Scholarly article on topic 'Some Weighted Estimates for Multilinear Fourier Multiplier Operators'

Some Weighted Estimates for Multilinear Fourier Multiplier Operators Academic research paper on "Mathematics"

CC BY
0
0
Share paper
Academic journal
Abstract and Applied Analysis
OECD Field of science
Keywords
{""}

Academic research paper on topic "Some Weighted Estimates for Multilinear Fourier Multiplier Operators"

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 987205,10 pages http://dx.doi.org/10.1155/2013/987205

Research Article

Some Weighted Estimates for Multilinear Fourier Multiplier Operators

Zengyan Si

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China Correspondence should be addressed to Zengyan Si; zengyan@hpu.edu.cn Received 3 August 2013; Revised 2 October 2013; Accepted 20 October 2013 Academic Editor: Mieczyslaw Mastylo

Copyright © 2013 Zengyan Si. 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.

We first provide a weighted Fourier multiplier theorem for multilinear operators which extends Theorem 1.2 in Fujita and Tomita (2012) by using Lr-based Sobolev spaces (1 < r < 2). Then, by using a different method, we obtain a result parallel to Theorem 6.2 which is an improvement of Theorem 1.2 under assumption (i) in Fujita and Tomita (2012).

1. Introduction

During the last several years, considerable attention has been paid to the study of multilinear Fourier multiplier operators. Let S(rd) be the Schwartz space of all rapidly decreasing smooth functions on rd, for some d e z+. The multilinear Fourier multiplier operator TCT associated with a symbol <r is defined by

X/l fe) '"/m

for /; e S(r"), i = 1,...,m.

Coifman and Meyer [1] proved that if a is a bounded function on rm" \ {0} that satisfies

^ • ••3£ a(Çi,...,Çm)|^Ca(|Çi| + ...+ |Çm|)"

-(|a1|+-+|am|) (2)

away from the origin for all sufficiently large multi-indices a^ then TCT is bounded from the product L^1 (r") x • • • x ia(r") to i/(r") for all 1 < pl,...,pm,p < m satisfying

1/pi H-----+ 1/pm = 1/^. The multiplier theorem of Coifman

and Meyer was extended to indices p < 1 (and larger than 1/rn) by Grafakos and Torres [2] and Kenig and Stein [3]

(when m = 2). Exploiting the idea of the proof of the Hormander multiplier theorem in [4], Tomita [5] gave a Hormander type theorem for multilinear Fourier multipliers with more weaker smoothness condition assumed on <r than (2). Grafakos and Si [6] gave similar results for p < 1 by using Lr-based Sobolev spaces (1 < r < 2). Grafakos et al. [7] proved the L -boundedness of TCT with multipliers of limited smoothness.

In order to state other known results, we first introduce some notations. The Laplacian on Rd is A^ = S^/Sxj, that is, the sum of the second partials of g in every variable. We define the operator (7 - A)r/2(^) = F-1(wyF(^)), where

(Ç) = (1+4^2|^|2)r/2 for y > 0. LetLry(rd) be the Lr-based Sobolev space with norm

= ||U-A)y/2 /

where 1 < r < ra.

Let s = (s1,...,sm) and let the product type Sobolev space WSl""'Sm(Rm") consist of all functions F such that the following norm of F is finite:

l|F||wsw

■WiD««'

(f Ä)2S1 •••(^m)2S™ ^(OM

VjRm" /

where Ç=(Çi,...,Çm) and (&> = (1 + |2)1/2.

Let y 6 S(rmn) be such that supp y c r 6 rmn : 1/2 < <2} and = 1 for £ = 0.

Let Sj(rd) be the set of all Schwartz functions W on rd, whose Fourier transform is supported in an annulus of the form {£ : q < |£| < c2}, is nonvanishing in a smaller annulus {£ : q' < |£| < c2} (for some choice of constants 0 < q < q' < c'2 < c2 < to), and satisfies

(2-j£) = constant, £ e rd \ {0}.

The weighted estimate for Ta is also an interesting topic in harmonic analysis. And it has attracted many authors in this area. Recently, Fujita and Tomita [8] established some weighted estimates of Ta under the Hormander condition and classical A„ weights. For other works about the weighted estimates for Ta, see [9,10] and the references therein.

TheoremA(see [8]). Let 1 < p^p2,...,pN < to, 1/pi + ••• + 1/Pn = 1/p, and Nn/2 < s < Nn. Assume

(i) min p1,...,pN > N«/S and W 6 A min plS/Nn,...,pNs/Nn;

(ii) minp1,...,pN > (Nn/s)' and 1 < p < to, «1-p 6

A p's/(Nn).

If a 6 Lm satisfies supfcg2||<T(2fc-)^||L2 < to, then Ta is bounded from Lpl («) x • • • x Lpm («) to Lp («).

An improvement of Theorem 1.2 is stated as follows.

Theorem B (see [8]). Let 1 < p^p2,. ..,pN < to, 1/p1 + • • • + 1/pN = 1/p, and n/2 < Sj < n, j = 1,..., N. Assume pj > n/sj and Wj 6 A ijSi/n for 1 < j < N. If a 6 Lm satisfies supfc6zlla(2fc0^Hw<si'-'sN>(£N") < to, then Ta is bounded from Lpl x • • • x Lim(wN) to L*(w), where w = of* • • • </pN.

The first purpose of this paper is to improve Theorem A by using Lr-based Sobolev spaces (1 < r < 2). The second purpose is to give a new proof of Theorem B. The following are the main results.

Theorem 1. For some 1 < r < 2, suppose that a 6 L° and W 6 S1(rmn) satisfy, for some mn/r < y < mn,

suplu2 -m , , = K < to.

fcez" "Lv(R '

If p1,..., pm, y, and the weights w satisfy one of the following two conditions:

(i) min|p^..., pm} > mn/y and w e

A min {p1y/(mn),...,pmy/(mn)},

(ii) min|p^..., pm} > (mn/y)', 1 < p < to, and «1-p e

A p'y/(mn),

then there is a number S = <?(mn, y, r) satisfying 0 < S < r - 1, such that the m-linear operator Ta, associated with the multiplier a, is bounded from Lp1 («) x • • • x Lpm («) to Lp(«), whenever r - 5 < < to for all j = 1,..., m, and p is given by 1/P= 1/Pl +•••+ 1/Pm.

Theorem 2. Let 1 < p1,...,pm < 2 and let s1 > n/p1,..., Sm > M/pm and S1 + • • • + Sm < «/p1 + • • • + «/pm + 1.f ^ 6 Lm(rmn) satisfies sup^H^^U- ,,sm (Rmn) < to, then Ta is bounded from Lqi (w^1 )x---xi?" (w^) to L?(w?), whenever 1 < < to, 1/^1 + ••• + 1/^m = 1/q, and

,...,^m™ ) 6 (A?I ,...,A?m ) with ^ = ^1 '^m. 2. The Proof of Theorem 1

In this section we discuss the proof of Theorem 1. We begin with some definitions for maximal operators. Throughout the paper, M denotes the Hardy-Littlewood maximal operator defined by

M(/)(x)= sup^ f (7)

xeQ M JQ

where Q moves over all cubes containing x. For S > 0, Ms is the maximal function defined by

Ms/(*) = M(|/|S)1/S (x) = (qupfQ |/(y)|Sdyy .

In addition, M® is the sharp maximal function of Fefferman and Stein:

M*/(x) = ^pf^ [ |/(y)-c|dy

Q3x c M JQ

" sUPÎH Í

Q3xIvl JQ

where /q denotes the average of f over Q and a variant of M® is given by

M»/M = M»(|/|S)1/S (x). (10)

We prepare some lemmas which will be used later.

Lemma 3 (see [11]). Let 1 < p < to and w 6 Ap. Then (1) 6 Api; (2) there exists q < p such that w 6 Aq.

Lemma 4 (see [12]). Let 1 < p, q < to, and w 6 AThen there exist positive finite constants C(p, such that

1/? 1/?

{l|M(/fc)r <C(p,^) ■ Il/.r

fceZ LP M .fceZ

Li(^) (11)

/or a// sequences {/¡¿1^ of locally integrable functions on r".

Lemma 5. Let A¡¡ be t^e Littlewood-Paley operator given by Afc (^)^(^) = fc e z, w^ere ¥ is a Schwartz

function whose Fourier transform is supported in the annulus {£ : < < 2bj, for some b e z+, and satisfies

ZfcsZ = c0, for some constant c0. Let 0 < p < to

and w e Am. Then there is a constant c = c(n, p, c0such that for L^ («) functions f one has

I|Afc (/)|:

Proof. The proof follows from similar steps in Lemma 4 of [6] and combines the method used in Remark2.6 of [8]. Let O be a Schwartz function with integral one. Then,

| /(x)| = lim|Ot */(x)| < sup |Ot */(x)|.

If «6 ATO, the weighted Hardy space H^(«) coincides with the weighted Triebel-Lizorkin space F^' for 0 < p < ra. Hence, if «6 ATO, we have

IIlp(W)

The proof is complete.

Now we give the proof of Theorem 1.

Proof. Since the proof follows from similar steps in Theorem 1 in [6], we just give the different parts. For each j = 1,..., m, we let be the set of points (£1;...,£m) in (r")m such that = max{|^1|,..., |£m|} and we introduce nonnegative smooth functions on [0, ra)m-1 that are supported in [0,11/10]m-1 such that

1 = jr(j^,...,^,...,^

for all ,..., £m) = 0, with the understanding that the variable with the hat is missing. These functions introduce a partition of unity of (r")m \ {0} subordinate to a conical neighborhood of the region £j.

Each region can be written as the union of sets:

Rjjc = {&,...,vs = j} (16)

with fc = 1,..., m. We need to work with a finer partition of unity, subordinate to each . To achieve this, for each j, we introduce smooth functions on [0, ra)m-2 supported in [0,11/10]m-2 such that

M M M N

for all ..., £m) in the support of with = 0.

We now have obtained the following partition of unity of

T \{0}:

1 = H0; (-)0;Jt (-).

j=1fc=1

where the dots indicate the variables of each function.

We now introduce a nonnegative smooth bump f supported in the interval [(10m)- ,2] and equal to 1 in the interval [(5m)- , 12/10] and we decompose <r into a finite number of multipliers:

< sup^f * /(%)|

f>0 where

= c ll/IU(o>) ~ ll/llpf («) (14)

/ ^ 1/2 fcj-Jt &

< c (iK (/)|2

VfceZ /

j=1fc=1

0,,fc (-)0;Jt (-)(1-y( M

We will prove the required assertion for each piece of this decomposition, that is, for the multipliers oO^j. and oWjj for each pair (j, fc) in the previous sum. In view of the symmetry of the decomposition, it suffices to consider the case of a fixed pair (j, fc) in the previous sum. To simplify notation, we fix the pair (m, m - 1); thus, for the rest of the proof we fix j = m and fc = m - 1 and we prove boundedness for the m-linear operators whose symbols are a1 = oOmm-1 and <r2 = <r¥mm-1. These correspond to the m-linear operators Ta and Ta, respectively.

We first prove Theorem 1 under assumption (i). Since 1 < mn/y < min{r, p1,..., _pm}, we can take p such that 1 < mn/y < p < min{r,p1 ,...,pm} and W 6 Amin{Pl/p'...'Pm/p}. We first consider T^ (/1,..., /m), where /j are fixed Schwartz functions. We fix a Schwartz radial function ^ whose Fourier transform is supported in the annulus 1 - (1/25) < |£| < 2 and satisfies

1^) = 1,

£ e r" \ {0}.

Associated with ^ we define the Littlewood-Paley operator Aj(/) = / * V2-j , where %(%) = for i > 0. We

also define an operator Sj by setting

= # * C2-J >

where £ is a smooth function whose Fourier transform is equal to 1 on the ball |z| < 3/5m and vanishes outside

the double of this ball. As in [6, page 143], by using Lemma 5 we get

IK (/l> • • • , /rn)|

XK (Si (/l) , • • • , (/m-l), A; (/m))|

L?(a>)

We will use the following estimate for Ta (see [6, page 145]):

K (Si (/l)>->Sj C/m-lMj (/m))|

m-1 . ,/

<C^n(M(M(/,)P))l/P(M(|A; (/jf))^

We now square the previous expression, we sum over j 6 z, and we take square roots. Since r - S = p, the hypothesis pj > r-5implies pj > ^,andthuseachterm (M(M(/)p))1/p is bounded on Lpj («). We obtain

I^i (/l' • • • ' /m-l' /m)|

xK (Si (/l)>->Sj (/m-l) a;(/m))

ima (/m)D xfl (m(m(/)p))1/p

L? (w)

IMA (/m)D

ni/.II

LPm'P(w)

< C" K^I/I

L? (w)>

L?i (w)

where the last step holds due to Lemma 4 with ^ = 2/p and the weighted Littlewood-Paley theorem.

Next we deal with <r2. Following [6, page 146], we write

^a2 (/1, . . . , /m-1, /m)

= Ira2 (Sj (/1) , . . . , Sj (/m-2) , A'j (/m-1) , A j (/m)} ,

K (Si (/l).....Si (/m-2) , A'i (/m-l) , A i (/m))|

m-2 . . ! /

<CKn(M(M(/,)P))l/P(M(|A'i (/m-l)|P))l/P i=l

x(M(|A; (/m)|P ))l/P,

for some other Littlewood-Paley operator A'j which is given on the Fourier transform by multiplication with a bump ©(2-j£), where 0 is equal to one on the annulus {£ 6 rn : (24/25)-(1/10m) < |£| < 4} and vanishes on a larger annulus. Also, Sj is given by convolution with ^-j, where is a smooth function whose Fourier transform is equal to 1 on the ball |z| < (22/10) and vanishes outside the double of this ball.

Summing over j and taking Lr(«) norms yield

IK (/l' • • • ' /m-l' /m^L?^)

n(M(M(/)P))l/PX(M(|A'; (/m-l)D)

x(M(|A; (/m)|P))

n(M(M(/)P))

n IKA(/)|p)

i=m-l jeZ

P || 2/ P

where the last step holds due to the Cauchy-Schwarz inequality and we omitted the prime from the term with i = m - 1 for the matter of simplicity. Applying Holder's inequality and using that p < 2 and Lemma 4 we obtain the conclusion that the expression above is bounded by

C'K II/ll

Il?I (w)

mIIL?" (w)'

We next prove Theorem 1 under assumption (ii). It was proven in [6, page 136] that condition (6) is invariant under the adjoints; that is, it is also valid for the symbols of the dual operators a*m(^1 ,...,£„) = ^(£1,..., ^m-1,-^1 +••• + £«)). To prove the required assertion, by duality, it is enough to prove that Ta.m and Ta.m are bounded from Lp1 («) x • • • x

Lfo-1 («) x Lr'(«1-p') to Lr™(«1-p™). We may assume that Pm = rnin{p!,...,pm}. Since pm < (mn/y)', we see 1/p',

1/pfc < 1/p' + 1/P2 + ••• + 1/pm = 1/pm < y/(mn).

Hence, mn/y < min{r, p', p1,..., pm-1}. Since p < pm and «1-p 6 Ayy/(mn) c Ay, we deduce that w 6 A^ c A„ ; then 6 A„/ . It is obvious that w(1-ft»)/rm =

w-1/V/ri • • • . Since pm < pfc, 1/p = 1/p1 +• • • + 1/pfc +

••• + 1/Pm-1 > 1/Pm + 1/Pfc ^ 2/Pfc. That is, P < Pfc/2; then w 6 Ar c Aft/2 c Afty/(mn). Therefore, we take a

positive number p such that 1 < mn/y < p < min{r, p', p1,

...,pm-1}, and y > rnn/p such that «1-p 6 Ay/p and w 6 Apk/p. We have

|K1*m (/1,..., /m-l, /m)||LP;„(^-fi)

X|rffi (Sj (/l ),•••, Sj (/m-l),

Aj (/m))|

X^(|Aj (/m)|P)2/P i

xn №(/>)P))l/P

L" (w1-Pm )

L? (w1-? )

L?i (w)

XM(A (/m)|P)

xnII/>L, (.) ¡=l

< C K H/l IIL?i (w) • • • H/mIl?"(»)• Similarly, we have

II^ff2*m (/l> • • • " /m-l> /m ) 11 L?rn (w1-?" )

n(M(M(/,)P))l/P ^

xX(M(|A'j (/m-l)|P))l/P

x (M(|Aj (/m)|P))l/P--l/i+l/^™-1

n(M(M(/,)P))l/P

L?" (w1-Pm )

L?i («i)

X|M(|Aj (/m-l )D

P || 2/ P

XK(|Aj (/m)|P)|2/P

< C K II/lIIL?1 (w) ••• II/mIIL?m(„)•

This concludes the proof of Theorem 1.

3. The Proof of Theorem 2

LP''^1-?')

(30) □

We begin with some lemmas which will be used in the proof of Theorem 2.

Lemma 6 (see [11]). Let 0 < p and S < to and let w be a weight in ATO. Then, there exists C > 0 (depending on the ATO constant of w) such that

(Mä/ (%) dx < C [ (31)

for all function / for which the left-hand side is finite.

Lemma 7 (see [13]). Let 0 < p^p2,p < to, and 1/pl + 1/p2 = 1/p. Let a be a multiplier satisfying

supfcezlH2fc-MIWs1,s2(R"") < TOforSi > max|n/2,n/pl-n/2|, s2 > max|n/2, n/p2 - n/2|, and sl + s2 > n/pl + n/p2 - n/2; then Tff is bounded from H^1 (r") x (r") to L^(r").

Remark 8. It should be pointed out that Lemma 7 can be extended to the case m > 3.

Lemma 9 (see [8]). Let r > 0, e [2, to), and

sl, • • •, sm > 0. Then there exists a constant C > 0 such that

(j -((j .....imJl*1 <«lP

\ Jr" \ V JR" 1 1 /

• tfm > "

< C||F||WS

'Sm/^m ( Rm" )

for all F 6 Wsi/gl--s"/<?™(rmn) with supp F c

{^|2 + • • • + |Xm|2 <f}.

Next, we give a pointwise control of M®Ta(/) which becomes very useful in the proof of Theorem 2.

Lemma 10. Let 1 < p1,...,pm < 2. Assume that o 6 Lm(rmn) which satisfies supfcgZ||o-(2fc-)y|| < TOwSl,...,Sm(Rm„) for

S1 > n/p1,...,Sm > «/pm and S1 +-" + sm < n/p1 +

Lpm-1/P(w)

----h n/pm + 1. For any 0 < S < 1/m, one has M®Ta(/)(x) <

cnm=1 ^ /j(^).

Proof. For simplicity, we only prove for the case m = 2, since there is no essential difference for the general case. Fix an x 6 rn and a cube Q with side length /, such that x 6 Q.

Let /, = +/T where ¿0 = /lXp, and = /lX(p,)C for « = 1,2 and Q* = 4V«Q. Since 0 < 5 < 1/2,we have

/ 1 f \1/5 (iQ|Jp |lJff (/1,/2)(z)|S-|c|S|dzJ

<C(i^iP lJff (/1,/2)(Z)-C|SdZ)l/S <C(^ip ^ W ^

+ (/r,/2°°)(z) + Tff (/i,,/2°°)(Z) (/10,/20)(z)-C|

+ (¿1 iP |Jff (/r, /2°) (z) + (/i>, /2°) (z)

5 \ 1/5 +T. (/°°,/2>)-C| dZ)

= ^1 +U2.

We first consider U1. By Kolmogorov's inequality, Holder's inequality, and Lemma 7, we have

(¿IpK (/?•/,>)!'<

<cn (|<HIp- |/j «f

<cnMft /j (*),

where 1/_p = 1/p1 + 1/_p2 with p > 5 and 1 < p1, p2 < to. Next we deal with U2. We choose C = £j=1 C;, where

C1 = rff (/r,/2°)(x),

C2 = Tff (/10,/2°)(x), (35)

Cs = rff (/r,/2°)M.

We may split U2 as U2 < U21 + U22 + U23, where

U21 = (¿! Ip ^ (/r, /2°) (z) - (^ /2°)

u22 = (¿1 Ip K (/0, /2°) (s) - (/0, /2°) tof^)1",

U23 = (^ Ip V" /20) - ^ (/°, /2) (*)f

Now we estimate U21 first. We decompose a as

a=X^('/2j) (37)

jeZ jeZ

Let = a(-)^(-/2j), where ^ 6 S(r2") with supp f c tf 6 r2" : 1/2 < < 2} and £jeZ ^(2-jO = 1, ? = 0. Thus, we have

K (/°°,/2°°)(*)-rff (/r,/2°°)(x)| < c£ k. or, /2°) (Z) - r*. (/r, /2°) (x)|

<CI I I L , * L, * kJ (z-^1,Z-^2)-^J h/1 00/2 00 ^2

jeZfc7=0i:;=^2t2+1p'\2'2p. J2fci+1p*\2fcip* | ; |

<ClIIlt1 „ L, „ K (z->,1,z->0-0'j fc-^*-.^ |/1 00/2 OO^tt ^2

fcT=0fc2=0 jeZ 2'2+1p.\2^2p. J2fci+1p*\2fcip* |; 7 |

= 1 ZJfcl'fc2. fc1=0fc2=0

Applying Holder's inequality we have

1 2 jeZ ^2fc2+1Q*\2fc2Q* J2fcl+1Q*\2fcl Q* | ^ ^ |

<CZ(L , iL , kV (z-^1'Z-^2)-^V O-^*-^)|Pl ^ ^2

jeZ\ J2t2+1Q«\2t2Q^ J2fci+1Q*\2fciQ* 1 ^ ^ 1 !

^ (J2..+1Q. ^ M"1 ^ )1/"ilJ.2.1Q. ^ W^ ^.f2

Let h = z - x and (J = % - Q*. Then we have

1 ^ \>J2t2+1Q'\2t2Q' \ J2^+1Q'\2^Q' 1 J J 1 !

< C(L - t -(L - „ -kV (h + ^1 'h + ^2)-^V (71>72)|"1 ^

y J2t2+1Q\2t2Q VJ2t1+1Q\2t1Q | l ' )

<C(| , _ , _(| , _ , _K (^1'^2)|"1 )

2l2+1Q\2t2Q \J2l1+1Q\2l1Q

<G(2fc1 /)-S1 (2fc2/)"* ( | ^ 1 _ ^ _ ( | ^ 1 ^ _ |^V

V 7 V 7 \ J2t2+1Q\2t2Q \ j2fc1+1Q*\2fc1Q | 7 |

s/2 \"2/"' s/.

<G(2fc1 if1 (2fc2/)-S2 ( | ( | |aV (2-jy1,2-jr2)|"1

s \"2/"I s \1/p2

x(1 + |2-j>1 |2)W22-j"x ) (1 + |2-,^2|2)^2-j"d72 )

< C(2fcI Z)—I (2fc2/}-S22-j(Sl+S2)2jn((1/Pl)+(1/p2))

v u-i mpI/, . |.. |2\SIp!/2,

, , xo'/o' , \1/r2

|2-2jnav (r^^f(1 + WT^) * '(1 + W2r%)

< c(2fcI/}-SI (2fc2/}-S22-j(SI+S2)2jn((1/rI)+(1/r2))||a (2j-) ^||wsI„

WSI'S2

where the last inequality holds due to Lemma 9.Supposethat x^(2ji/} SI (2fc2/} S22j(n/rI+n/r2 SI S2)

2-B < / < 2-B+1. Since n/p1 + n/p2 - s1 - s2 <0, we have j>£

<C sup||a (2j-) ^||wSI>S22-fcISI2-fc2S2Z-n/rIZ-n/r2. j

^ < sup|-(2j-}^|w^ _ , ^ (41)

j>b j On the other hand _I

^ < ( Lq^Q ( l2,I+IQ5\2^IQ |aV ^ + ^ * + ^ ^ ^ ^^ ) ^ j

(fx / \r2/rI x 1/r2

<C( l^Q (i^^^Q (io ^^ + ^ + 0h)|*O

< C 10 ( (L^Q ^ • V ^ ^ + ^ » + ^ *

where h = (h, h) 6 r2n. Since h • V(aV)(y1, y2) =

hr9r(aV)(^1, ^2), we have

2n /r /r „' \p2/ri V

<c£(U*Q(UIs-w^N **)

<c;|i(2'Ii}-S'(2'-¡}-S- (fR, (iR_ K(.v}(2-'.1,2-j,!}r:(1 + |2-j„|2)"PI,12-jnd,1)

2 Sr'/2 \1/p2 x(1 + |2-j^2|2f 2 2-jn)

x (1 + b2|2)52) 2-j(si +s>V«1^«1^»

<C^<2fci/)-Sl (2fc2/)-2 2-i(^i+^2)2i««1/Pi)+(1/P2))2i ||a(2j-)^||wsi,s2,

where in the last inequality Lemma 9 was used again and hence

<C supfa^.)^ 2-fclSl 2-fc2S2 Z^1 Fk/^2 .

Combining the above arguments we have

|K (/r,/2")(z)-r„ (/r,/2")(*)|

< c X Z 2-fcisi2-fc2s2rn/^ rn/ft fcl=0fc2=0

x ( I t |/1 (^F

V J2li+lQ*

J2fc2+lQ! œ

< C Z 2-fci(si-"/^i)

|/2 OOf^

fc, =0

x Z2-l2(s2-"'?2»Mfi/1Mi2/2

< CMfi/1Mf2/2.

Thus, we obtain U21 < cm^i/1m^2/2. What remain to be considered are U22 and U23. We just estimate Ï722 since the same arguments can be applied to U23:

|K (/10,/200)(^)-rff (/10,/200)(x)| <cZ|rffj (/0, /2° )(*)-r„, (/0,./r)(*)|

œ /• /•

CZ Z J2t2+l . ^2 (^2)| J . (z - Z - ^2) - - * - ^2)| (^1)| ^2

j ¿2=0

<CZZ( ( L |ajV (z-^1'Z-^2 )-°j O-^*-^ ^2

jezfc2=Aj2l2+lQ*\2l2Q^JQ!

x(I |/1

2l2+lQ'

|/2 (^F^

Then by similar arguments as the above mentioned we get that

2l2+lQ'

|/2 OOP^:

|K (/1,,/2>0)(*)-'rff (/0, /2

<CZ2-fc2S2rn/pirn/p2(I |/1 (* r^

¿2=0 VJQ'

<CZ2-l!(s2-i2)Mfi /1Mf2/2

< CMfi/1Mf2/2. The proof of Lemma 10 is complete.

(47) □

Now we are ready to give the proof of Theorem 2.

Proof. By Lemma 3, we can choose 1 < p1 < and 1 < p2 < <?2 such that wf 6 A^ and «f 6 A^. Then by the Holder inequality, Lemma 10, and the weighted boundedness of M, we deduce that

|K (/1,/2)LM <||M5Ta (/1,/2)||

<C||M»Ta (/1,/2)||L^)

<C||MPi /1MP2 /2 ^^ (48)

< C||MPI Z1 ||L,I (^I ) ||Mri Z1 ||L,2 (m|2 )

< C||/1||№ « )||/2||L«2 ).

The proof of Theorem 2 is complete. □

[10] W. Li, Q. Xue, and K. Yabuta, "Weighted version of Carleson measure and multilinear Fourier multiplier," Forum Mathe-maticum, 2012.

[11] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993.

[12] K. F. Andersen and R. T. John, "Weighted inequalities for vector-valued maximal functions and singular integrals," Studia Mathematica, vol. 69, no. 1, pp. 19-31,1980/81.

[13] A. Miyachi and N. Tomita, "Minimal smoothness conditions for bilinear Fourier multipliers," Revista Matematica Iberoamericana, vol. 29, no. 2, pp. 495-530, 2013.

Conflict of Interests

The author declares that there is no conflict of interests

regarding the publication of this paper.

Acknowledgments

This work was supported by the National Nature Science

Foundation of China (nos. 11226102 and 11226103) and Doctor

Foundation of Henan Polytechnic University (no. B2012-

References

[1] R. R. Coifman and Y. Meyer, "Au delà des opérateurs pseudo-differentiels," Asterisque, vol. 57, pp. 1-185,1978.

[2] L. Grafakos and R. H. Torres, "Multilinear Calderén-Zygmund theory," Advances in Mathematics, vol. 165, no. 1, pp. 124-164, 2002.

[3] C. E. Kenig and E. M. Stein, "Multilinear estimates and fractional integration," Mathematical Research Letters, vol. 6, no. 1, pp. 1-15, 1999.

[4] J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2001.

[5] N. Tomita, "A Hormander type multiplier theorem for multilinear operators," Journal of Functional Analysis, vol. 259, no. 8, pp. 2028-2044, 2010.

[6] L. Grafakos and Z. Si, "The Hormander multiplier theorem for multilinear operators," Journal fiir die Reine und Angewandte Mathematik, vol. 668, pp. 133-147, 2012.

[7] L. Grafakos, A. Miyachi, and N. Tomita, "On multilinear Fourier multipliers of limited smoothness," Canadian Journal of Mathematics, vol. 65, no. 2, pp. 299-330, 2013.

[8] M. Fujita and N. Tomita, "Weighted norm inequalities for multilinear Fourier multipliers," Transactions of the American Mathematical Society, vol. 364, no. 12, pp. 6335-6353, 2012.

[9] G. Hu and C. C. Lin, "Weighted norm inequalities for multilinear singular integral operators and applications," Analysis and Applications. In press, http://arxiv.org/abs/1208.6346.