Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 606504, 10 pages http://dx.doi.org/10.1155/2014/606504

Research Article

Compactness of the Commutator of Multilinear Fourier Multiplier Operator on Weighted Lebesgue Space

Jiang Zhou and Peng Li

College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China Correspondence should be addressed to Jiang Zhou; zhoujiangshuxue@126.com Received 19 January 2014; Accepted 12 March 2014; Published 1 June 2014 Academic Editor: Yongqiang Fu

Copyright © 2014 J. Zhou and P. Li. 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.

Let Ta be the multilinear Fourier multiplier operator associated with multiplier a satisfying the Sobolev regularity that supl€Z||o,l||ws1,...,sm (Rrnn) < m for some sk e («/2,«] (fc = 1,... ,m). The authors prove that if bt ,...,bm e BMO(R") and T e nr=iAPk/tk(h = »/%), then the commutator TaXb is bounded from Lp1 (R™, ) x • • • x LPm(R™, wm) to LP(R", v-). Moreover, the authors also prove that if ... ,i>m e VMO(R") and u; e n™=1^ pk!tk k = «/%), then the commutator Tffa is compact operator from Lp1 (R™, Wl) x • • • x LPm(R", wm) to L^R", v-).

1. Introduction

The study of the multilinear Fourier multiplier operator was originated by Coifman and Meyer in their celebrated work [1, 2]. Let a e Lm(rm"); the multilinear Fourier multiplier operator Ta is defined by

Ta (/)(*):= [ exp (2m* + ••• + ?„))

jRm" (1)

for all f1,...,fm e s(r"), where 4 = cH)1 • • • and / is the Fourier transform of /. Coifman and Meyer [2] proved that if a e Cs(rm" \ {0}) satisfies

-c^...^ (|Ç1| + ... + |Çm I)

-(iaii+-jat!)

for all |a1| + ••• |am| < s with s > 2mn + 1, then Ta is bounded from Lp1 (r")x^ • • (r") to Lp(r") for all 1 < p, p1,...,pm < m with 1/p = 1/pfc. For the case of

s > mn + 1, Kenig and Stein [3] and Grafakos and Torres [4] improved Coifman and Meyer's multiplier theorem to the indices 1/m < p < 1 by the multilinear Calderon-Zygmund

operator theory. In the last several years, considerable attention has been paid to the behavior on function spaces for Ta when the multiplier satisfies certain Sobolev regularity condition. Let O e S(Rm") satisfy

supp O c

& I I - 2

For le z, set Ol ft,..., ÇJ := a (2"%..., 2-1 ) O ft,..., ÇJ ,

lHws(Rm")

(L (i + l^il2 + - +

x|âft,...,^m fdt)

Tomita [5] proved that if

sup||^l|lw's(R™>) < œ

for some s e (mn/2, m), then TCT is bounded from L^1 (r") x • • • x L^»(r") to L^(r") provided that 1 < p, pv ..., < m

and l/_p = 1/pj.. Grafakos and Si [6] considered the mapping properties from L^ (r") x • • • x (r") to L^(r") for TCT when p < 1. Let a satisfy the Sobolev regularity that

:=(i (^i)2Sl -(O*2

x|a£ 1/2,

where := (1 + |£fc|2)1/2. Miyachi and Tomita [7] proved that if

for some e (n/2, n] (fc = 1,..., m), then TCT is bounded from L^1 (r") x ••• x Lim(r") to L^(r") provided that 1 < pi,..., pm < to with 1/p = 1/Pfc.

As well known, when a satisfies (3) for some s > mn + 1, then TCT is a standard multilinear Calderon-Zygmund operator, and then by the weighted estimates with multiple weights for multilinear Calderon-Zygmund operators, which were estimated by Lerner et al. [8], we know that, for any pi,...,pm e [1,to) and p e (0,to) with 1/p = £™=1 1/pfc and weights such that u> = (w1,...,^m) e

IK (/i,...,/ffl

fclllft )•

By a suitable kernel estimate and the theory of multilinear singular integral operator, Bui and Duong [9] established the weighted estimates with multiple weights for TCT when a satisfies (3) for m = 2 and s e (n, 2n]. Hu and Yi [10] considered the behavior on L^ (r") x • • • x L^"(r") for TCT>a when <r satisfies (5) for s1,..., sm e (n/2, n] and showed that enjoys the same L^(r") x ••• x ia(r") ^ L^(r") mapping properties as that of the operator TCT.

Now, considerable attention has been paid to the behavior on the compactness of multilinear Fourier multipliers operator with Sobolev regularity. Let VMO(r") be the closure of C™ in the BMO(r") topology, which coincides with the space of functions of vanishing mean oscillation (see [11,12]). Benyi and Torres [13] proved that if fc1,..., fcm e VMO(r") and T is multilinear Calderon-Zygmund operator, then, for p1 ,...,pm e (1,to),£ e [1,to) with 1/p = 1/pfc, the commutator is compact operator from L^1 (r") x • • • x ¿fe (r«) to i/(r"). Hu [14] proved that if a is a multilinear multiplier which satisfies (5) for some s e (mn/2,mn], i1,...,im e [1,2) ifc = n/sfc, fc1,...,fcm e VMO(r"), and e (ifc, to) for fc = 1,..., m and p e (1, to) with 1/_p = Zi=1 1/pfc, then is compact operators from L^1 (r") x ••• x Lim(r") to L^(r"). Benyi et al. [15] proved that if S e VMO(r") x • • • x VMO(r") and T is multilinear Calderon-Zygmund operator, w e Ap x • • • x A^, then, for p1,..., pm e (1,to), p e (1, to) with 1/_p = 1/pfc, the commutator is compact operatorfrom L^1 (r", ^)x- • •xLim(r", u>m) to L^(r", va).

Inspired by the above, we consider the weighted compactness of the commutator of the multilinear Fourier multiplier operator on L^ (r").

Given a multilinear Fourier multiplier operator TCT, fc1,..., fcm e BMO(r") and b = (fc1,..., fcm), the commutator To, zfc(/)(x) is defined by

(/) (*) :=! fe, Tjfc (/1,..., /m) (x), (9)

(/1,...,/fc,...,/m)(x) :=fcfc (x)Tff (/1,...,/fc,...,/m)(x) (10) -rff (/1,...,fefc/fc,...,/m)(x).

Our main results are stated as follows.

Theorem 1. Suppose that a be a multilinear multiplier which satisfies (7) /or some e (n/2, n] (fc = 1,..., m) and i1,..., im e [1,2). Let = n/sfc, e (ifc, to) /or fc = 1,..., m and 1/_p = 1/pfc with p > 1. /the weights u^,..., satis/y w e nr=1Aft/tt, then,./or any ^..., fcm e BMO(r"), Tff>a is bounded from L^ (r", ) x • • • x (r", wm) to L^ (r", v£).

Theorem 2. Suppose that a be a multilinear multiplier which satisfies (7) /or some e (n/2, n] (fc = 1,..., m) and i1,..., im e [1,2). Let = n/sfc, e (ifc, to) /or fc = 1,..., m and 1/p = 1/pfc with p > 1. /the weights ,..., satis/y e ni^ , then,/or any &1,...,fcm e VMO(r"), Tff>2, is a compact operator from L^1 (r", ) x • • • x Lim (r", wm) to

Because the regularity condition ||oiNw's(Rm") < to is stronger than NalNws1- -sm(Rmn) < to, we have the following corollaries.

Corollary 3. Suppose that a be a multilinear multiplier which satisfies (5) /or some s e (mn/2, mn]. Let r = mn/s, e (rnn/s, to) /or fc = 1,..., m and 1/_p = 1/pfc with p > 1. |/the weights ..., satis/y w e n^U ^pk/r, then,/or any bl,...,bm e BMO(r"), is bounded/rom L^1 (r",w1) x • • • x L^m(r",^m) to L^(r", vd).

Corollary 4. Suppose that a be a multilinear multiplier which satisfies (5) /or some s e (mn/2, mn]. Let r = rn«/s, e (rnn/s, to) /or fc = 1,..., m and 1/_p = 1/pfc with _p > 1. Tf the weights ..., satis/y w e n^-Apt/r, then, /or any fc1,..., fcm e VMO(r"), TCT is a compact operator/rom L^1 (r", w1) x • • • x Lim(r", wm) to L^(r", v^).

The paper is organized as follows. In Section 2, we give some necessary notion and lemmas. In Section 3, we prove our main results, Theorems 1 and 2. Throughout the paper, C always denotes a positive constant that may vary from line to line but remains independent of the main variables. We use the symbol A < B to indicate that there exists a positive constant C such that A < GB. We use B(x, .R) to denote a ball

centered at x with radius R. For a ball Be r" and A > 0, we use AB to denote the ball concentric with B whose radius is A times of B's. As usual, |£| denotes the Lebesgue measure of a measurable set E in r" and denotes the characteristic function of E. For p > 1, we denote by p' = p/(p-1) the dual exponent of p.

2. Some Notations and Lemmas

Let us first introduce some definitions below.

Definition 5. Let m > 1 be an integer, and let wl,...,wm be weights, p1,...,pm, p e (0, to), with 1/p = 1/pk, rk e (0,pk] (1 > k > m). Set w = (w1,...,wtn) and v^ :=

UZiwkPk .Then

we A pJn x---xA pm lrm

a — m1-(Pklrk)' e A ,

Vk .- Wk e Am{pkirk),,

v,?, e A

k - 1,. ,.,m,

Definition 6. For any f := (f1,..., fm), r := (r1,..., rm) and p > 1, m is defined by

m I 1 ( \1/rk

Mf(f)(x):= sup (ykTdyk) . (12)

xeQ k=\\ |Q| JQ J

For S > 0, Ms is the maximal function Ms (f)(x) := M(|/|S)1/S (x)

suP^ f If^fdy

xeQ M JQ

The sharp maximal function M* of Fefferman-Stein is defined by

{f) (x) .- sup mf-^ f \f {y) - c\ dy

xeQ c M JQ

- suPl^ f \f{y)-fQ\dy>

xeQ IUI JQ

where /q :=(1/|Q|)Jq f(y)dy. Next, we give some symbols.

Let a e Lm(Rmn) and O e s(rm") satisfy (3). For 1 e z, define

ai (^...¿m):.= ®(ftl,...,2%n)o(tl,...Xm). (15) Then

= 2-l™ae(2-eSi,...,2-lQ,

where f denotes the inverse Fourier transform of f.

For N e N, let

and denote by TaN the multiplier operator associated with aN. It is obvious that TaN is an m-linear singular operator with kernel

KN (x;yi,...,ym) := aN (x - yu ... ,x - ym) . (18) For an integer k with 1 < k <2 and x,x ,y1,..., ym e r", let

WN {x,x';y1,...,ym) .- KN {x;yl,...,ym)

-KN (x';y1,...,ym).

Assume that T is a multilinear operator initially defined on the m-fold product of Schwartz spaces, and, taking values in the space of tempered distributions,

T . S (r") x • • • x s (r") S' (r").

By the associated kernel K(x, y1,..., ym), we mean that K is a function defined off the diagonal x = y1 = ••• = ym in n+1)n, satisfying

T(fi,...,fm)(x):= f K(x;y1,...,ym)f1 (y^ • • • fm

bm )dy,

for all functions fk e s(r") and all x i supp fk. It is easy to see that the associated kernel K(x, y1,..., ym) to Fourier multiplier operator Ta is given by

K(x,y1,...,ym) :=a(x-y1,...,x-ym). (22)

To prove main results, we need the following lemmas. By the reverse Holder inequality, we have the first lemma.

Lemma 7. Assume that w e Ofc!1^pk/tk, with t1,...,tm e [1,2), pk e (tk,rn) (k = 1,...,m), and 1/p = ^k=1 1/pk with p > 1. Let sk e (n/2, n]; then there exists a constant ek e (1, min[pk/tk,sk/(sk - 1),2sk/n}) such that u>k e APkKtj£ky

For p1,...,pm e (0, rn) and s1,...,sm e r, the weighted Lebesgue space of mixed type L(pl'""pm\to(s )) is defined by the norm

•••(\ (f \F(x)\p1 (Xl)Sl dXl)P2'qi Jr" \JR" \ JR" '

X{X2)S dx2) •••(XmYm dXm where x :- (x1,... ,xm) e r" x • • • x r".

Lemma 8 (see [16]). Let r > 0, 2 < pj < <x>, and Sj > 0 for 1 < j <m. Then there exists a constant C > 0 such that

for all F e w^i-5™^(rm") with suppF c {ixj2••• + \xj2 < r}.

The following lemma is the key to our main lemma.

Lemma 9. Suppose that a be a multilinear multiplier which satisfies (7) for some sk e (n/2, n] (k = 1,..., m). Let 0 < S < r, 1/r = 1/r1 + ■■ ■ + 1/rm, rj = £jtj, and £j is the same as that

appears in Lemma 7. Thenfor all f e Lp1 (r") x-^x Lpm (r") with rj < pj < >x> for 1 < j <m,

M* (Ta (f))(x]<C.Mf(f)(x), where r = (r1,..., rm).

Proof. By Lemma 8, 1 < tjej < 2; then rj/m < 1. Fix a point x and a cube Q such that x e Q.It suffices to prove

I 1 C S \

\]Q\ h\Ta (Z) - CqI dz) < CM (?) (x), (26)

for some constant Cq. We decompose fj = f0 + f°° with fj = IjXq* for all ] = 1,...,m and Q* = 4^nQ. Then

m (yj)=n(fj (yj)+fr (yj)) j=i

= I /T &)■■■& (ym)

= n0 (yj)+ I ft (yi )-ftmm (ym),

where I = {ax,...,am : there is at least one aj = 0}. Then we can write

Ta (f)(z) = Ta (f°)(z)+ I Ta (#■ ■ )(z) := I + II.

Applying Kolmogorov's inequality to I, we have

¿JQ K (/»I'*

< dlT (f0)ll

- ll a\J 'llLr,^(Q4x/|Q|)

<cn (^Jq. ('j)i"

< CMf (f) (x),

since Ta is bounded from V1 x ■■ ■x Vm to V.

cq = I Ta(j?■■■fmmr)(x).

We claim that, for any z e Q,

I k(ft1 ■■■/mm-)(z)-Ta(ft1 -ft)wi

< CMr (f) (x).

Wl (x,z;yi,...,ym) = ae(x-yi,...,x-ym)

-oe(z-yi ,...,z-ym). At first, we consider the case al = ■■ ■ = am. We get

K (!?■■ ■ft)(z)-To (!?■■ ■OWl -IK (!?■■ ■ft)(z)-Tai (/?■■ ■OMl

m \we (x,z;yi ,...,ym)\Ufj (yj)dy

prn^n^m I \ A. A. J \ J /

TO /• /•

iiZfc=o^(2l+lQ)\(^Q)'

to m / f

IZn (I(l,+iQ,. l/j Ml* "I

k=0leZ;=l VJ(2 Q)

QT' j=i

< > > " K('■i;yf"-ym)\Ufj(y)dr

J (J "(J

\w£(x,z;yi■...■ymt dyi

rm/rm-i

to m / /•

(IQ,,\fj (yjf ^

Denote h = z - x, Q = x - Q*, and £(Q) the side length of a cube Q; it follows from Lemma 7 that

(I (I "(i

\ J(2fc+iQ)\(2fcQ^ \ J(2fc+iQ)\(2fcQ) V J(2t+iQ)\(2tQ)

\ae(h + yi,...,h + ym) - öl ...■ 7m)fi dyi

, r2/ri

rm/rrn-l

q 2fcl(Q)<|ym |<ci2t+1£(Q)

q2fc«(Q)<|ym-1 |<c32fc+1 «(Q)

c12tl(Q)<|y1|<c22t+1l(Q)

< c(2fcl(Q)}

-(S1+-+Sm)

q2fc £(Q)<|ym |<C22t+1£(Q)

c12fcl(Q)<|ym-1 |<c22fc+1l(Q)

c12fcl(Q)<|y1 |<c22fc+1l(Q)

li^(yi>--->7rn)|r1 (ri)51"1

m' m-1 /

(ym)smrm c(2fcl(Q)}-(S1+'"+Sm)2l(s1+-+sm)

i(Q)S|,m,<,.1«Q,

c12fcl(Q)<|ym-1 |<c22fc+1l(Q)

c12fcl(Q)<|y1 |<c22fc+1l(Q)

-im«- ,-,-« N|ri

|2 ^(2 y1>--->2 ^

/m/^m-1

/--l \smrm ,

(2 7rn)

c(2fcl(Q)) (S1 +"'+Sm)2i(s1+"'+sm)2-'m"2-'("/ri+'"+"/rm)

Jc12t£(Q)<|ym|<£2 2fc+1l(Q)

Jc12fcl(Q)<|ym-1 |<c22fc+1l(Q)

:12fcl(Q)<|^1 |<C22fc+1l(Q) ^ (Z1, . . . , Zm)|f1

x (Z1)S1r1 dz1

r2 /ri

4/rm~1

(zm) m

< C(2fcl (Q))-(S1 +'''+Sm)2-l("/r1+'''+"/rm-51 5m)

lHWs1'-'sm*

Given that 2l° < l(Q) < 2lo+1, we have that

< C sup||ai||WS1,...,Sm «

x £ (2fcl(Q))-(s1+'''+Sm)2-l("/r1+'''+"/r1-s1-'''-sm)

«<«0

< C SUp||al||WS1,...,Sm 2-fc(S1+ ' ' '^(Q)-^1* '' '+«/>m ).

On the other hand, a similar process follows that in [17]; we get that

(2fc+1Q)\(2fcqAJ(21+1q)\(2'Q) \ J(2fc+1Q)\(2fcQ)

(h + ,..., h + yj - <7« ..., yjf1

4/rm~1

we have

Jc12t£(Q)<|ym|<c22t+1 l(Q)

Jc12l£(Q)<|ym_1|<c22l+1£(Q)

y Jc12fcl(Q)<|y1|<c22fc+1l(Q)

(|0 ^V^fo +Qh,...,yn

+0h)| ddj dy1

m' m-1

0 y Jc12t£(Q)<|ym|<C22t+1 l(Q)

c12fcl(Q)<|ym_1 |<c22fc+1l(Q)

c12l£(Q)<|y1|<c22l+1£(Q)

I h • Va£ (^1 +dh,...,ym + 0h)|r 1

m' m-1

C12t£(Q)<|ym|<C22t+1£(Q)

c12fcl(Q)<|ym-1 |<c2 2fc+1l(Q)

c12fcl(Q)<|y1 |<c22fc+1l(Q)

VOt(yi,...,ym)\1 d^i

r2 /ri

rm/rm-1

where h = (h,...,h) e (r")m Since

h • V<^ ..... 7m) = Yhid}Vö^ ..... 7m i=1

<Ii(Q)l

~1\ Jc12t£(Q)<|ym|<C22t+1£(Q)

c12t£(Q)<|ym-1 |<c22 l(Q)

c12fcl(Q)<|y1 |<c22fc+1l(Q)

rm/rm-1

<£l(Q) (2fcl(Q)} i=1

-(S1+-+Sm)

C12t£(Q)<|ym|<c22t+1£(Q)

c12fcl(Q)<|ym-1 |<c2 2fc+1l(Q)

c12fcl(Q)<|y1 |<c22fc+1l(Q)

rm/rm-1 ,

c(2fcKQ))-(S1+'"+Sm)2l(s1+-+s") "(j

\ Jq2fcl(Q)<|ym |<C22t+1£(Q)

Jc12fcl(Q)<|y1 |<c22fc+1l(Q) I --fmn -v - \|r

|2 ^M2 ^1.....2 7m)|

/--I \smrrn j

X (2 7rn) ^

< c(2fcl(Q)) (Sl+"'+Sm)2i(Sl+'"+s™)2-im"2-i("/ri' +"'+"/r'») "(l

i2t£(Q)<|ym|<c22t+1£(Q)

c12fcl(Q)<|ym_1 |<c22fc+1l(Q)

Jq2fcl(Q)<|y1 |<c22fc+1l(Q)

|a..â^(zi,...,zm)ri (zi)51"1 dzi

r2 /ri

m' m-1

(zm) "

<c(2fcl(Q)}......2

(s1+"'+s»L-f(»/r1+-'t»/r„t1-i1-----sm)

FromLemma8, n/^ h-----hm/^ > s1 h-----+sm-1. It is deduced

«>«0

It remains to consider the case that there exists a proper subset iii'-'-'iyl of {1,...,rn}, 1 < y < m, such that o^ = ••• = a; = 0. We have

K (/r1,...,/:- )(Z)-rff (/r1)(X)|

œ m / r \ 1/r,-

<zxn(|2tto, |/, MP ^

"(l -il (J

\ J2fc+1Q*\2fcQ* \ J2fc+1Q*\2fcQ* \ Jq*

|W|y (x,z;yi,...,ym)|r1 dyi

\ ry+1/ry

• • • dyy ) dyr+i ) • • • dyn

By the same argument as that of the case a1 = • • have that

= am> we

(/r1,...,/:- )(z)-rff (/r1,...,/:- )(x)i

< Cm? (/) (%).

This completes the proof.

Lemma 10. Suppose that a be a multilinear multiplier which satisfies (7) for some e (n/2, n] (fc = 1,..., m). Let 0 < 5 < r, 1/r = 1/r1 + • • • + 1/rm, and Tj = e-tj, and is the same as

that appears in Lemma 7. Then, for b e (BMO(R"))m and any y > r, that is, > Tj, j = 1,..., m, there exists some constant C > 0 such that

m^ (rff>zb (/))(*)

< C||S|La» (M- (r* (/)) (*) + M (/) (*)) •

I K (/f1 (/r •••/:» )M|

-fc(s1+'"+sm-K/r1-----rc/rm)

< Cm? (/) (%).

/or a// m-tuples f = (/1; ...,/m) of bounded measurable functions with compact support.

The proof of the above lemma is standard. A statement similar to Lemma 2.7 in [17] with minor modifications (/j (x) ( ) deduces the estimates. We omit the details here.

Lemma 11. Let a be a multilinear multiplier which satisfies (7) for some e (n/2, n] (fc = 1,..., m), r1,..., rm e [1,2) such

that rjSj > n (j = 2,...,m). Then, for every i e (0, sx ], R > 0 and x e r" \ 2R,

Jr" JR" Jj

fx-yl,...,x-ym)

xfi {yi)---fm {ym)\dyi •••dym

i 2-iu-n/ri^nMrkfk \x\ i=2

x(x)9(R)(1/ri)-(1/Pi) .

By a similar way in the proof of the Lemma 2.4 in [14], with slight changes, we can get the conclusion of Lemma 11 and we omit the details.

About the proof of compactness, as in [18] we will rely on the Frechet-Kolmogorov theorem characterizing the precompactness of a set in Lp. More precisely, see Yosida's book [19]. For more about compactness, we refer to [20, 21].

Lemma 12. A set H is precompact in Lp, < p < to if and only if

(i) suph€H\MLP < TO

(ii) limA^m\Mlp(\x\>a) = 0 uniformly in he H,

(iii) limt^0\\h(- +1) - h(^)\\LP = 0 uniformly in he H.

3. Proof of Theorems 1 and 2

Proof of Theorem 1. We only present the case that m = 2. We have by Lemma 10 and Theorem 3.2 in [8]

\\Tl.b {flfm)\\Le(K",va)

<\\Ms (t» (mim^) <K (T-b (/))IUva)

< IISIL. „, (\\.M9(f)(x)\\

ÏÏBMO2

< \\b\\

\Lt(R",v.)

+ IK (T)(f)(x)l{R^ + H (T)(f)Ml(R^

+ ||m (f)(x)

sllfclL.„, n\\Â

which completes the proof of Theorem 1.

ÏÏBMO2 i i "JkÏÏLPk (R",wk)' k=l

Proof of Theorem 2. We will employ some ideas of Benyi and Torres [13]. Without loss of the generality, we only prove the case m = 2. Let pk e (rk,ix>) (k = 1,2), p e (1, to), with lip = £1/pk,andbl,b2 e C°°(r").Notethat,forany fl,f2 e

s(r") and almost every % e r",

lim T^zb {fi> fi)(x) = Ta,i.b {fi> fi)(x). (46)

It is enough to prove that the following conditions hold:

(a) TaN is bounded from Lpl (r", w1) x Lp2 (r", w2) to

Lp(R", va);

(b) for each fixed e > 0, there exists a constant A = A(e) which is independent of N, fl, and f2 such that

( VP J-r

(fiJi )\P]a(x)dx) <en\\fk

Jjxj>A ) k=1

kÏÏLPk (R",wk)'

(c) for each fixed e > 0, there exists a constant p = p(e) which is independent of N, fl, and f2 such that, for all t with 0 < \t\ < p,

\KNxb {fi> fi) (0 - Ta"zb {fi'fi) (■ +

nïï/k

kÏÏLPk (R",wk)'

Then by the Fatou Lemma, the conditions (a), (b), and (c) still hold true if TcrNXb(fi, f2) is replaced by TaXb(fi, f2).

It is clear that the first condition (a) holds according to Theorem 1.

Then, we prove the conclusion (b). Let D > 0 be large enough such that supp bl c BD(0) := B(0, D) and let A > max|2D, 1}. Then for every x with \x\ > 2A, we have by Lemma 11 that

\kn {x; yi ,yi) fi {yi) fi {yi)\ dyidyi

JR" J supp b

JR" J supp b1

X aN{x-yi'x-yi)

xfi {yi)fi {yi.

dyidyi

1 --¿(Sj-n/r)

iIIl^i (r",^) xmrj f2 (x)9(bd (0))(1/r)-(1/^

Y 1 2-l(8-"/^)

-N<£<0 M

xMr2 f2 (x)9(bd (0))(1/r')-(1/^

< 1 + ) l^1^1 (R" xM,2 f2 (x)9(bd (0))(1/^)-(1/a),

if we choose i = s1 and i = 9 e (n/(p1e1 ),n/r1) in Lemma 11 respectively. Therefore,

(j \Va (Â> fl)\P]w (X) dx)

Mi~(R») nll/tlliÄ(R-^)9(£d (0))(1/ri

(1/r) -(1/pl)

1ll~(R") n llJklliPk (R»

)9(5d (0))(1/ri

(1/r) -(1/A)

W1 (x)

W1 (x)

y1lll™(R") I I lUjlllA(R",u>t)' j=1

where the last inequality holds by the fact (see [22, 23]) that for v e A p > 1

(1 + M)np

dx < œ>.

This in turn leads to conclusion (b) directly.

We turn our attention to conclusion (c). We write

Va (ÀJi) (x) - VA (fv f2) (x + t) = YEj (X. t).

with E1 (x,t)

= \ kn (x;y1>y2)

l<k<2 'k '

x (X + t)- \ (X)) /1 (^1) /2 (y2) dy1dy2,

E2 (x, t)

= j (KN (x;y1,y2)-KN (x + t;y1,y2))

i<k<2 'k '

x (h1 (yô -h1 (x + t)) /1 (yù /2 (y2) dy1dy2,

E3 (x, t)

= j KN (x-.y1.y2)

Jmax |x-v(.|<5, l<k<2 'k '

x (h1 (yô - h1(x)) /1 (yù /2 (y2) dy1dy2.

E4 (x. t)

max |x-v(.|<5, l<k<2 'k '

K (x + t;y1 .y2)

X {bi (x + t)- \ (yx)) fi (yj f2 (y2) dyxdy2,

with St > 4\t\ a convenient choice to be determined later.

In a completely same way in the proof of Theorem 1.1 in [14], we can obtain the estimate of\Ej(x,t)\ (j = 1,...,4).We only list the results and omit the details.

\Ei M^II^Ur»)

x(jl Mrk fk (*)+Ms (T(f1.f2))(x)j.

xn(Mrt fj (X)+Mrt fj (X + t)), j=1

\£3 (x.t)\ <5tllV6illiM(RB) n^rt fj (x).

\E4 (x. t)\ < 5tHV6illiM(RB) n Mrt fj (x + t). Fix each e > 0, set P =

2 (1 + llV^1Hl^(R„))'

A = min

2^ + NUr^

and St = |i|A-1 for each t e r", where constant q > 0. Our estimates for those terms of Ej (j = 1,..., 4) then lead to that when 0 < |i| <

f^A (/1-/2) (') - TaN,Ub (/l' /2) ( + i)||LP(R»>].)

s((|i|+5t)||vfci||Lra(R„) + (|i|5t-1)e||fci||Lra(R„))

n«/»

fc=1 2

fclllft

this establishes conclusion (c) and we conclude that T^n^ is compact.

In a completely analogous way, if fc2 s C™, then T^n^ is compact. Moreover, then T0.w>2i is compact, thus we complete the proof of Theorem 2. □

Conflict of Interests

The authors declare that they do not have any commercial or associative interest that represents a conflict of interests in connection with the work submitted.

Acknowledgments

The authors would like to thank the referee for his/her helpful suggestions. The paper is supported by the National Natural Science Foundation of China (11261055), the Natural Science Foundation Project of Xinjiang (2011211A005), and Xinjiang University Foundation Project (BS120104).

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