R E S E A R C H
Open Access
Weighted estimates for vector-valued
multilinear square function
Wenjuan Li and Manli Song
**Correspondence: [email protected] School of Natural and Applied Sciences, Northwest Polytechnical University, Xi’an, Shaanxi 710129, People’s Republic of China
Abstract
LetTbe the multilinear square operator, respectively with certain smooth kernels and non-smooth kernels defined in (Xue and Yan in J. Math. Anal. Appl. 422:1342-1362, 2015) and (Hormoziet al.in arXiv preprint, 2015), and letT∗be its corresponding maximal operator. In this paper, we prove the vector-valued weighted norm boundedness forTandT∗and also establish multiple weighted inequalities for their corresponding iterated commutator generated by the vector-valued multilinear operator and BMO function.
MSC: Primary 42B20; secondary 42B25; 47G10
Keywords: multilinear square function; vector-valued; weighted estimates; commutators
1 Introduction
The importance of the multilinear Littlewood-Paleyg-function and related multilinear Littlewood-Paley type estimates was shown in PDE and other fields, one can see the works by Coifmanet al.[, ], David and Journe [], and also by Fabeset al.[–]. Moreover, a class of multilinear square functions was considered in [], which was used for Kato’s problem.
Recently, Xueet al.[] introduced the multilinear-Paleyg-function with a convolution-type kernel in the following way:
g(f)(x) =
∞
tmn
(Rn)m
ψ
y
t, . . . , ym
t m
j=
fj(x–yj)dy
dt t
/ ,
and obtained the strongLp(ω
)× · · · ×Lpm(ωm) toLp(vω) boundedness and the weak type
results. Later, Xue and Yan [] studied a class of multilinear square functions associated with the following more general non-convolution-type kernels.
Definition (Integral smooth condition of C-Z type I) (see []) For anyv∈(,∞), let
Kv(x,y, . . . ,ym) be a locally integrable function defined away from the diagonalx=y= · · ·=ymin (Rn)m+and denotey= (y, . . . ,ym). We say thatKvsatisfies the integral
condi-tion of C-Z type I, if for some positive constantsγ,A, andB> , the following inequalities
hold:
∞
Kv(x,y)
/
dv v ≤
A
( mj=|x–yj|)mn
, (.)
∞
Kv(z,y) –Kv(x,y)dv
v /
≤ A|z–x|γ
( mj=|x–yj|)mn+γ, (.)
whenever|z–x| ≤ Bmaxmj=|x–yj|; and
∞
Kv(x,y) –Kvx,y, . . . ,yi, . . . ,ym
dv
v /
≤ A|yi–yi|γ
( mj=|x–yj|)mn+γ
(.)
for anyi∈ {, . . . ,m}, whenever|yi–yi| ≤B|x–yi|.
We define the multilinear square functionTby
T(f)(x) =
∞
(Rn)m
Kv(x,y, . . . ,ym) m
j=
fj(yj)dy
dv v
/
(.)
for anyf = (f, . . . ,fm)∈S(Rn)× · · · ×S(Rn) and for allx∈/ m
j=suppfj.
In order to state their results, we first give the definition of multiple weightsAp.
Definition (Multiple weights) (see []) Let ≤p, . . . ,pm<∞, and p =p +· · ·+pm. For anyω= (ω, . . . ,ωm), denotevω=
m i=ω
p/pi
i . If
sup B
|B|
B vω
/p m
i=
|B|
B ω–pi
i /pi
<∞ (.)
holds, we say thatωsatisfies theAp condition. Specially, whenpi= , (|B|
Bω
–pi i )/p
i is
understood as (infBωi)–.
We will need the easy fact: if eachωj∈Apj, thenjm=Apj⊂Ap.
In [], the multilinear maximal operatorMwas defined by
M(f)(x) =sup x∈Q
m
i= |Q|
Q
fi(yi)dyi, (.)
where the supremum is taken over all cubesQcontainingx. The easy fact is thatM(f)(x)≤
m
i=Mfi(x), whereMis the Hardy-Littlewood maximal operator.
Theorem A(see []) Let T be the multilinear square operator defined in(.)with the
kernel satisfying the integral smooth condition of C-Z typeI.Let <p,p, . . . ,pm<∞and
p=
p+· · ·+
pm.Ifωsatisfies the Apcondition,there exists a constant C such that
T(f)Lp(v
ω)≤C
m
i=
Theorem B(see []) Let T be the operator defined in(.)with the kernel satisfying the integral smooth condition of C-Z typeI.Let <δ< /m,the following inequality holds:
MδT(f)(x)≤CM(f)(x) (.)
for any bounded and compact supported functions fi,i= , . . . ,m.
In order to extend it to a more general case, we recall a class of integral operators{At}t> defined in [], where the operatorsAtassociated with the kernelsat(x,y) are defined by
Atf(x) =
Rnat(x,y)f(y)dy
for every functionf∈Lp(Rn), ≤p≤ ∞, anda
t(x,y) satisfies the following size condition:
at(x,y)≤ht(x,y) :=t–n/sh |
x–y| t/s
for a fixed constants> , (.)
wherehis a positive, bounded, decreasing function satisfying
lim r→r
n+η
hrs= (.)
for someη> . The above conditions indicate that for someC> and all <η≤η, the kernelsat(x,y) satisfy
at(x,y)≤Ct–n/s +t–/s|x–y|–n–η.
Assumption (H) Assume that for eachi= , . . . ,m, there exist operators{A(ti)}t> with kernelsa(ti)(x,y) satisfying conditions (.) and (.) with constantssandηand that for
everyi= , . . . ,m, there exist kernelsKt(,iv)such that
Tf, . . . ,A(ti)fi, . . . ,fm
,g
=
Rn
∞
(Rn)mK
(i)
t,v(x,y, . . . ,ym)
m
i=
fi(yi)dy
dv v
/
g(x)dx (.)
for all Schwartz functionsf, . . . ,fm,gwith
m
k=suppfk∩suppg=∅.
There exists a functionφ∈C(R) withsuppφ∈[–, ] and a constant > so that, for everyi= , . . . ,m, we have
∞
Kv(x,y) –Kt(,iv)(x,y)
dv
v /
≤ A
( mj=|x–yj|)mn m
k=,k=i φ
|yi–yk| t/s
+ At
/s
( mj=|x–yj|)mn+ , (.)
Assumption (H) Assume that there exist operators{At}t>with kernelsat(x,y) that sat-isfy conditions (.) and (.) with constants sandη, and there exist kernelsKt(),v(x,y)
such that
Kt(),v(x,y) =
RnKv(z,y)at(x,z)dz (.)
makes sense for all (x,y)∈(Rn)m+ andt> . Assume also that there exists a function
φ∈C(R) andsuppφ⊂[–, ] and a constant > such that
∞
Kt(),v(x,y)
dv
v /
≤ A
( mk=|x–yk|)mn
, (.)
whenever t/s≤min
≤j≤m|x–yj|and
∞
Kv(x,y) –Kt(),v(x,y)
dv
v /
≤ A
( mj=|x–yj|)mn m
k=,k=i φ
| x–yk|
t/s
+ At
/s
( mj=|x–yj|)mn+ (.)
for someA> , whenever t/s≤max
≤j≤m|x–yj|.
Assumption (H) Assume that there exist operators{At}t>with kernelsat(x,y) that sat-isfy condition (.) and (.) with constantsandη. Also assume that there exist kernels
Kt(),v satisfying (.) and positive constantsAand such that
∞
Kt(),v(x,y) –Kt(),vx,ydv v
/
≤ At/s
( mj=|x–yj|)mn+ , (.)
whenever t/s≤min
≤j≤m|x–yj|and |x–x| ≤t/s.
We say that the kernelsKvgeneralized the square function kernels if they satisfy (.), (.), and (.) with parameters m,A, s,η, , and we denote their collection by m–
GSFK(A,s,η, ). We say thatTis of classm–GSFO(A,s,η, ) ifThas an associated kernel
Kvinm–GSFK(A,s,η, ).
Theorem C(see []) Let T be a multilinear operator in m–GSFO(A,s,η, )with a kernel satisfying Assumptions(H)and(H).For <p, . . . ,pm<∞,p≥withp=p+· · ·+pm,
ω∈Ap,the following inequality holds:
T(f)Lp(ω)≤C
m
i=
fi Lpi(ω). (.)
Theorem D(see []) Let <δ< /m and T be a multilinear operator in m–GSFO(A,s,
η, )with a kernel satisfying Assumptions(H)and(H).Then there exists a constant C such that
MδT(f)(x)≤C
m
j=
Mfj(x) (.)
Moreover,the corresponding multilinear maximal square function T∗is defined by
T∗(f)(x) =sup
δ>
∞
m
i=|x–yi|>δ
Kv(x,y)
m
k=
fj(yj)dy
dv v
/
. (.)
Theorem E(see []) Let T be a multilinear operator in m–GSFO(A,s,η, )with a kernel
satisfying Assumptions(H)and(H).For <p, . . . ,pm<∞,p≥withp=p+· · ·+pm,
ω∈Ap,the following inequality holds:
T∗(f)
Lp(ω)≤C
m
i=
fi Lpi(ω). (.)
Theorem F(see []) Let T be a multilinear operator in m–GSFO(A,s,η, )with a kernel
satisfying Assumptions(H)and(H).For anyη> ,there is a constant C<∞depending onηsuch that the following inequality holds:
T∗(f)(x)≤C
MηT(f)(x) +
m
j=
Mfj(x)
, ∀x∈Rn (.)
for allf in any product of Lqj(Rn)spaces,with≤qj<∞.
2 Main results
In this section, we first list some results about vector-valued multilinear operatorTqand the corresponding vector-valued maximal multilinear operatorTq∗which are defined, re-spectively, by
Tq(f)(x) =T(f)(x)q=
∞
k=
T(fk, . . . ,fmk)(x)q /q
, (.)
Tq∗(f)(x) =T∗(f)(x)q=
∞
k=
T∗(f
k, . . . ,fmk)(x) q
/q
, (.)
wheref = (f, . . . ,fm) withfi={fik}∞k=.
Theorem Assume that T is a multilinear square operator defined in(.)with the kernel
satisfying the integral condition of C-Z typeI.Let <p,p, . . . ,pm<∞, <q,q, . . . ,qm< ∞, and /m<p,q<∞ with p = p
+· · ·+
pm,
q =
q +· · ·+
qm. If (ω
p , . . . ,ω
pm
m )∈
(Ap, . . . ,Apm),the following inequality holds:
Tq(f)Lp(m j=ω
p j)≤C
m
j=
fj qj
Lpj(ωpjj ). (.)
Theorem Assume that T is a multilinear square operator defined in(.)with the kernel
satisfying the integral condition of C-Z typeI.Let≤p,p, . . . ,pm<∞, <q,q, . . . ,qm<
∞,and <p,q<∞withp =p +· · ·+
pm,
q=
q+· · ·+
(i) If≤p,p, . . . ,pm<∞andω∈Ap∩ · · · ∩Apm,there exists a constantC> such
that
Tq(f)Lp(ω)≤C
m
j=
fj qjLpj(ω). (.)
(ii) If at least onepj= andω∈A,there exists a constantC> such that
Tq(f)Lp,∞(ω)≤C m
j=
fj qjLpj(ω). (.)
Next, we show the results for the multilinear square operatorT with non-smooth ker-nels and its corresponding maximal operator T∗. Meanwhile, we also establish multi-ple weighted inequalities for their corresponding iterated commutator generated by the vector-valued multilinear operator and BMO function. We will state our results as follows.
Theorem Let T be a multilinear operator in m–GSFO(A,s,η, )with a kernel satisfying Assumptions(H)and(H).Let <p,p, . . . ,pm<∞, <q,q, . . . ,qm <∞and/m< p,q<∞with p =p
+· · ·+
pm,
q=
q +· · ·+
qm.If(ω
p , . . . ,ω
pm
m )∈(Ap, . . . ,Apm),there
exists a constant C> such that
Tq(f)Lp(m j=ω
p j)≤C
m
j=
fj qjLpj(ωpj j )
. (.)
A similar estimate also holds true for the corresponding maximal operator T∗.
Theorem Let T be a multilinear operator in m–GSFO(A,s,η, )with a kernel satisfying Assumptions(H)and(H).Let≤p,p, . . . ,pm<∞, <q,q, . . . ,qm<∞and <p,q<
∞withp =p +· · ·+
pm,
q =
q+· · ·+
qm.
(i) If≤p,p, . . . ,pm<∞andω∈Ap∩ · · · ∩Apm,the following inequality holds:
Tq(f)Lp(ω)≤C
m
j=
fj qjLpj(ω). (.)
(ii) If at least onepj= andω∈A,the following inequality holds:
Tq(f)Lp,∞(ω)≤C
m
j=
fj qjLpj(ω). (.)
Similar estimates also hold true for the corresponding maximal operators T∗.
The commutator associated withTis given by
Tb(f)(x) =
b,
b, . . .
b–, [b,T]
–· · ·
(f)(x)
=
(Rn)m
j=
bj(x) –bj(y)K(x,y, . . . ,ym)
m
i=
fi(yi)dy,
For simplicity of notation, for the sequence{fk}∞k=={fk, . . . ,fmk}∞k=of vector functions, the commutator associated with a vector-valuedTqcan be defined by
Tb,q(f)(x) =Tb(f)(x)q =
∞
k=
Tb(fk)(x) q
/q
. (.)
Theorem Assume that T is a multilinear operator in m–GSFO(A,s,η, )with kernel sat-isfying Assumptions(H)and(H).Let/m<p<∞,
p=
p+· · ·+
pm with <p, . . . ,pm<
∞, /m<q<∞,and q=q +· · ·+
qm with <q, . . . ,qm<∞.Suppose thatω∈Apand
b∈(BMO).
(i) There exists a constantC> such that
Tb,q(f)Lp(vω)≤
j=
bj BMO m
j=
f qjLpj(Mωj). (.)
(ii) Ifωj∈Apj,there exists a constantC> such that
Tb,q(f)Lp(vω)≤
j=
bj BMO m
j=
f qjLpj(ωj). (.)
3 The proof of Theorem 3
Since ωj∈Apj, by the previous statement,
m j=ω
pj
j ∈Ap. Writingvω =
m j=(ω
pj
j )p/pj = m
j=ω
p
j, Theorem A implies that
T(f)Lp(m j=ω
p j)≤C
m
j=
fj Lpjωjpj. (.)
We will apply the following lemma to get the desirable result.
Lemma (see []) LetT be an m-linear operator,and let <s, . . . ,sm<∞and/m<s< ∞be fixed indices such that
s=
s+· · ·+
sm.For(ω
s
, . . . ,ωsmm)∈(As, . . . ,Asm),the following
estimate holds:
T(f)Ls(m j=ωsj)
m
j=
fj Lsj
ωjsj. (.)
Then, for all indices, <p, . . . ,pm <∞ and /m<p<∞ satisfy p = p +· · ·+ pm, <q, . . . ,qm<∞,and/m<q<∞such that q =q +· · ·+qm,and all(ωp, . . . ,ω
pm
m )∈
(Ap, . . . ,Apm).Then the following inequality holds:
Tq(f)Lp(m j=ω
p j)≤C
m
j=
fj qj
Lpj(ωpjj ). (.)
4 The proof of Theorem 5
Lemma (see []) Let <p,δ<∞andωbe any Mockenhaupt A∞weight.Then there exists a constant C independent of f such that the inequality
Rn
Mδf(x)
p
ω(x)dx≤C
Rn
Mδf(x)
p
ω(x)dx, (.)
holds for any function f for which the left-hand side is finite.
Lemma (see []) For(ω, . . . ,ωm)∈(Ap, . . . ,Apm)with≤p, . . . ,pm<∞and for <
θ, . . . ,θm< such thatθ+· · ·+θm= ,we haveωθ· · ·ω θ
∈Amax{p,...,pm}.
Note that (ωp , . . . ,ω
pm
m ) ∈ (Ap, . . . ,Apm), and Lemma indicates that
m j=ω
p j = m
j=(ω
pj
j )p/pj∈Amax{p,...,pm}⊂A∞.
Exploiting Lemma . in [] and the standard argument, we obtain MδT(f) Lp(m j=ω
p j)<
∞. Together with Theorem D, we have
T(f)Lp(m j=ω
p j)≤
MδT(f)Lp(m j=ω
p j)
≤CMδT(f)Lp(m j=ω
p j)
≤C
m
j=
Mfj
Lp(m j=ω
p j)
≤C m
j=
Mfj Lpj(ωpjj )
≤C m
j=
fj Lpj(ωpjj ).
By Lemma , we finish the proof of Theorem .
The estimate forT∗will follow from Lemma , Theorem F, and the following argument:
T∗(f)Lp(m j=ω
p j)≤C
MηT(f)Lp(m j=ω
p j)+
m
j=
Mfj
Lp(m j=ω
p j)
≤C
MT(f)η η
L
p
η(m j=ω
p j)
+
m
j=
Mfj
Lp(m j=ω
p j)
≤C
T(f)η η
L
p
η(m j=ω
p j)
+
m
j=
Mfj
Lp(m j=ω
p j)
≤C
T(f)Lp(m j=ω
p j)+
m
j=
Mfj
Lp(m j=ω
p j)
≤C
m
j=
Mfj
Lp(m j=ω
p j)
. (.)
5 The proofs of Theorem 4 and Theorem 6
LetFdenote a family of ordered pairs of non-negative, measurable functions (f,g), if we say that for somep, <p<∞, andω∈A∞,
Rnf(x)
pω(x)dx≤C
Rng(x)
pω(x)dx, (.)
and we denote it by (f,g)∈F.
Lemma (see []) Given a familyF,suppose that for some p, <p<∞,and for every
weightω∈A∞, (f,g)∈F.Then we have,for all <p,q<∞andω∈A∞,
k
(fk)q
/q
Lp(ω)≤
C k
(gk)q
/q
Lp(ω)
, (fk,gk)k⊂F. (.)
For all <p,q<∞, <s≤ ∞,andω∈A∞,
k
(fk)q
/q
Lp,s(ω)≤
C k
(gk)q
/q
Lp,s(ω)
, (fk,gk)k⊂F. (.)
Lemma (see [])
(i) Let <q<∞and≤p<∞,there is a constantCr,psuch that
k
Mfk(x)q
/q
Lp,∞(ω)≤Cq,p
k fk(x)q
/q
Lp(ω)
(.)
if and only ifω∈Ap.
(ii) Let <q<∞and <p<∞,there is a constantCq,psuch that
k
Mfk(x)q
/q
Lp(ω)≤
Cq,p
k fk(x)q
/q
Lp(ω)
(.)
if and only ifω∈Ap.
By Theorem B, Theorem D, and Theorem F, together with the argument from Section and (.), we have
T(f)Lp(ω)≤C
m
j=
Mfj
Lp(ω)
. (.)
HereT can be replaced byTandT∗which are from Theorem and Theorem . We apply Lemma to (T(f),mj=Mfj)∈F, and by Lemma we get the desirable re-sults.
6 The proof of Theorem 7
In order to prove Theorem , first we will list some notations and lemmas:
M f q(x) :=sup
x∈Q m
j= |Q|
Q
ML(logL)
f q(x) :=sup
x∈Q m
j=
fj qjL(logL),Q, (.)
Mρ
f q(x)
:=sup x∈Q
∞
ν=
–νn
j∈ρ |Q|
Q
fj(yj)qjdyj
j∈ρ |νQ|
νQ
fj(yj)qjdyj, (.)
whereρ={j, . . . ,j} ⊂ {, . . . ,m}, ≤<mandρ={, . . . ,m}\ρ.
Lemma (see []) Let <p, . . . ,pm<∞,p=p+· · ·+pm,P= (p, . . . ,pm),ω∈AP,and ρ={j, . . . ,j} ⊂ {, . . . ,m}, ≤<m.ThenM,ML(logL),Mρare bounded from Lp(ω)× · · · ×Lpm(ω
m)to Lp(vω).
Lemma Let T be a multilinear operator in m–GSFO(A,s,η, )with kernel satisfying Assumptions(H)and(H).Assume that≤<m,ρ={j, . . . ,j},and/m<q<∞, ≤
q, . . . ,qm<∞withq=q+· · ·+qm.Then there exists a constant C> such that
MδTq(f)(x)≤C
M f q(x) +Mρ f q(x). (.)
Proof For a pointxand a cubeQx, to obtain (.), it suffices to prove for <δ< /m,
|Q|
Q
T(f)(z) –cδqdz
/δ
≤CM f q(x) +Mρ f q(x) (.)
for some constantcto be determined later.
Writefk=fk+fk∞, where{fk}∞k=={fkχQ∗}∞k=={fkχQ∗, . . . ,fmkχQ∗}andQ∗= (
√
n+)Q. Letc= α,...,αmT(f
α)(x) and in the sum eachα
j= or∞and in each term there is at least
oneαj=∞. Then
|Q|
Q
T(f)(z) –cδqdz
/δ
≤C
|Q|
Q
Tqf(z)δdz /δ
+C
α,...,αm
|Q|
Q
Tfα(z) –Tfα(x)δ qdz
/δ
:=I+ α,...,αm
IIα,...,αm,
where in each term of the last sum there is at least oneαj=∞.
Kolmogorov’s inequality and Theorem implies that
I≤CTqfL/m,∞(Q,dz
|Q|)
≤C m
j= |Q|
Q
fj(z)qjdz
We proceed to the estimate forIIα,...,αm. Here we chooset= [
√
n(Q)]s. Ifα
=· · ·αm=
∞, we have
II∞,...,∞≤
C
|Q|
Q
Tfα(z) –Tfα(x)qdz
≤|C
Q| Q ∞ k=
Tfk∞(z) –Tfk∞(x)q
/q dz
≤|C
Q| Q ∞ k= ∞
(Rn\Q∗)m
Kv(z,y) –Kv(x,y) m
j=
fjk(yj)dy dv v
q//q dz,
applying Minkowski’s inequality, we get
C
|Q| Q ∞ k= ∞
(Rn\Q∗)m
Kv(z,y) –Kv(x,y) m
j=
fjk(yj)dy dv v
q//q dz
≤ C |Q|
Q ∞ k=
(Rn\Q∗)m
∞
Kv(z,y) –Kv(x,y)dv
v
/m
j=
fjk(yj)dy
q/q
dz
≤|C
Q| Q ∞ k=
(Rn\Q∗)m
∞
Kv(z,y) –Kt(),v(z,y)
dv
v /m
j=
fjk(yj)dy
q/q dz
+ C |Q|
Q ∞ k=
(Rn\Q∗)m
∞
Kt(),v(z,y) –Kt(),v(x,y)
dv v / × m j=
fjk(yj)dy
q/q
dz
+ C |Q|
Q ∞ k=
(Rn\Q∗)m
∞
Kt(),v(x,y) –Kv(x,y)
dv
v /m
j=
fjk(yj)dy
q/q dz
:=II∞,...,∞+II∞,...,∞+II∞,...,∞.
Because ofz∈Qandyj∈Rn\(√n+ )Q, we obtain|yj–z|> (√n+ )(Q) > t/sfor allj= , . . . ,m. Assumption (H) gives
II∞,...,∞≤ C
|Q|
Q
(Rn\Q∗)m
At/s
( mj=|z–yj|)mn+
m
j=
fj(yj)qjdy dz
≤ ∞ k= k m j= (k+)n|Q∗|
k+Q∗fj(yj)qjdyj
≤CM f q(x).
Sincex,z∈Q,|z–x| ≤√n(Q)≤/t/s. Noting that|yj–z|> (√n+ )(Q) > t/sfor allj= , . . . ,m, applying Assumption (H) and a similar argument toII, we haveII∞,...,∞≤ CM( f q)(x). Similarly, we also getII∞
Now let us consider the typical case ofIIα,...,αm, that is,α=· · ·=αh=∞andαh+=· · ·=
αm= , ≤h<m,
II∞,...,≤
C
|Q|
Q ∞
k=
(Rn)m
∞
Kv(z,y) –Kt(),v(z,y)
dv
v /
×
h
j=
fjk∞ m
j=h+
fjkdy
q/q
dz
+ C |Q|
Q ∞
k=
(Rn)m
∞
K()
t,v(z,y) –K
()
t,v(x,y)
dv
v /
×
h
j=
fjk∞ m
j=h+
fjkdy
q/q dz
+ C |Q|
Q ∞
k=
(Rn)m
∞
Kt(),v(x,y) –Kv(x,y)
dv
v /
×
h
j=
fjk∞ m
j=h+
fjkdy
q/q
dz
:=II∞,...,+II∞,...,+II∞,...,.
ForII
∞,...,, by Assumption (H), we have
II∞,...,≤ C
|Q|
Q
(Rn\Q∗)h
At/sh
j= fj qjdyj ( mj∈{,...,h}|z–yj|)mn+
+
(Rn\Q∗)h
Ahj= fj qjdyj
( mj∈{,...,h}|z–yj|)mn m
j=h+
Q∗
fj qjdyjdz
≤
∞
k=
A|Q∗| /n
(k|Q∗|/n)mn+
(kQ∗)h
h
j=
fj qjdyj
+
∞
k=
A
(k|Q∗|/n)mn
(kQ∗)h
h
j=
fj qjdyj m
j=h+
Q∗
fj qjdyj
≤C ∞
k=
m
j= k
(k+|Q∗|/n)n
k+Q∗ fj qjdyj
+C ∞
k= kn(m–h)
m
j=h+ |Q∗|
Q∗
fj qjdyj
×
h
j= (k+|Q∗|/n)n
k+Q∗ fj qjdyj
≤CM f q(x) +Mρ f q(x).
By a similar argument, we deduce that II∞,...,≤CM( f q)(x) +Mρ( f q)(x) and
Given any positive integerm,∀≤j≤m, letCjm denote the family of all finite subset
σ={σ(), . . . ,σ(j)}ofjdifferent elements. For anyσ∈Cjmwe associate the complementary sequenceσgiven byσ={, , . . . ,m}\σ.
Lemma Let <δ< < /m, /m<q<∞,andq=q +· · ·+
qm with <q, . . . ,qm<∞.
Suppose thatb∈(BMO).Then there exists a constant C> depending only onδ and
such that
Mδ(Tb,qf)(x)≤C
j=
bj BMO
ML(logL) f q(x) +MTq(f)(x)
+C
–
i=
σ∈C
i
Ci,
j∈σ
bj BMOM(Tbσ,qf)(x) (.)
for any smooth vector function{fk}∞k=for any x∈Rn.
Proof For simplicity of notation, we replacemj=fj(yj) byF(y) and letλj=|Q|
Qbj(z)dz,
forj= , . . . ,. Letx∈RnandQbe a cube centered atx. We have
Tb(f)(x) =
∞
(Rn)m
j=
bj(x) –bj(yj)
Kv(x,y)F(y)dy
dv v
/
=
∞
(Rn)m
j=
bj(x) –λj
–bj(yj) –λj
Kv(x,y)F(y)dy
dv v
/
≤
i=
σ∈Ci
j∈σ
bj(x) –λj
×
∞
(Rn)m
j∈σ
bj(yj) –λjKv(x,y)F(y)dy
dv v
/
=
j=
bj(x) –λjT(f)(x) +T
j=
bj(·j) –λj
f
(x)
+ –
i=
σ∈Ci
j∈σ
bj(x) –λj
×
∞
(Rn)m
j∈σ
bj(yj) –λj
Kv(x,y)F(y)dy
dvv /.
Noting thatbj(yj) –λj= (bj(yj) –bj(x)) + (bj(x) –λj), we get
Tb,q(f)(z) =
j=
bj(z) –λjTq(f)(z) +Tq
j=
bj(·j) –λj
f
(z)
+ –
i=
σ∈Ci
Ci,
j∈σ
bj(z) –λjTbσ,qf(z).
Letc= c q= ( ∞
k=|ck|q)/q. Since <δ< /m< , we have
|Q|
Q
Tb,qf(z)
δ
–|c|δdz
/δ
≤C
|Q|
Q
Tb(f)(z) –c
δ qdz /δ ≤C |Q|
Q j=
bj(x) –λjT(f)(z) δ q dz /δ +C – i=
σ∈Ci
Ci,
|Q|
Q
j∈σ
bj(z) –λjTbσ,qf(z) δ dz /δ +C |Q|
Q T j=
bj(·j) –λj
f
(z) –c δ q dz /δ
:=I+II+III.
Now exploiting the standard Hölder inequality for finitely many functions with <p< /δ, it follows that
I≤C
j=
bj BMOM(Tqf)(x),
II≤C
–
i=
σ∈Ci
Cj,
j∈σ
bj BMOM (Tbσf)(x).
Next let us address partIII. Setfj=fj+fj∞, wherefj=fjχQ∗. Letfα=fα· · ·fmαm and Q∗= (√n+ )Q. Takingc= α,...,αm T((b(·) –λ)· · ·(b(·) –λ))f
α
· · ·fmαm(x) q, we
have T j=
bj(·j) –λj
f
(z) –c
q
≤Tq
j=
bj(·j) –λj
f
(z)
+C
α,...,αm
T j=
bj(·j) –λj
fα
(z) –T
j=
bj(·j) –λj
fα
(x)
q
,
where in the last sum eachαj= or∞and in each term there is at least oneαj=∞.
Ifα=· · ·=αm= , Kolmogorov’s inequality and Theorem imply
|Q|
Tq j=
bj(·j) –λj
f
(z)
δ/δ
≤C Tq j=
bj(·j) –λj
f
L/m,∞(Q,dz
≤C
j=
bj BMO fj qjL(logL),Q m
j=+ |Q|
Q
fj qjdz
≤C
j=
bj BMOML(logL)
fj qj(x).
Ifα=· · ·=αm=∞, applying Hölder’s inequality and Minkowski’s inequality, then we
get
|Q|
Q T j=
bj(·j) –λj
fα
(z) –T
j=
bj(yj) –λj
fα
(x)
δ q /δ
≤|C
Q| Q ∞ k= ∞
(Rn\Q∗)m
Kv(z,y) –Kv(x,y)
×
j=
bj(yj) –λj
fk(y)· · ·fmk(ym)dy
dv v
q//q dz
≤ C |Q|
Q ∞ k=
(Rn\Q∗)m
∞
Kv(z,y) –Kv(x,y)dv
v / × j=
bj(yj) –λj
fk(y)· · ·fmk(ym)dy
q/q
dz
≤|C
Q| Q ∞ k=
(Rn\Q∗)m
∞
Kv(z,y) –Kv,t(z,y)dv
v / × j=
bj(yj) –λj
fk(y)· · ·fmk(ym)dy
q/q dz
+ C |Q|
Q ∞ k=
(Rn\Q∗)m
∞
Kv,t(z,y) –Kv,t(x,y)dv
v / × j=
bj(yj) –λj
fk(y)· · ·fmk(ym)dy
q/q
dz
+ C |Q|
Q ∞ k=
(Rn\Q∗)m
∞
Kv,t(x,y) –Kv(x,y)dv
v / × j=
bj(yj) –λj
fk(y)· · ·fmk(ym)dy
q/q
dz
:=III+III+III.
First we considerIII. Takingt= [ √
n(Q)]s, by Assumption (H) we have
III≤
C
|Q|
Q
(Rn\Q∗)m
At /s
( mj=|z–yj|)mn+
j=
bj(yj) –λj f q· · · fm qmdy dz
≤C ∞ k= k j= (k+)n|Q∗|
×
m
j=+ (k+)n|Q∗|
k+Q∗ fj qjdyj
≤C ∞
k= k
j=
bj BMO fj qjL(logL),k+Q∗
×
m
j=+ (k+)n|Q∗|
k+Q∗ fj qjdyj
≤CML(logL)
f q(x).
Similarly, we haveIII≤CML(logL)( f q)(x) andIII≤CML(logL)( f q)(x). Now it
re-mains to consider the typical case ofIII,
|Q|
Q T
j=
bj(·j) –λj
f∞, . . . ,f∞,f+, . . . ,fm
(z)
–T
j=
bj(yj) –λj
f∞, . . . ,f∞,f+, . . . ,fm
(x)
δ
q
/δ
≤ C |Q|
Q ∞
k=
∞
(Rn\Q∗)m
Kv(z,y) –Kv(x,y)
j=
bj(yj) –λj
×f∞k(y)· · ·f∞k(y)f+,k(y+)· · ·fmk (ym)dy
dv v
q//q dz
≤ (Rn\Q∗)
t/s
j=(bj(yj) –λj) fj(yj) qjdyj
( j=|z–yj|)mn+
+
(Rn\Q∗)
j=(bj(yj) –λj) fj(yj) qjdyj
( j=|z–yj|)mn
m
j=+
Q∗
f(yj)qjdyj
≤C
j=
bj BMOML(logL)
f q(x).
Lemma Let <p<∞, /m<q<∞,andq=q +· · ·+
qm with <q, . . . ,qm<∞and
letω∈A∞.Suppose thatb∈(BMO).Then there exists a constant C> such that
Rn|Tb,q
f|pω(x)dx≤C
j=
bj pBMO
Rn
ML(logL)
f q(x)pω(x)dx. (.)
The proof is similar to [], so we omit it here.
Based on the above lemmas, the proof of Theorem . in [] provides the main ideas for the proof of Theorem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Acknowledgements
The first author was supported by the National Natural Science Foundation of China (No. 11401175). The second author was supported by the Fundamental Research Funds for the Central Universities (No. 3102015ZY068). The authors thank the referees for carefully reading the manuscript and providing many valuable suggestions, which have improved the article.
Received: 15 October 2015 Accepted: 26 November 2015
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