

xl\2k+1 B\—m+jWa(j)(2k+1 B)—U^\\faU)X*+4

2k+1B\\i qaO

,q<r(j)

x(\bv (X)-(bv)2B\ + \\bv ||bmo)

x\2B\wv(2B)—1/*] \\fvX2B\\L*

<q n \\fiX2Bhwm)-1"'

ier\{]}

*niwa(,)(2k+1B)—1^ WUtX.

i=1k=1 to

xl2—k(m—j^waU)(2k+1B)—1/^('^\faU)X2k+iB\\

,q<r(j)

x ( | bv (x) - (bv)

2B\ 1 Wuv\IBMO^

xwv(2B)—1/q]\\f]X2B\\Lq],

which satisfies (21) in the case of that ft = 0 and y = 0 or ft = m - j and y = 0.

For Tnl, j e{1,...,m}, a e Dm, a = |a(l),...,a(j)} and t = {l, ...,m} \a,we have

(f)(x)

= 1 U(bt (X) - bi (y))K(x,y1,...,ym)

x Wi (yi') X(2B)C (yd nfi (yd X2B (yd dy1 ••• dyn

■=11 (f)(*)>

r,=0 o0eD"

■= n (bi M-(bi)u,)

x I U((bi)2B -bi (yi))K(x,y1,...,ym) x nfi (yd X(2B)C (yd Wi (yi') X2B (yd dy1 ••• dym.

ito itT

For fixed x e B,q e {0, ...,m} and a0 e D», we set

t1 ■= m t0, t2 ■= t0 \t, t3 ■= t\ t0. (53) Then by (25) and (38), we have

\Tjo0 (f)(*)\

<c(n\bt (x)-(b)w\

x f m (n\(b (y) - (bdw) fi (y)X2B (yi )\ x(u\b, (yd-(b,)2B\)(n\f (y)X2B (y,)\

\i6T2 ) \i6T3 ,

x n \fi (yi) X(2B)C (yd\ \K(x,y1,..., ym)\dy1 ••• dyn

<G(n|bi (x)-(bb)w\

xiiii \bt (z)-(bt)w\\ft (z)\dz

J—1 TO

x(]j\ \f, W\dz

ter3 j2b

nf M-ikUlf.

n 1 (2B)C

I y-z\n I f, (z) |

^UUj)}) J(2B)C

fo(j) (z)\ (2B)C ¡y -Z|(m-j+1)"

It is easy to check that

f I (b, (z)-(b,)2B)fi (z)\

J(2B)C

<C^(k+1)\ k=1

k+1R)-1/® (55)

^,(2K+1B)

WfiX2k+i

for any i e {1,...,m}. This combining (39)-(41), (45) with (54) yields that

<c(n\b, M-()2B| jmiNI

x( n^,(2£)-1/?'\\f,X2B\\

x( n^,(25)-1/?'Wf,X:

2B||L«¡.

x|ni(fc+1)^, (2fc+1 B)-1/q'\\f,

ter2 k=1

x n iM^By^WM^

teCT\(T2U{CT(j)})fc=1

xfr^^1 B)-l/q^\\fa(j)X,

^t+1B|| tMj) .

This together with (51) implies (24) in the case of that ft = 0 and y = 1, or ft = 0 and y = 0,or ft = m - j and y = 0, and completes the proof of Theorem 6. □

3. Applications

3.1. On the Multilinear Calderon-Zygmund Operators. An m-linear operator T associated with K is said to be an m-linear Calderon-Zygmund operator if, for some 1 < qj < rn, it

extends to a bounded multilinear operator from Lqi x---x Lqm

to Lq, where 1/q = 1/q1 +-----+1/qm and the kernel K satisfies

(25) and the regularity conditions

I K (yo'...yj,...,ym)-K(yo,...,y'j..... ym) I

A\yj -y¡I

< -LJ-il-

- /„m I ,xm«+e

(Z«=o \yk -yi\)

for some e > 0 and all 0 < j < m, whenever - y'j\ < 1/2 max0sfcsm\y;- - yk\. We denote by m - CZK(A,e) the collection of all kernels K satisfying (25) and (57).

As is well known, the multilinear version of the Calderon-Zygmund theory originated in the works of Coifman and Meyer in the 1970s; see, for example, [13, 14], and it was oriented towards the study of the Calderón commutator. Later on the topic was retaken by several authors, including Christ and Journó [15], Kenig and Stein [16], and Grafakos and Torres [17, 18]. Moreover, commutators of multilinear singular integral operators with BMO functions have been the subject of many recent articles (see [11,19-21] et al.). The following results, which will be used in the next theorem, follow from [11, 20].

Lemma 9 (cf. [11, 20]). Let me N with m > 1 and T be an m-linear Calderon-Zygmund operator. Let 1/p = Y!m=1 1/pi with 1 < p, pj < rn, and j = l,...,m. Suppose that

b = (b1,..., bm) e BMOm and u> = (w1,..., wm) satisfies the Ap condition. Then

T(flkm <cUWfj

jhis, (r")

|rzS (f)WLt (r") < C ( ZINIbmo(r») /1 wn\LrJ. (r")' "" \j=1 / j=1 J

nifií» (

\\Tnb wil^ (r") < C ( oinibmoor") / 1 l\\J j\\Lr¿. (r")' "" \j=1 / j=1 J

where v,7. = n? 1 wpfp'. w 1 lj=l j

This lemma together with Theorems 3-6 directly leads to the following result.

Theorem 10. Let me N with m > 2 and T be an m-linear

Calderon-Zygmund operator. Let 1 < q^ < Oj < pj <

rn (j = 1,...,m) satisfy 1/q = 1/qit 1/a = 1/at,

1/P = Tih 1/pf and p/pj = q/qj = a/aj (j = 1,..., m).

1 ÊCT

Suppose that w = (w1,...,wm) e n=1A , vÄ = Ulirf''1',

_ JW(L% ,Lp')"' (r")'

and b = (bl,..., bm) e BMOm. Then

\\T(f)\\alam <CH\\fi

\\T-Lb {f)\\(L\7i,Lff(r")

I\h ILo(r") ^\\fj Ki Wllo^,Lef(Rn)

1=1 INLmo(R") ) I! ]ifj\\(LiJ,. ,LP> )V")'

where C depends only on m, n, p, q, a and [w]A

> llbmo(r") IL W iWLi Lp' )"' (r")' j_1 / j_1 >

Furthermore, by Remark 7 and Lemma 9, we have the following.

Theorem 11. Let me N with m > 2 and T be an m-linear Calderon-Zygmund operator. Let 1 < q^ < <Xj < pj < >x> (j = !,...,m) satisfy 1/q = 1/li> 1/a = Ym=i 1/ai and 1/p = J™1 1/pr Assume that w e Aq and b = (b1, ...,bm) e BMOm. Then

l|r(/)|| IM/)I

(li,lpy( r")

* cn\\fj\

j\\(L\i ,Lpi) j(r")'

(Lqw,Lf) (R")

(m \ m

XINLmO(R") ) n\\fj\\(LÜ ,Lp')' (r")'

\\Tnb (f)\\(Li,Lpf(R")

where C depends only on m, n, p, q, a and [w]A .

jll£MO(r") ) nj/jas ,Lp')' (r")' j_l / j_l

3.2. On the Multilinear Singular Integrals with Nonsmoothness Kernels. Let [At]t>Q be a class of integral operators, which play the roles of approximate identities (see [22]). We always assumed that the operators At are associated with kernels at(x, y) in the sense that for all f e Ui<p<m Lp and x e R",

Atf(x)=[ at (x,y)f(y)dy, (61)

and the kernels at(x, y) satisfy the following conditions:

K (*, y)\<K (x, y) := t-n/sh ( ) , (62)

where s is a positive fixed constant and h is a positive, bounded, decreasing function satisfying that for some q > 0,

lim rn+nh (rs) = 0.

Recall that the jth transpose T*'j of the mth linear operator T is defined via

(fi'...'fm)'9) = (T(fl'...'fj-l'h'fj+i'...'fm)'fj)

for all f1,..., fm, h in S(R"). Notice that the kernel K*'j of T*'j is related to the kernel K of T via the identity

x'7l '...'/j-l'/j'/j+l'

= K (yj' yl'...' yj-l'x' yj+l' ...'ym).

If an m-linear operator T maps a product of Banach spaces X1 x---x Xm to another Banach space X, then the transpose T*'j maps X1 x ■■■ x Xj-1 x X x Xj+1 x ■■■ x Xm to Xj. Moreover, the norm of T and T*'j is equal. To maintain uniform notation, we may occasionally denote T by T*'Q and K by K-'Q.

Assumption 12. Assume that for each i e [l,...,m], there exist operators {A^} Q with kernels at'\x,y) that satisfy conditions (62)-(63) with constants s, q and that for every j e {0,1,..., m}, there exist kernels K*'j'(t\x, y1,..., ym) such that

(T-'l (f1....,A?f,..,fm).f)

K^ (X,yl'...'ym)

Xfl (yi)---fm 0Ü 9 (X) fyl ■■■ dym dx

for all f1,...,fm in S(R") with nm=1 supp(A) n supp(^) = 0. Also assume that there exists a function 0 e C(R) with supp 0 c [-1,1] and a constant e > 0 so that for every j e {0,...,m} and every i e {1,..., m}, we have

\K-'j (X,y1,...,ym)-K-'j'(i) (X,y1,...,ym)\ A

(Hl \x-yk\)mn b; -yk\

k_l,k_i

tl" > (im_-Ax-yk\T

whenever t1/s <\x - y\/2.

We say that T is an m-linear operator with generalized Calderon-Zygmund kernel K if T satisfies Assumption 12. We denote by m-GCZK(A, s,q,e) the set of functions K satisfying (25), (66)-(67) with parameters m, A, s, rj, and e. We also say T is of class m-GCZO(A, s, q, e) if T has an associated kernel K in m-GCZK(A, s, t], e).

Assumption 13. Assume that there exist operators |Bjt>0 with kernels bt(x,y) that satisfy conditions (66)-(67) with constants s and Let

Kl0) (x,yi,---, ym) = I K(z,y1,...,ym )bt (x,z)dz,

whenever 2t1/s < min1SjSm|x - yj\, and \K (x , yu ... ym)- K(0) [x,yi,...,ym)\

(lZAx-yk\T

k=1,k=i

t1/s ) (Zt=i \x-yk\)mn+e'

whenever 2\x - x'\ < t1/s and 2t1/s < max1SjSm|x - yt\.

It should be pointed out that the condition (67) is weaker than the condition (57) (see [23, Proposition 2.1]). Similarly, we can verify that Assumption 13 is weaker than the condition (57). These assumptions were introduced by Duong et al. in [23, 24]. An important example for satisfying these assumptions is the mth Calderon commutator. For T in m-GCZO(A, s, q, e) with kernel K satisfying Assumption 13 and the corresponding commutators T2g and Tnp lots of attention has been given (e.g., see [23-29] et al.). In particular, following from [26, 28], we have the following.

Lemma 14 (cf. [26, 28]). Assume that T is a multilinear operator in m-GCZO(A, s, q, e) with kernel K satisfying Assumption 13, b = (b1,..., bm) e BMOm. Ifthere exist some 1 < q1,...,qm < x and some 0 < q < x with 1/q = jm=1 1/%, such that T maps Lqi x ■■■ x Lqm to Lq'm, then for 1/m < p < rn, 1 < p1,...,pm < x with 1/p = J™=1 1/pi, P = (p1,...,pm), and w e Ap,

\\T(f)\\LÎ

\KÁf)\\ KMW

,-(Ill N

i n 11 fj \ \ l% (r»)'

Im \ m

*c(n I I bj I I i^Jnil/jl \j=1 / j=1

where vä = n¡=1 wP/P'

Invoking this lemma and Theorems 3-6, one has the following results.

Theorem 15. Let me N with m > 2 and T be an m-linear operator in m-GCZO(A, s, q, e) with the kernel K satisfying Assumption 13. Let 1 < qj < aj < pj < x (j = 1,...,m) satisfy 1/q = h 1/q,, 1/a = h 1/at, 1/p = h 1/p,,

and p/pj = q/qj = a/aj (j = 1,...,m). Assume that

W = (W1,...,Wm) e nm=1Aqj, Vw = U"-, and b =

(b1,...,bm) e BMOm. Ifthere exist some 1 < T1,..., rm < x and some 0 < r < x with 1/r = 1/rt, such that T maps V1 (R") x^ ■ ■ x V"(R") to Lr'm(Rn), then

(.LV„ LP) (R»)

\Ki(f) \Ki(f)

* Cn|/j|(L,j. ,LP' )' (r»)' j=1 '

(Lqv, ,LP) (R»)

j II BMO(R ») I I í \\JjW(Lq¿. ,LP' f (r») ' j=1 / j=1 >

(Lt LP) (R»)

n\\fj\

jwbmo(R») I Í \\(Lq¿. ,LP' ) ' (r»)' j=1 / j=1 >

where C depends only on m, n, p, q, a and [w]A .

In particular, by Remark 7 again and Lemma 14, one has the following.

Theorem 16. Let me N with m > 2 and T be an m-linear operator in m-GCZO(A, s, q, e) with the kernel K satisfying Assumption 13. Let 1 < qj < aj < pj < x (j = 1,...,m) satisfy 1/q = J™=1 1/q,, 1/a = J™=1 1/at, and 1/p = J™=1 1/pt. Assume that w e A , b = (b1,.. .,bm) e BMOm. If there exist some 1 < r1,...,rm < x and some 0 < r < x with 1/r = 1Z\ 1/ri, such that T maps V1 (R") x^x Lr"(R") to Lr'm(Rn), then

II^(-0II(lí,lP)'*(r»)

\\T-Lb (/^(L^LPf(R»)

* cU\\fj\

j\\(L¿ LP') j(r»)'

j llBMO(r») / 1 W j\\(Lqi ,LP' ) ' (r»)' j=1 / j=1

ll(L%,LP) (r»)

'jwbmo(r») i i w j\\(Lqi ,LP' f (r»)' j=1 / j=1

where C depends only on m, n, p, q, a and [w]A .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the NNSF of China (nos.

11071200,11371295).

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