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

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

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 570450, 10 pages http://dx.doi.org/10.1155/2014/570450

Research Article

Weighted Morrey Estimates for Multilinear Fourier Multiplier Operators

Songbai Wang, Yinsheng Jiang, and Peng Li

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China Correspondence should be addressed to Yinsheng Jiang; ysjiang@xju.edu.cn

Received 7 December 2013; Revised 10 February 2014; Accepted 13 February 2014; Published 31 March 2014 Academic Editor: Dashan Fan

Copyright © 2014 Songbai Wang 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.

The multilinear Fourier multipliers and their commutators with Sobolev regularity are studied. The purpose of this paper is to establish that these operators are bounded on certain product Morrey spaces L^R"). Based on theboundedness of these operators from Lp1 (toj) x ••• x Lim(wm) to L^n™^-^'), we obtained that they are also bounded from L^toj) x ••• x Lim'fc(wm) to ¿M(nZiwi/ii), with0 < k < 1, 1 < pj < ot, Hp = 1/p! +••• + 1/pm,andwj e A ,j = 1,...,m.

1. Introduction

Recently some authors have taken so much interest in the text of multilinear Fourier multipliers with Sobolev regularity. To state some interesting results, we recall some necessary notations and definitions. Let <r e Lm(Rm"); the multilinear Fourier multiplier operator TCT is defined by

Tff (/)(*)= f exp+ --- + U)

}Rm" (1)

for all f = (/1 ,...,/„) e S(R")m, where 4 = ^ •••#„ and f is the Fourier transform of /. It is well known that [1] theboundedness of Tm fromL^1 (R")x- • -xL^(R") to L^R") holds if a e Cs(Rm" \ {0}) satisfying the condition

for all multi-indice |a| < s with s > 2m« + 1 and all 1 < p, p1,..., pm < ot with 1/p1 + • • • + 1/pm = 1/p. Grafakos and Torres [2] improved the multiplier theorem of Coifman and Meyer to the indices 1/m < p < 1 by the multilinear Calderon-Zygmund operator theory in the case of s > m«+1.

An important progress in this topic was given by Tomita. Let O e S(Rm") satisfy

supp O c

for all Rm" \{0}.

Ji|lws(Rm")

f (1 + Ki|2 +-- + M2)

2-,\i/2

Tomita [3] proved that if

sup Ha

'2|lws(Rm")

for some s e (mn/2, ot), then Ta is bounded from Lpl (R") x • • • x Lpn(R") to Lp(Rn) provided that 1 < p,p1,...,pm < ot and 1/p = Y!k=i 1/pk. Grafakos and Si in [4] obtained that Ta maps from Lpl(R") x ••• x Lp™(R") to Lp(Rn), if a satisfies (5) and 1/m < p < 1. Miyachi and Tomita [5] considered the problem to find minimal smoothness condition for multilinear Fourier multiplier. Let

Jl\\yfH,~¿m (Rm")

/ ( \1/2 (I (U2Sl■■■(U2Sm\¿i&,...,sm)\24] ,

VJr2" /

where := (1 + \^\2)1/2. Miyachi and Tomita [5] proved that if

suplid Wsi-s2 (R2") < OT,

for each Sj e (n/2,n], then Ta is bounded from Lpl(R") x Lp2(R") to Lp(Rn) provided that 1 < p1, p2 < ot, and p > 2/3 with 1/p = Xk=i 1/pk. Moreover, they also gave minimal smoothness condition for which Ta is bounded from Hpi (Rn)xHp2(R") to Lp(R").

Let mn/2 < s < mn, mn/s < p1,..., pm < ot, and 1/p1 +

----+ 1/pm = 1/p. Fujita and Tomita [6] proved the following

inequality:

\\Ta(fi>---JJ\W&) ^cnWMUw

if II Ol\\wslm,...,slm(Rmn) < OT and to = (Wv...,Wm) 6 A piS/(mn) X

••• X A pms/(mn) , where and in what follows = nm=i^l'Pk.

Li et al. [7] obtained the endpoint cases. Hu and Lin [8] also obtained this result from another approach. Replacing by Ws, Bui and Duong [9] and Li and Sun [10]

proved that if to = (Ui,...,Um) 6 A(Pis/(mn),..,pns/(mn))>

then (8) also holds. Jiao [11] gave a generalization of the above inequality with the class Ap/Q which generalizes the class Ap introduced by Lerner et al. [12]. Fujita and Tomita showed a counterexample to answer the question whether the inequality (8) holds under the conditions to = (w1,..., tom) 6

A(p1 s/(mn),...,pms/(mn)) and ||al||Ws|m-'s|m(Rm") < OT.

We still recall the weighted Morrey spaces which were introduced by Komori and Shirai [13]. A weight w is a nonnegative, locally integrable function on Rn. Let 1 < p < ot; a weight function w is said to belong to the class Ap, if there is a constant C such that for any cube Q,

Definition 1 (See [13]). Let 1 < p < ot, let 0 < k < 1, and let to be a weight function on R". The weighted Morrey space is defined by

Lp,K (to) = [feL

loc • IU Wu>-K(u>)

L-M = sgP(to(Q^ \Q\f(x)fto(x)dx) ■ (12)

Q V to(Q)K }Q Our main results can be stated as follows. Theorem 2. Let a be a multiplier satisfying

Jl\\wsi-"sm (Rm")

for s1,...,sm e (n/2, n] and let Ta be the operator defined by (1) and 0 < k < 1. Set tj = n/sj. Ifpj e (tj, ot) and the weight toj e A pi t (R") for 1 < j < m and p e [1,ot) such that 1/p = 1/pi + •••+ 1/pm, then

IMv.JJL^) icn\\fj\ i=i

LPi''(wj)'

where v¿ = n7=itoplP'

Given a multilinear Fourier multiplier operator Ta and b = (b1,...,bm) e BMO(R")m, we define the commutators TaXh(f)(x) to be

TaXb {f](x)=Z [bj,Ta] (fi,...,fm)(x), (15)

[bj, Ta] (fifm) (x) = bj (x) Ta (fi,...,fj,..., fm) (x)

-Ta (fi,...,bjfj,...,fm)(x)

Theorem 3. Let o be a multiplier satisfying

H^W5!-"5™(Rm") <

for s1,...,sm e (n/2, n] and let Ta be the operator defined by (1) and 0 < k < 1. Set tj = n/sj. Ifpj e (tj, ot) and the weight toj e A pi t (R") for 1 < j < m and p e [1,ot) such that

f 1 c \f 1 { ' \p-i 6 ^p¡/1¡(R ) for 1 < J < m and p 6 [1,ot) such that

VIQUq W(X)1~P dX) <C' (9) Hp=1l'pi + ---+Hpm,thenforanybl,...,bm 6 BMO(Rn),

and to belongs to the class A1, if there is a constant C such that, for any cube Q,

m) IlLP-^R".^)

— I to(x)dx<Cinfto(x).

\Q\ Jq ^eq

< C\\b\\

j\\LPi-K(R",w¡y

Wedenote A m = Up^ A p.

where \\b\\BMOm = n™ y bj\\BM0 and v¿ = n™=itoF:lFi.

Because the regularity condition ||CT|||Ws(

stronger than that of following corollaries

< m is

< m, we have the

Corollary 4. Let a be a multiplier satisfying

'l\\ws(M.mn)

for s e (mn/2,mn] and let Ta be the operator defined by (1) and 0 < k < 1. Set r = mn/s. If pj e (mn/s,m) and the weight Wj e Ap /r(f") for 1 < j <m and p e [1, m) such that 1/p = 1/p1 +■■■+ 1/pm, then

№i>->fr

ml HL^ivs,)

Hl^IO,)'

where = n"Liw

Similarly, we have the responding lemma on weighted Morrey spaces as a consequent result.

Lemma 8. Let 0 < k < 1, 0 < p,S < m, and w e AThen there exists some constant CnpSa such that

HMsWW^) <cn„sjMl(f)\\K

tf*(w)

For f = (f1,..fm ), rt > 0, i = 1,. ,.,m, and set r = (r1,... ,rm), we define

m / If \ 1/ri

Mf)M = -pn[M\qMP^ . (27)

Corollary 5. Let a be a multiplier satisfying

'l\\Ws(M.mn)

for s e (mn/2,mn] and let Ta be the operator defined by (1) and 0 < k < 1. Set r = mn/s. If pj e (mn/s,m) and the weight Wj e Ap,/r(fn) for 1 < j <m and p e [1, m) such that 1/p = 1/p1 + ■ ■ ■ + 1/pm, then, for any b1,...,bm e BMO(fn),

This maximal function is the generalization of M which is introduced by Lerner et al. [12], we refer to [11] for some properties of it. The following lemma is the special example of [11, Theorem 2.1].

Lemma 9. Let p1,...,pm,p e rj e (0,pj), and Wj e

Ap/r for 1 < j < m and \/p1 + ■ ■ ■ + l/pm = 1/p. Then we have

\KXh(fl,---Jr

m'IILM(va) < C\\b\\BM^ J"! \\/j\\LPi.-(„.)>

K^U-j < CU\\fiLw

where wMbmct = n"jLi\\bjWBMO and vs = n"jLi^P/Pi.

Remark 6. For m = 1 and w e Ap, we also extend Hormander's theorem [14] to the weighted Morrey spaces.

2. Some Notations and Lemmas

We begin with the definitions of Hardy-Littlewood maximal function,

Mf(x) = sup-M 1/(^1^ (23)

Qix M JQ

and of the sharp maximal function,

M* (f) (x) = sup mf-L [ \f (y) - c\ dy. (24)

For S > 0, we also define the following maximal functions

Ms(f) = M(lff)l/S and Ml(f) = MHlff)l/s. The following classical result belongs to Fefferman and Stein [15].

Lemma 7. Let 0 < p,S < m, and w e AThen there exists some constant Cnp S w such that

\\MS(f)\\Lnw) <Cn,pAa\\Ml(f)\\mwy (25)

and if at least one rt = pi, then

\Kw\U^ <cl\\\fjl,^

where = n'=l^i/P'.

Lemma 10. Let k e (0,1), pl,.. .,pm, p e (0, m), rj e (0, pl), and Wj e Ap,/r, for 1 < j < m and 1/pl + ■ ■ ■ + 1/pm = 1/p. Then we have

j\\LPi'K(W: )'

Proof. From [11], there exists some q e (0,1) such that

Mr(f) (x)<Cn

* ((^ )X)

1/(ipj)

where M^ is the weighted centered maximal operator. Then by the Holder inequality and [13, Theorem 3.1], we get

||m?(/)(*)|L

< cni/,

i/(<?p)

i/(<?p)

L^Vö)

[\/jf ]

1/(<?Pj)

1/(1pj)

Ll/i.t(!.)

Lemma 11 (See [6]). Let 1 < p1,...,pm < ot and 1/p = 1/p1 + • • • + 1/pm. Suppose that a e Lm(Rm") satis/es

||al||Ws1-"sm (Rm") < (33)

"ften Tff is bounded from Lp1 <R") x • • • x LP™ <R") to LP(R").

For ..., e (0, ot) and s1,..., sm e R, the weighted Lebesgue space of mixed type L(ll""'1m)(w(si>_>s )) is defined by the norm

llFllL<«."-*m>(a>(sl,..,Sm))

= [j •••{[ (J l^Ml* (*l)Sl ^l)^

|_jr" [ jr» \ jr" /

x (^2)

(^m) ^m

Lemma 12 (See [6]). Let r > 0, 2 < ^ < ot, and Sj > 0/or 1 < j < m. Then there exists a constant C > 0 such that

||^?||L(«i.....im)(№ 0 < CyFyws1/,1,..,sm/,m, (35)

/or all F e Wsi/«i'""sm/«m(Rm") with suppF c {ixj2••• + i^mi2 < rj.

By the reverse Holder inequality, we have the following lemma.

Lemma 13. Assume that w e p , with 1 < p^ ..., pm <

ot. Let n/2 < Sj < n; then there exist constants 1 < ej < min{pj, Sj/(Sj - 1), 2sj/nj such that e Ap/e .

The following lemma is the key to our main results. Lemma 14. Let "a" be a multplier satis/ying

l||W<si-~';

/or s1,..., sm e (n/2, n] and let TCT be the operator defined by (1). I/1 < pj < ot, ij = n/sj and 0 < 5 < r/m, where 1/r = 1/r1 + • • • +1/rm, = Cjtj and 1 < < min{p;-,s;-/(s;- -

1), 2^/nj. Then/or all f e Lp1 (R") x • • • x Lpm(R") with < pj < ot /or 1 < j < m,

M» (rff (/))w<CM? (/}(*),

where r = (r1,..., rm).

Proo/. By Lemma 13, 1 < tjSj < 2; then r^/m < 1. Fix a point x and a cube Q such that x e Q.It suffices to prove that

/ir g \ 1/5

ViQi jq I1" (Z) - cq' < ^ W' (38)

for some constant cq. We decompose // = /0 + with /0 = //xq* for all j = 1,..., m and Q* = 4^«Q. Then

n/i (*) = nw W+/T (*)) /=1 i=1

I /r (7m)

= n/0M + I /rW-/mmGJ.

where I = {a1,..., am : there is at least one Oj = 0j. Then we can write

(/>) = rff(/°)(Z)+ i rff(/r1 --/mr)(Z) «1

:= 7 + 77.

(34) Applying Kolmogorov's inequality to 7, we have

iH iT-

<cllr (f°)ll

- ll /llLr'~(Q>dx/|Q|)

(¡^JQ. i/j (,J)\"

< CM? (/) (x), since TCT is bounded from Lri x • • • x Lr™ to Lr by Lemma 11.

Taking Let

= I (/f-.com,

we claim that, for any zeQ, „ ,

I k(/r •••/m™)(*)-?;(/r •••/m™)wi

< CM? (/) (x). At first we consider the case a1 =••• = «„

to m / c

, \r2/ri \rm/rm_i \1/rm

XII ti t II ti t •••II t |Wi (X;Z;y1)...;ym)ridy1) ••• ) d^m

J(2lt1Q*)\(2»Q*nJ(2lt1Q*)\(2lQ*) V J(2t+1Q*)\(2tQ*) / f

to m / r \ 1/ri

Iin(|2,.iQ* 1/,Ml"/is-.

k (/r •••r )(^)-rff (/r •••/m )wi

< I K ar •••/mo) (-) - ar •••/mo) < I i(Q*r iwi .....7m)i n/, M^

ZeZ ZeZ m \(Q ) ,=1

ifcZK=o -2 Q \2 Q ) ,= 1

1/r, (45)

Denote fo = z - x and Q = x - Q* ;it follows from Lemma 12 that

r, v2/r; \r»/r-i

. t _ . . t _ . , t _ l<Mk+y1>...>k + ym)-<T; (71,...,ym)ri ) ••• )

J(2fc+1Q)\(2fcQ^ \ J(2fc+iQ)\(2fcQ) V J(2t+iQ)\(2'Q) / /

Iii \ 1/r™

' / ' \ r„/r„_i v m , N r2/ri X m mi

Y Jci2t£(Q)<|ym|<c^+^Q) \ Jci2t£(Q)<|ym_i|<C22t+i£(Q) \ Jc^QX^Kc^.^«) <C(2fc/(Q))-(S i+-+SJ

, \r2/ri

'/-' \ rrn/rm_i

\Jci2t£(Q)<|yi|<C22t+i£(Q) / /

yrw \ Siri

Jci2t£(Q)<|ym|<C22t+i£(QA Jci2t£(Q)<|ym_i|<C22t+i£(Q) V Jci2t£(Q)<|yi|<c22t+i£(Q) '

s 1/rm

X (7m)S"r™^m )

<C( 2fc/(Q))-(Sl+"'+Sm)2i(s'+* * ^

Jc12t£(Q)<|ym|<c22t+1£(Q)

U-Zmn- /^-Z \|r1 A- \si'i.

ci2fcl(Q)<|ym-i |«*2t+1£(Q) VJci2t£(Q)<|yi|<C22t+1£(Q) 1 '

/--I \5mrrn j

X (2 7rn) ^

fcZ (n)) (Sl +'''+s>»)2*(S1+'''+sm)2-'^»2-'("/ri+'''+"/r™)

<c(2fc/(Q))

C12t£(Q)<|ym|<C22t+1£(QU Jc12*£(Q)<|;Vm-1|«*2*+1£(Q) \ .MMQXW«^1««)

|a;(zi,...,zm)jr1 (zi)51"1 dZi

/r' \rm/r, I I \r2'r1

k1 m\\ (S1+'''+S™^-i(«/r1+'''+K/rm-S1-'''-Sm)

<c(2fc/(Q)f 2

r dlws1'-'sm.

Given that 2'° < Z(Q) < 2'0+1,then we have that

< Csup||a;||WS1,...,Sm£(2fcZ(Q))

fci ^\\-(s1+'''+s™)

-;(n/r1+'"+n/rm-S1-----sm)

< Csup||a;||WS1,...,Sm2-fc(S1+'''+s")/(Q)

-fc(s1+ ' ' '+sm)7//-)\-(»/r1+' ' '+M/rm)

On the other hand, a similar process follows in [10]; we get that

(2t+1Q)\(2*Q) (l(2t+1Q)\(2tQ) (l(^1Q)\(^Q)

+ ..., h + 7m) - 5,(71,. • • > 7m)P

C12'i(Q)<|ym|<£22'+1 i(Q) \ JC12'i(Q)<|ym_1|<C22'+1i(Q)

1 r; y2/r1 Y™/r'-1

(f t t1 (f I^-Vi,(n +0h,...,7m + 0h)|de)1d^1 ) • •• ) dym

yJc12ti(Q)<|y1|<£i2t+1i(QAJo 1 ' J )

' / ' rm/r„

Jo \ hi2ke(Q)<\ym\<C22k+1e(Q)

([ „ „ •••([ „ „ |h-vt^yi +eh,...,ym + eh)fdylX2"'1 • ••) dym) de

q2Ke(Q)<\ytn\<C22k+1e(Q)\lcl2k e(Q)<\ym-i\<C2 2t+1e(Q) \)^2ke(Q)<\n\<C22k+1e(Q)

([ t t1 |,...,ym)r!dyX'"1 •••)

\h12ke(Q)<\y1\<C22k+1e(Q) 1 1 ) )

1,1 \ 1/rm

/ ' \ rm/rm-1 m

where h = (h,...,h) e Rmn. Since we have

(yi'---'ym)' (49) __I

0, , ^\ 1/r'm

t ([ t •••([ * ,, IV * (yi>-->ymfdyi]2i/l • dy

Ci2ke(Q)<\ym\<c22k+1e(Q)\Sci2ke(Q)<\ytn-i\<c12k+1e(Q) \ Jci2 e(Q)<\yi\<Ci2k+1e(Q) 1 1 > )

< .................-(si+-+sm)

Yi(Oi) (2kl(Q)Y

I "I \ rm/rm-1

. .,ii. . , ii. . , i J l Vy 1' ** ' ' / m!

hi2ke(Q)<\ym\<c22k+1e(Q)\ici2ke(Q)<\y,n-i\<c22k+1 l(Q) \ Jc^e(Q)<\yi\<c22k+1 l(Q) 1

•([ , „ ^ (yi,..;ym)f {yi)Sir1 dyX"1 • )

\hi2ke(Q)<\yi\<c22k+1e(Q) 1 ! )

x (ymYmrmdym

<c(2kl(Q))-(s 1+"'+Sm)2l (s

hi2ke(Q)<\ym\< c22k+1((Q)

Jci2ke(Q)<\ym-i\<c22k+1e(Q) \Jci2ke(Q)<\yi\<c22k+1e(Q)

([ t t1 ^ (2-lyv...,2-lym)f (2-lyi)Sir1 dyi)

\Jc12ke(Q)<\y1\<c22k+1e(Q)1 1 /

2 ir[ \r"'ir

< c(2^Z(Q)) <Sl+"'+Sm'2i<Sl+'"+s™)2-im™2-i<"/r' +'"+"/r'»)

(l;12ti(Q)<jymj<£22t+1i(Q) (lc

Ci2*i(Q)<jymj<c22*+1i(Qn Jci2*i(Q)<jym_ij<c22*+1 i(Q) \ Jci2*i(Q)<jyij<c22*+1i(Q)

x (¿J5™1™

(zi,...,zm)l <Zi>Siridzi

< c(2fcZ(Q)) (Si+'''+Sm)2-i("/ri+'''+"/r™+1-si-'''-s™)||a';

From Lemma 13, n/rr +-----+ w/>*m > +-----+ sm -1,itdeduces

(ZT•••/T)(*)-Tff (/r -OWl

< c£2-fc(si+-+s"-"/ri---"/rJM?(/) (x)

< CM? (/) (%) .

It remains to consider the case that there exists a proper subset |j1,...,jy| of |1,...,m|, 1 < y < m, such that o^ = ••• = O; = 0. Without loss of generality, we write, for the case

Ij1,...,jy} = |i,...,yl,

K (/r1)(Z)-rff (/r1)(x)i

r m / c \ 1/r)

r m / (■

iin(U u, wi* »

J t t •••(J t t (J •••(J |Wj (x,^,...,^)^) 21 •••dyy) dyy+1) •••dy,

^^Q^Q* \ J2fc+1^\2fcQ* \Jq* VJq* ! ) 1

The same argument as the case a1 = • • • = am = ot computes that

K or1,...,/:- or1,...,/:- )wi

< CM? (/) (%).

This completes the proof.

Lemma 15. "Let a" be a multplier satisfying

for sr,..., sm e (n/2, n] and let TCT be the operator defined by (1). If 1 < pj < OT ij = n/sj and 0 < S < e < r/m, where 1/r = 1/r1 + ... + 1/rm, = e-tj and 1 < < min|_pj, Sj/(Sj -

1),2sj/n}, and let b e BMOm. Then for any y > r, that is,

y,- > = 1,...,m, there exists some constant C > 0 such that

M* (Tff>zb (/))(*)

<c||i|Lr (M- (r* (/))(*)+Mr(/>)),

for all m-tuples f = (/r,...,/m) of bounded measurable functions with compact support.

Proof. By linearity it is sufficient to consider the particular case when b = b e BMO. Fix b e BMO and consider the operator

(/) (x) = b (x) Tff (/) (*) - Tff (fc/1, /2,..., /J (x).

Fix x e R", for any cube Q with center at x; set A = &q., where Q* = 4V«Q. We have

(/)(*)

= (fc(x) - A) T(/) (*) - TCT ((fc - A)/1,/2,...,/m) (*).

Since 0 < 5 < r/rn < 1, 1

!Q! Jq

(/>)| -!c|;

¿Jq u»-^

|(fc(Z)-A)Tff (/)(z)|Sdz

(^Jq K ((fc-A)/i;...;/m)(Z)-C|SdZ)

:= A + B.

By the John-Nirenberg inequality and Holder inequality, one has, for 1 < q < e/5 such that > 1,

KkUq ^f^f (60)

< CHfcllBMoM^ (rff (/))(*) <CyfcyBMcMe (rff (/))(*).

To estimate term B, we split each function /j as /j = /j0 + where /j0 = /j^Q* for j = 1,..., m. We also have the same decomposition,

n/j (*)

=nw (y}))

= I /r oo-./m-(^m)

= n/0 (>1) + I /r 0Ü'

j=1 ai,...,«me1

Taking c = Z^...*^ W - A)/!*1 ■ ■ ■ ym-), we have

ß < C

„ e A M Jq

-rff ((fc-A)/11 ,...,/mm)

x (x)dz|s

:= B1 + ^

By using Kolmogorov's inequality and Holder's inequality, one has

m'v 'NLV'~(Q,dx/iQi)

£1 <c||Tff((fc-A)/1,'...'/f0

KKiIL |(fe-A)/1°(Z)r 1 d)

xn (¿rJQ w «J

<CyfcyBMoMi(/).

By the same argument in the proof of Lemma 14, we have the following estimate:

I K (/f1 -ft )(Z)-rff Of1 • ••/;- )(x)|

-fc(s 1+-+Sm-n/r1-----n/r-)

m / 1 c \1/r>

xn (12^ Lq. w «H

< CMr (/) (%).

where I = {«j,..., am : there is atleast one a.- = 0}.

This completes the proof.

3. Proof of Theorems

Proof of Theorem 2. By Lemmas 8 and 14, we have

IK (/)L.*(]3) * cIK(w))|Lfe)

^IK^IU

* cni/il

LPj,K-

Proof of Theorem 3. By Lemma 13, there are e'j < pj/rj such that Wj e Ap,/(r,£i). Let = r^.; by Lemmas 8 and 15, one

and then

IK^IU,) scnwu 1= 1

IK,zb(/)

LM(„a)

* cni/j

J^'V,-)'

(67) □

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Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors would like to thank the referee for some very valuable suggestions. This research was supported by the NSF of China (no. 11161044 and no. 11261055) and by XJUBSCX-2012004.

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