Let Tσ be the multilinear Fourier multiplier operator associated with multiplier σ satisfying the Sobolev regularity that supl∈ZσlWs1,…,sm(Rmn)<∞ for some sk∈(n/2,n] (k=1,…,m). The authors prove that if b1,…,bm∈BMO(Rn) and w⃗∈∏k=1mApk/tk(tk=n/sk), then the commutator Tσ,Σb is bounded from Lp1(Rn,w1)×⋯×Lpm(Rn,wm) to Lp(Rn,νw⃗). Moreover, the authors also prove that if b1,…,bm∈VMO(Rn) and w⃗∈∏k=1mApk/tk(tk=n/sk), then the commutator Tσ,Σb is compact operator from Lp1(Rn,w1)×⋯×Lpm(Rn,wm) to Lp(Rn,νw⃗).
1. Introduction
The study of the multilinear Fourier multiplier operator was originated by Coifman and Meyer in their celebrated work [1, 2]. Let σ∈L∞(Rmn); the multilinear Fourier multiplier operator Tσ is defined by
(1)Tσ(f→)(x)≔∫Rmnexp(2πix(ξl+⋯+ξm))×σ(ξl,…,ξm)f^1(ξl)⋯f^m(ξm)dξ→
for all f1,…,fm∈S(Rn), where dξ→=dξ1⋯dξm and f^ is the Fourier transform of f. Coifman and Meyer [2] proved that if σ∈Cs(Rmn∖{0}) satisfies
(2)|∂ξ1α1⋯ξkαkσ(ξ1,…,ξm)|≤Cα1,…,αm(|ξ1|+⋯+|ξm|)-(|α1|+⋯|αk|)
for all |α1|+⋯|αm|≤s with s≥2mn+1, then Tσ is bounded from Lp1(Rn)×⋯×Lpm(Rn) to Lp(Rn) for all 1<p, p1,…,pm<∞ with 1/p=∑k=1m1/pk. 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 Calderón-Zygmund operator theory. In the last several years, considerable attention has been paid to the behavior on function spaces for Tσ when the multiplier satisfies certain Sobolev regularity condition. Let Φ∈S(Rmn) satisfy
(3)suppΦ⊂{(ξ1,…,ξm):12≤∑k=1m|ξk|≤2},∑l∈ZΦ(2lξ1,…,2lξm)=1∀(ξ1,…,ξm)∈Rmn∖{0}.
For l∈Z, set
(4)σl(ξ1,…,ξm):=σ(2-lξ1,…,2-lξm)Φ(ξ1,…,ξm),∥σl∥Ws(Rmn)≔(∫R2n(1+|ξ1|2+⋯+|ξm|2)s∫R2n×|σ^(ξ1,…,ξm)|2dξ→)1/2.
Tomita [5] proved that if
(5)supl∈Z∥σl∥Ws(Rmn)<∞
for some s∈(mn/2,∞), then Tσ is bounded from Lp1(Rn)×⋯×Lpm(Rn) to Lp(Rn) provided that 1<p,p1,…,pm<∞ and 1/p=∑k=1m1/pk. Grafakos and Si [6] considered the mapping properties from Lp1(Rn)×⋯×Lpm(Rn) to Lp(Rn) for Tσ when p≤1. Let σ satisfy the Sobolev regularity that
(6)∥σl∥Ws1,…,sm(Rmn)≔(∫Rmn〈ξ1〉2s1⋯〈ξm〉2s2∫Rmn×|σ^l(ξ1,…,ξm)|2dξ→)1/2,
where 〈ξk〉:=(1+|ξk|2)1/2. Miyachi and Tomita [7] proved that if
(7)supl∈Z∥σl∥Ws1,…,sm(Rmn)<∞
for some sk∈(n/2,n](k=1,…,m), then Tσ is bounded from Lp1(Rn)×⋯×Lpm(Rn) to Lp(Rn) provided that 1<p1,…,pm<∞ with 1/p=∑k=1m1/pk.
As well known, when σ satisfies (3) for some s≥mn+1, then Tσ is a standard multilinear Calderón-Zygmund operator, and then by the weighted estimates with multiple weights for multilinear Calderón-Zygmund operators, which were estimated by Lerner et al. [8], we know that, for any p1,…,pm∈[1,∞) and p∈(0,∞) with 1/p=∑k=1m1/pk and weights w1,…,wm such that w→=(w1,…,wm)∈Ap→(Rmn),
(8)∥Tσ(f1,…,fm)∥Lp,∞(Rn,νw→)≲∏k=1m∥fk∥Lpk(Rn,wk).
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 Tσ when σ satisfies (3) for m=2 and s∈(n,2n]. Hu and Yi [10] considered the behavior on Lp1(Rn)×⋯×Lpm(Rn) for Tσ,Σb when σ satisfies (5) for s1,…,sm∈(n/2,n] and showed that Tσ,Σb enjoys the same Lp1(Rn)×⋯×Lpm(Rn)→Lp(Rn) mapping properties as that of the operator Tσ.
Now, considerable attention has been paid to the behavior on the compactness of multilinear Fourier multipliers operator with Sobolev regularity. Let VMO(Rn) be the closure of Cc∞ in the BMO(Rn) topology, which coincides with the space of functions of vanishing mean oscillation (see [11, 12]). Bényi and Torres [13] proved that if b1,…,bm∈VMO(Rn) and T is multilinear Calderón-Zygmund operator, then, for p1,…,pm∈(1,∞),p∈[1,∞) with 1/p=∑k=1m1/pk, the commutator Tσ,Σb is compact operator from Lp1(Rn)×⋯×Lpm(Rn) to Lp(Rn). Hu [14] proved that if σ is a multilinear multiplier which satisfies (5) for some s∈(mn/2,mn], t1,…,tm∈[1,2)tk=n/sk, b1,…,bm∈VMO(Rn), and pk∈(tk,∞) for k=1,…,m and p∈(1,∞) with 1/p=∑k=1m1/pk, then Tσ,Σb is compact operators from Lp1(Rn)×⋯×Lpm(Rn) to Lp(Rn). Bényi et al. [15] proved that if b→∈VMO(Rn)×⋯×VMO(Rn) and T is multilinear Calderón-Zygmund operator, w→∈Ap×⋯×Ap, then, for p1,…,pm∈(1,∞),p∈(1,∞) with 1/p=∑k=1m1/pk, the commutator Tσ,Σb is compact operator from Lp1(Rn,w1)×⋯×Lpm(Rn,wm) to Lp(Rn,νw→).
Inspired by the above, we consider the weighted compactness of the commutator Tσ,Σb of the multilinear Fourier multiplier operator on Lp(Rn).
Given a multilinear Fourier multiplier operator Tσ, b1,…,bm∈BMO(Rn) and b→=(b1,…,bm), the commutator Tσ,Σb(f→)(x) is defined by
(9)Tσ,Σb(f→)(x)≔∑k=1m[bk,Tσ]k(f1,…,fm)(x),
with
(10)[bk,Tσ]k(f1,…,fk,…,fm)(x)≔bk(x)Tσ(f1,…,fk,…,fm)(x)-Tσ(f1,…,bkfk,…,fm)(x).
Our main results are stated as follows.
Theorem 1.
Suppose that σ be a multilinear multiplier which satisfies (7) for some sk∈(n/2,n](k=1,…,m) and t1,…,tm∈[1,2). Let tk=n/sk, pk∈(tk,∞) for k=1,…,m and 1/p=∑k=1m1/pk with p>1. If the weights w1,…,wm satisfy w→∈∏k=1mApk/tk, then, for any b1,…,bm∈BMO(Rn), Tσ,Σb is bounded from Lp1(Rn,w1)×⋯×Lpm(Rn,wm) to Lp(Rn,νw→).
Theorem 2.
Suppose that σ be a multilinear multiplier which satisfies (7) for some sk∈(n/2,n](k=1,…,m) and t1,…,tm∈[1,2). Let tk=n/sk, pk∈(tk,∞) for k=1,…,m and 1/p=∑k=1m1/pk with p>1. If the weights w1,…,wm satisfy w→∈∏k=1mApk/tk, then, for any b1,…,bm∈VMO(Rn), Tσ,Σb is a compact operator from Lp1(Rn,w1)×⋯×Lpm(Rn,wm) to Lp(Rn,νw→).
Because the regularity condition ∥σl∥Ws(Rmn)<∞ is stronger than ∥σl∥Ws1,…,sm(Rmn)<∞, we have the following corollaries.
Corollary 3.
Suppose that σ be a multilinear multiplier which satisfies (5) for some s∈(mn/2,mn]. Let r=mn/s, pk∈(mn/s,∞) for k=1,…,m and 1/p=∑k=1m1/pk with p>1. If the weights w1,…,wm satisfy w→∈∏k=1mApk/r, then, for any b1,…,bm∈BMO(Rn), Tσ,Σb is bounded from Lp1(Rn,w1)×⋯×Lpm(Rn,wm) to Lp(Rn,νw→).
Corollary 4.
Suppose that σ be a multilinear multiplier which satisfies (5) for some s∈(mn/2,mn]. Let r=mn/s, pk∈(mn/s,∞) for k=1,…,m and 1/p=∑k=1m1/pk with p>1. If the weights w1,…,wm satisfy w→∈∏k=1mApk/r, then, for any b1,…,bm∈VMO(Rn), Tσ,Σb is a compact operator from Lp1(Rn,w1)×⋯×Lpm(Rn,wm) to Lp(Rn,νw→).
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≤CB. We use B(x,R) to denote a ball centered at x with radius R. For a ball B⊂Rn and λ>0, we use λB to denote the ball concentric with B whose radius is λ times of B′s. As usual, |E| denotes the Lebesgue measure of a measurable set E in Rn and χE 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 w1,…,wm be weights, p1,…,pm, p∈(0,∞), with 1/p=∑k=1m1/pk, rk∈(0,pk](1≥k≥m). Set w→=(w1,…,wm) and νw→:=∏k=1mwkp/pk. Then
(11)w→∈Ap1/r1×⋯×Apm/rm⟹{ϑk≔wk1-(pk/rk)′∈Am(pk/rk)′,k=1,…,m,νw→∈Amp/r.
Definition 6.
For any f→:=(f1,…,fm), r→:=(r1,…,rm) and p≥1, M is defined by
(12)Mr→(f→)(x)≔supx∈Q∏k=1m(1|Q|∫Q|fk(yk)|rkdyk)1/rk.
For δ>0, Mδ is the maximal function
(13)Mδ(f)(x)≔M(|f|δ)1/δ(x)=(supx∈Q1|Q|∫Q|f(y)|δdy)1/δ.
The sharp maximal function M# of Fefferman-Stein is defined by
(14)M#(f)(x)≔supx∈Qinfc1|Q|∫Q|f(y)-c|dy=supx∈Q1|Q|∫Q|f(y)-fQ|dy,
where fQ:=(1/|Q|)∫Qf(y)dy.
Next, we give some symbols.
Let σ∈L∞(Rmn) and Φ∈S(Rmn) satisfy (3). For l∈Z, define
(15)σ~l(ξ1,…,ξm)≔Φ(2lξ1,…,2lξm)σ(ξ1,…,ξm).
Then
(16)σ~l(ξ1,…,ξm)=Φ(2lξ1,…,2lξm),σ~ˇl=2-lmnσˇl(2-lξ1,…,2-lξm),
where fˇ denotes the inverse Fourier transform of f.
For N∈N, let
(17)σN(ξ1,…,ξm)≔∑|l|≤Nσ~l(ξ1,…,ξm)
and denote by TσN the multiplier operator associated with σN. It is obvious that TσN is an m-linear singular operator with kernel
(18)KN(x;y1,…,ym):=σˇN(x-y1,…,x-ym).
For an integer k with 1≤k≤2 and x,x′,y1,…,ym∈Rn, let
(19)WN(x,x′;y1,…,ym)≔KN(x;y1,…,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,
(20)T:S(Rn)×⋯×S(Rn)⟶S′(Rn).
By the associated kernel K(x,y1,…,ym), we mean that K is a function defined off the diagonal x=y1=⋯=ym in R(m+1)n, satisfying
(21)T(f1,…,fm)(x)≔∫RmnK(x;y1,…,ym)f1(y1)⋯fm×(ym)dy→,
for all functions fk∈S(Rn) and all x∉⋂k=1msuppfk. It is easy to see that the associated kernel K(x,y1,…,ym) to Fourier multiplier operator Tσ is given by
(22)K(x,y1,…,ym)≔σˇ(x-y1,…,x-ym).
To prove main results, we need the following lemmas.
By the reverse Hölder inequality, we have the first lemma.
Lemma 7.
Assume that w→∈∏k=1mApk/tk, with t1,…,tm∈[1,2), pk∈(tk,∞)(k=1,…,m), and 1/p=∑k=1m1/pk with p>1. Let sk∈(n/2,n]; then there exists a constant ϵk∈(1,min{pk/tk,sk/(sk-1),2sk/n}) such that w→k∈Apk/(tjϵk).
For p1,…,pm∈(0,∞) and s1,…,sm∈R, the weighted Lebesgue space of mixed type L(p1,…,pm)(ω(s1,…,sm)) is defined by the norm
(23)∥F∥L(p1,…,pm)(ω(s1,…,sm))=((∫Rn|F(x)|p1〈x1〉s1dx1)p2/q1〈x2〉s2dx2)p3/q2∫Rn⋯(∫Rn(∫Rn|F(x)|p1〈x1〉s1dx1)p2/q100000(∫Rn|F(x)|p1〈x1〉s1dx1)p2/q100000×〈x2〉s2dx2)p3/q2⋯〈xm〉smdxm)1/pm,
where x:=(x1,…,xm)∈Rn×⋯×Rn.
Lemma 8 (see [16]).
Let r>0, 2≤pj<∞, and sj≥0 for 1≤j≤m. Then there exists a constant C>0 such that
(24)∥F^∥L(p1,…,pm)(ω(s1,…,sm))≤C∥F∥Ws1/p1,…,sm/pm
for all F∈Ws1/p1,…,sm/pm(Rmn) with suppF⊂{|x1|2⋯+|xm|2≤r}.
The following lemma is the key to our main lemma.
Lemma 9.
Suppose that σ be a multilinear multiplier which satisfies (7) for some sk∈(n/2,n](k=1,…,m). Let 0<δ<r, 1/r=1/r1+⋯+1/rm, rj=ϵjtj, and ϵj is the same as that appears in Lemma 7. Then for all f→∈Lp1(Rn)×⋯×Lpm(Rn) with rj≤pj<∞ for 1≤j≤m,
(25)Mδ#(Tσ(f→))(x)≤CMr→(f→)(x),
where r→=(r1,…,rm).
Proof.
By Lemma 8, 1<tjϵj≤2; then rj/m≤1. Fix a point x and a cube Q such that x∈Q. It suffices to prove
(26)(1|Q|∫Q|Tσ(f→)(z)-cQ|δdz)1/δ≤CMr→(f→)(x),
for some constant cQ. We decompose fj=fj0+fj∞ with fj0=fjχQ⋆ for all j=1,…,m and Q⋆=4nQ. Then
(27)∏j=1mfj(yj)=∏j=1m(fj0(yj)+fj∞(yj))=∑α1,…,αm∈{0,∞}f1α1(y1)⋯fmαm(ym)=∏j=1mfj0(yj)+∑α1,…,αm∈If1α1(y1)⋯fmαm(ym),
where I={α1,…,αm:thereisatleastoneαj≠0}. Then we can write
(28)Tσ(f→)(z)=Tσ(f→0)(z)+∑α1,…,αm∈ITσ(f1α1⋯fmαm)(z)≔I+II.
Applying Kolmogorov’s inequality to I, we have
(29)(1|Q|∫Q|Tσ(f→0)(z)|δdz)1/δ≤C∥Tσ(f→0)∥Lr,∞(Q,dx/|Q|)≤C∏j=1m(1|Q⋆|∫Q⋆|fj(yj)|rldyj)rj≤CMr→(f→)(x),
since Tσ is bounded from Lr1×⋯×Lrm to Lr.
Take
(30)cQ=∑α1,…,αm∈ITσ(f1α1⋯fmαm)(x).
We claim that, for any z∈Q,
(31)∑α1,…,αm∈I|Tσ(f1α1⋯fmαm)(z)-Tσ(f1α1⋯fmαm)(x)|≤CMr→(f→)(x).
Let
(32)Wl(x,z;y1,…,ym)=σˇl(x-y1,…,x-ym)-σˇl(z-y1,…,z-ym).
At first, we consider the case α1=⋯=αm. We get
(33)|Tσ(f1∞⋯fm∞)(z)-Tσ(f1∞⋯fm∞)(x)|≤∑l∈Z|Tσl(f1∞⋯fm∞)(z)-Tσl(f1∞⋯fm∞)(x)|≤∑l∈Z∫Rmn∖(Q⋆)m|Wl(x,z;y1,…,ym)|∏j=1mfj(yj)dy→≤∑l∈Z∑k=0∞∬(2k+1Q⋆)∖(2kQ⋆)|Wl(x,z;y1,…,ym)|∏j=1mfj(yj)dy→≤∑k=0∞∑l∈Z∏j=1m(∫(2k+1Q⋆)m|fj(yj)|rjdyj)1/rj×(∫(2k+1Q⋆)∖(2kQ⋆)Wl×(x,z;y1,…,ym)|r1′dy1)r2′/r1′⋯)rm′/rm-1′∫(2k+1Q⋆)∖(2kQ⋆)(∫(2k+1Q⋆)∖(2kQ⋆)Wl×(x,z;y1,…,ym)|r1′dy1)r2′/r1′∫(2k+1Q⋆)∖(2kQ⋆)⋯(∫(2k+1Q⋆)∖(2kQ⋆)|Wl(x,z;y1,…,ym)|r1′∫(2k+1Q⋆)∖(2kQ⋆)dy1)r2′/r1′⋯)rm′/rm-1′dym)1/rm′≔∑k=0∞∑l∈Z∏j=1m(∫(2k+1Q⋆)m|fj(yj)|rjdyj)1/rjIIk,l∞,…,∞.
Denote h=z-x, Q~=x-Q⋆, and l(Q) the side length of a cube Q; it follows from Lemma 7 that
(34)IIk,l∞,…,∞=(∫(2k+1Q~)∖(2kQ~)⋯(∫(2k+1Q~)∖(2kQ~)⋯)rm′/rm-1′∫(2k+1Q~)∖(2kQ~)(∫(2k+1Q~)∖(2kQ~)⋯(∫(2k+1Q~)∖(2kQ~)∫(2k+1Q~)∖(2kQ~)|σˇl(h+y1,…,h+ym)-σˇl(y1,…,ym)|r1′dy1)r2′/r1′∫(2k+1Q~)∖(2kQ~)⋯(∫(2k+1Q~)∖(2kQ~)⋯)rm′/rm-1′dym)1/rm′≤2(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)∫c12kl(Q)≤|y1|<c22k+1l(Q)|σˇl(y1,…,ym)|r1′|σˇl(y1,…,ym)|r1′dy1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯)rm′/rm-1′dym)1/rm′≤C(2kl(Q))-(s1+⋯+sm)×(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)∫c12kl(Q)≤|y1|<c22k+1l(Q)|σˇl(y1,…,ym)|r1′〈y1〉s1r1′dy1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯⋯)rm′/rm-1′〈ym〉smrm′dym)1/rm′≤C(2kl(Q))-(s1+⋯+sm)2l(s1+⋯+sm)×(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)|2-lmnσˇl(2-ly1,…,2-lym)|r1′∫c12kl(Q)≤|y1|<c22k+1l(Q)×〈2-ly1〉s1r1′dy1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯)rm′/rm-1′〈2-lym〉smrm′dym)1/rm′≤C(2kl(Q))-(s1+⋯+sm)2l(s1+⋯+sm)2-lmn2-l(n/r1′+⋯+n/rm′)×(∫c12kl(Q)≤|y1|<c22k+1l(Q)|σˇl(z1,…,zm)|r1′×〈z1〉s1r1′dz1)r2′/r1′⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|y1|<c22k+1l(Q)|σˇl(z1,…,zm)|r1′×〈z1〉s1r1′dz1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)|σˇl(z1,…,zm)|r1′∫c12kl(Q)≤|y1|<c22k+1l(Q)|σˇl(z1,…,zm)|r1′×〈z1〉s1r1′dz1)r2′/r1′⋯)rm′/rm-1′〈zm〉smrm′dzm)1/rm′≤C(2kl(Q))-(s1+⋯+sm)2-l(n/r1+⋯+n/rm-s1-⋯-sm)∥σl∥Ws1,…,sm.
Given that 2l0≤l(Q)≤2l0+1, we have that
(35)∑l<l0IIk,l∞,…,∞≤Csupl∥σl∥Ws1,…,sm×∑l<l0(2kl(Q))-(s1+⋯+sm)2-l(n/r1+⋯+n/r1-s1-⋯-sm)≤Csupl∥σl∥Ws1,…,sm2-k(s1+⋯+sm)l(Q)-(n/r1+⋯+n/rm).
On the other hand, a similar process follows that in [17]; we get that
(36)IIk,l∞,…,∞=(∫(2k+1Q~)∖(2kQ~)⋯(∫(2k+1Q~)∖(2kQ~)⋯)rm′/rm-1′∫(2k+1Q~)∖(2kQ~)(∫(2k+1Q~)∖(2kQ~)⋯(∫(2k+1Q~)∖(2kQ~)∫(2k+1Q~)∖(2kQ~)|σˇl(h+y1,…,h+ym)-σˇl(y1,…,ym)|r1′dy1)r2′/r1′∫(2k+1Q~)∖(2kQ~)⋯(∫(2k+1Q~)∖(2kQ~)⋯)rm′/rm-1′dym)1/rm′≤(∫c12kl(Q)≤|y1|<c22k+1l(Q)dy1)r2′/r1′⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|y1|<c22k+1l(Q)dy1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)(∫01|h→·∇σˇl(y1+θh,…,ym∫01|h→·∇σˇlh→+θh)|dθ)r1′dy1)r2′/r1′∫c12kl(Q)≤|y1|<c22k+1l(Q)dy1)r2′/r1′⋯)rm′/rm-1′dym)1/rm′≤∫01(∫c12kl(Q)≤|y1|<c22k+1l(Q)|h→·∇σˇl(y1+θh,…,ym+θh)|r1′dy1)(r_2^′)/(r_1^′)⋯)(r_m^′)/(r_(m-1)^′)∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|y1|<c22k+1l(Q)|h→·∇σˇl(y1+θh,…,ym+θh)|r1′dy1)(r_2^′)/(r_1^′)∫c12kl(Q)≤|ym-1|<c22k+1l(Q)(∫c12kl(Q)≤|y1|<c22k+1l(Q)∫c12kl(Q)≤|y1|<c22k+1l(Q)|h→·∇σˇl(y1+θh,…,ym+θh)|r1′|h→·∇σˇl(y1+θh,…,ym+θh)|r1′dy1)r2′/r1′∫c12kl(Q)≤|y1|<c22k+1l(Q)|h→·∇σˇl(y1+θh,…,ym+θh)|r1′dy1)(r_2^′)/(r_1^′)⋯)rm′/rm-1′dym)1/rm′dθ≤(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)∫c12kl(Q)≤|y1|<c22k+1l(Q)|h→·∇σˇl(y1,…,ym)|r1′dy1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯)rm′/rm-1′dym)1/rm′,
where h→=(h,…,h)∈(Rn)m. Since
(37)h→·∇σˇl(y1,…,ym)=∑j=1mhj∂j∇σˇl(y1,…,ym),
we have
(38)IIk,l∞,…,∞≤∑j=1ml(Q)(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)∫c12kl(Q)≤|y1|<c22k+1l(Q)|∂j·σˇl(y1,…,ym)|r1′dy1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯⋯)rm′/rm-1′dym)1/rm′≤∑j=1ml(Q)(2kl(Q))-(s1+⋯+sm)×(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)∫c12kl(Q)≤|y1|<c22k+1l(Q)|∂jσˇl(y1,…,ym)|r1′〈y1〉s1r1′dy1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯⋯)rm′/rm-1′〈ym〉smrm′dym)1/rm′≤C(2kl(Q))-(s1+⋯+sm)2l(s1+⋯+sm)×(×∫c12kl(Q)≤|y1|<c22k+1l(Q)〈2-ly1〉s1r1′dy1)r2′/r1′⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|y1|<c22k+1l(Q)×〈2-ly1〉s1r1′dy1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)|2-lmn·∂jσˇl(2-ly1,…,2-lym)|r1′∫c12kl(Q)≤|y1|<c22k+1l(Q)×〈2-ly1〉s1r1′dy1)r2′/r1′⋯)rm′/rm-1′×∫c12kl(Q)≤|ym|<c22k+1l(Q)〈2-lym〉smrm′dym×∫c12kl(Q)≤|y1|<c22k+1l(Q)〈2-ly1〉s1r1′dy1)r2′/r1′⋯)rm′/rm-1′)1/rm′≤C(2kl(Q))-(s1+⋯+sm)2l(s1+⋯+sm)2-lmn2-l(n/r1′+⋯+n/rm′)×(∫c12kl(Q)≤|y1|<c22k+1l(Q)dz1)r2′/r1′⋯)rm′/rm-1′∫c12kl(Q)≤|ym|<c22k+1l(Q)(∫c12kl(Q)≤|y1|<c22k+1l(Q)dz1)r2′/r1′∫c12kl(Q)≤|ym-1|<c22k+1l(Q)⋯(∫c12kl(Q)≤|y1|<c22k+1l(Q)∫c12kl(Q)≤|y1|<c22k+1l(Q)|∂jσˇl(z1,…,zm)|r1′〈z1〉s1r1′dz1)r2′/r1′∫c12kl(Q)≤|y1|<c22k+1l(Q)dz1)r2′/r1′⋯)rm′/rm-1′〈zm〉smrm′dzm)1/rm′≤C(2kl(Q))-(s1+⋯+sm)2-l(n/r1+⋯+n/rm+1-s1-⋯-sm)×∥σl∥Ws1,…,sm.
From Lemma 8, n/r1+⋯+n/r1>s1+⋯+sm-1. It is deduced that
(39)∑l≥l0IIk,l∞,…,∞≤Csupl∥σl∥Ws1,…,sm2-l(s1+⋯+sm)l(Q)-(n/r1+⋯+n/rm).
So
(40)∑α1,…,αm∈I|Tσ(f1α1⋯fmαm)(z)-Tσ(f1α1⋯fmαm)(x)|≤C∑k=0∞2-k(s1+⋯+sm-n/r1-⋯-n/rm)Mr→(f→)(x)≤CMr→(f→)(x).
It remains to consider the case that there exists a proper subset {j1,…,jγ} of {1,…,m}, 1≤γ<m, such that αj1=⋯=αjγ=0. We have
(41)|Tσ(f1α1,…,fmαm)(z)-Tσ(f1α1,…,fmαm)(x)|≤∑k=0∞∑l∈Z∏j=1m(∫2k+1Q⋆|fj(yj)|rjdy→)1/rj×(∫2k+1Q⋆∖2kQ⋆⋯(∫2k+1Q⋆∖2kQ⋆((∫Q⋆|W0,l(x,z;y1,…,ym)|r1′dy1)r2′/r1′∫Q⋆⋯(∫Q⋆|W0,l(x,z;y1,…,ym)|r1′dy1)r2′/r1′∫Q⋆⋯⋯dyγ(∫Q⋆|W0,l(x,z;y1,…,ym)|r1′dy1)r2′/r1′)rγ+1′/rγ′dyγ+1)⋯dym)1/rm′.
By the same argument as that of the case α1=⋯=αm, we have that
(42)|Tσ(f1α1,…,fmαm)(z)-Tσ(f1α1,…,fmαm)(x)|≤CMr→(f→)(x).
This completes the proof.
Lemma 10.
Suppose that σ be a multilinear multiplier which satisfies (7) for some sk∈(n/2,n](k=1,…,m). Let 0<δ<r, 1/r=1/r1+⋯+1/rm, and rj=ϵjtj, and ϵj is the same as that appears in Lemma 7. Then, for b→∈(BMO(Rn))m and any γ→>r→, that is, γj>rj,j=1,…,m, there exists some constant C>0 such that
(43)Mδ#(Tσ,Σb(f→))(x)≤C∥b→∥BMOm(Mε(Tσ(f→))(x)+Mγ→(f→)(x)),
for all m-tuples f→=(f1,…,fm) 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 deduces the estimates. We omit the details here.
Lemma 11.
Let σ be a multilinear multiplier which satisfies (7) for some sk∈(n/2,n](k=1,…,m), r1,…,rm∈[1,2) such that rjsj>n(j=2,…,m). Then, for every ι∈(0,s1], R>0 and x∈Rn∖2R,
(44)∫Rn⋯∫Rn∫|y1|<R|σ~˘l(x-y1,…,x-ym)×f1(y1)⋯fm(ym)|dy1⋯dym≲1|x|ι2-l(ι-n/r1)∥f1∥Lp1(Rn,w1)∏i=2mMrkfk1|x|ι2-l(ι-n/r1)∥f1∥Lp1(Rn,w1)×(x)ϑ(R)(1/r1)-(1/p1).
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 Fréchet-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<∞ if and only if
suph∈H∥h∥Lp<∞,
limA→∞∥h∥Lp(|x|>A)=0 uniformly in h∈H,
limt→0∥h(·+t)-h(·)∥Lp=0 uniformly in h∈H.
3. Proof of Theorems 1 and 2Proof of Theorem 1.
We only present the case that m=2. We have by Lemma 10 and Theorem 3.2 in [8]
(45)∥TΣb(f1,…,fm)∥Lp(Rn,νw→)≲∥Mδ(TΣb(f→))∥Lp(Rn,νw→)≲∥Mδ#(TΣb(f→))∥Lp(Rn,νw→)≲∥b→∥BMO2(∥Mγ→(f→)(x)∥Lp(Rn,νw→)+∥Mγ→(f→)(x)∥Lp(Rn,νw→)∥Mϵ(T)(f→)(x)∥Lp(Rn,νw→))≲∥b→∥BMO2(∥Mγ→(f→)(x)∥Lp(Rn,νw→)+∥Mγ→(f→)(x)∥Lp(Rn,νw→)∥Mϵ#(T)(f→)(x)∥Lp(Rn,νw→))≲∥b→∥BMO2(∥Mγ→(f→)(x)∥Lp(Rn,νw→)+∥Mγ→(f→)(x)∥Lp(Rn,νw→)∥M(f→)(x)∥Lp(Rn,νw→))≲∥b→∥BMO2∏k=1m∥fk∥Lpk(Rn,wk),
which completes the proof of Theorem 1.
Proof of Theorem 2.
We will employ some ideas of Bényi and Torres [13]. Without loss of the generality, we only prove the case m=2. Let pk∈(rk,∞)(k=1,2), p∈(1,∞), with 1/p=∑1/pk, and b1,b2∈Cc∞(Rn). Note that, for any f1,f2∈S(Rn) and almost every x∈Rn,
(46)limN→∞TσN,Σb(f1,f2)(x)=Tσ,Σb(f1,f2)(x).
It is enough to prove that the following conditions hold:
TσN,Σb is bounded from Lp1(Rn,w1)×Lp2(Rn,w2) to Lp(Rn,νw→);
for each fixed ϵ>0, there exists a constant A=A(ϵ) which is independent of N, f1, and f2 such that
(47)(∫|x|>A|TσN,Σb(f1,f2)|pνw→(x)dx)1/p≲ϵ∏k=12∥fk∥Lpk(Rn,wk);
for each fixed ϵ>0, there exists a constant ρ=ρ(ϵ) which is independent of N, f1, and f2 such that, for all t with 0<|t|<ρ,
(48)∥TσN,Σb(f1,f2)(·)-TσN,Σb(f1,f2)(·+t)∥Lp(Rn,νw→)≲ϵ∏k=12∥fk∥Lpk(Rn,wk).
Then by the Fatou Lemma, the conditions (a), (b), and (c) still hold true if TσN,Σb(f1,f2) is replaced by Tσ,Σb(f1,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 suppb1⊂BD(0)≔B(0,D) and let A≥max{2D,1}. Then for every x with |x|>2A, we have by Lemma 11 that
(49)∫Rn∫suppb1|KN(x;y1,y2)f1(y1)f1(y1)|dy1dy2=∫Rn∫suppb1|∑|l|≤NσN~˘(x-y1,x-y2)∑|l|≤N×f1(y1)f1(y1)|dy1dy2≲∑0≤l≤N1|x|s12-l(s1-n/r1)∥f1∥Lp1(Rn,w1)×Mr2f2(x)ϑ(BD(0))(1/r1)-(1/p1)+∑-N≤l<01|x|θ2-l(θ-n/r1)∥f1∥Lp1(Rn,w1)×Mr2f2(x)ϑ(BD(0))(1/r1)-(1/p1)≲(1|x|s1+1|x|θ)∥f1∥Lp1(Rn,w1)×Mr2f2(x)ϑ(BD(0))(1/r1)-(1/p1),
if we choose ι=s1 and ι=θ∈(n/(p1ϵ1),n/r1) in Lemma 11 respectively. Therefore,
(50)(∫|x|>A|TσN,b1(f1,f2)|pνw→(x)dx)1/p≲∥b1∥L∞(Rn)∏k=12∥fk∥Lpk(Rn,wk)ϑ(BD(0))(1/r1)-(1/p1)×((∫|x|>Aw1(x)|x|s1p1dx)1/p1+(∫|x|>Aw1(x)|x|θp1dx)1/p1)≲∥b1∥L∞(Rn)∏k=12∥fk∥Lpk(Rn,wk)ϑ(BD(0))(1/r1)-(1/p1)×((∫|x|>Aw1(x)|x|n(p1/t1)dx)1/p1+(∫|x|>Aw1(x)|x|n(p1/(t1ϵ1))dx)1/p1)≲ϵ∥b1∥L∞(Rn)∏k=12∥fk∥Lpk(Rn,wk),
where the last inequality holds by the fact (see [22, 23]) that for υ∈Ap, p>1(51)∫Rnυ(x)(1+|x|)npdx<∞.
This in turn leads to conclusion (b) directly.
We turn our attention to conclusion (c). We write
(52)TσN,b1(f1,f2)(x)-TσN,b1(f1,f2)(x+t)=∑j=14Ej(x,t),
with
(53)E1(x,t)=∫max1≤k≤2|x-yk|≥δtKN(x;y1,y2)×(b1(x+t)-b1(x))f1(y1)f2(y2)dy1dy2,E2(x,t)=∫max1≤k≤2|x-yk|≥δt(KN(x;y1,y2)-KN(x+t;y1,y2))×(b1(y1)-b1(x+t))f1(y1)f2(y2)dy1dy2,E3(x,t)=∫max1≤k≤2|x-yk|<δtKN(x;y1,y2)×(b1(y1)-b1(x))f1(y1)f2(y2)dy1dy2,E4(x,t)=∫max1≤k≤2|x-yk|<δtKN(x+t;y1,y2)×(b1(x+t)-b1(y1))f1(y1)f2(y2)dy1dy2,
with δt>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. (54)|E1(x,t)|≲|t|∥∇b1∥L∞(Rn)|E1(x,t)|≲×(∏k=12Mrkfk(x)+Mδ(T(f1,f2))(x)),|E2(x,t)|≲(|t|δt-1)ϱ∥b1∥L∞(Rn)|E1(x,t)|≲×∏k=12(Mrkfk(x)+Mrkfk(x+t)),|E3(x,t)|≲δt∥∇b1∥L∞(Rn)∏k=12Mrkfk(x),|E4(x,t)|≲δt∥∇b1∥L∞(Rn)∏k=12Mrkfk(x+t).
Fix each ϵ>0, set
(55)ρ=Aϵ2(1+∥∇b1∥L∞(Rn)),
with
(56)A=min{1,(ϵ2(1+∥b1∥L∞(Rn)))1/ϱ}
and δt=|t|A-1 for each t∈Rn, where constant ϱ>0. Our estimates for those terms of Ej(j=1,…,4) then lead to that when 0<|t|<ρ,
(57)∥TσN,b1(f1,f2)(·)-TσN,∏b(f1,f2)(·+t)∥Lp(Rn,νw→)≲((|t|+δt)∥∇b1∥L∞(Rn)+(|t|δt-1)ϱ∥b1∥L∞(Rn))×∏k=12∥fk∥Lpk(Rn,wk)≲ϵ∏k=12∥fk∥Lpk(Rn,wk);
this establishes conclusion (c) and we conclude that TσN,b1 is compact.
In a completely analogous way, if b2∈Cc∞, then TσN,b2 is compact. Moreover, then TσN,Σb 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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