R E S E A R C H
Open Access
Sharp boundedness for vector-valued
multilinear integral operators
Jinsong Pan
*and Lijuan Tong
*Correspondence:
Hunan Mechanical and Electrical Polytechnic, Changsha, 410151, P.R. China
Abstract
In this paper, the sharp inequalities for some vector-valued multilinear integral operators are obtained. As applications, we get the weightedLp(p> 1) norm inequalities and anLlogL-type estimate for the vector-valued multilinear operators.
MSC: 42B20; 42B25
Keywords: vector-valued multilinear operator; Littlewood-Paley operator; Marcinkiewicz operator; Bochner-Riesz operator; sharp estimate; BMO;Ap-weight
1 Preliminaries and theorems
As the development of singular integral operators and their commutators, multilinear sin-gular integral operators have been well studied (see [–]). In this paper, we study some vector-valued multilinear integral operators as follows.
Suppose thatmjare positive integers (j= , . . . ,l),m+· · ·+ml=mandAjare functions
onRn(j= , . . . ,l). LetF
t(x,y) be defined onRn×Rn×[, +∞). Set
Ft(f)(x) =
Rn
Ft(x,y)f(y)dy
and
FtA(f)(x) =
Rn
l
j=Rmj+(Aj;x,y)
|x–y|m Ft(x,y)f(y)dy
for every bounded and compactly supported functionf, where
Rmj+(Aj;x,y) =Aj(x) –
|α|≤mj
α!D
α
Aj(y)(x–y)α.
LetHbe the Banach spaceH={h:h<∞}such that, for each fixedx∈Rn,F
t(f)(x) and FA
t (f)(x) may be viewed as a mapping from [, +∞) toH. For <s<∞, the vector-valued multilinear operator related toFtis defined by
TA(f)(x)s=
∞
i=
TA(fi)(x)
s
/s ,
where
TA(fi)(x) =FtA(fi)(x),
andFt satisfies: for fixedε> ,
Ft(x,y)≤C|x–y|–n
and
Ft(y,x) –Ft(z,x)≤C|y–z|ε|x–z|–n–ε
if |y–z| ≤ |x–z|. Set
T(f)(x)s=
∞
i=
T(fi)(x) s
/s
and |f|s=
∞
i= fi(x)
s
/s .
Suppose that|T|sis bounded onLp(Rn) for <p<∞and weak (L,L)-bounded. Note that whenm= ,TAis just a vector-valued multilinear commutator ofT andA (see []). While whenm> ,TAis a non-trivial generalization of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been studied by many authors (see [–]). In [], Hu and Yang proved a variant sharp estimate for the multilinear singular integral operators. In [], Pérez and Trujillo-Gonzalez proved a sharp estimate for some multilinear commutator. The main purpose of this paper is to prove a sharp inequality for the vector-valued multilinear integral operators. As applica-tions, we obtain the weightedLp(p> ) norm inequalities and anLlogL-type estimate for the vector-valued multilinear operators.
First, let us introduce some notations. Throughout this paper,Qwill denote a cube of Rnwith sides parallel to the axes. For any locally integrable functionf, the sharp function off is defined by
f#(x) =sup x∈Q
|Q|
Q
f(y) –fQdy,
where, and in what follows,fQ=|Q|–
Qf(x)dx. It is well known that (see [])
f#(x)≈sup x∈Q
inf c∈C
|Q|
Q
f(y) –cdy.
We say thatf belongs toBMO(Rn) iff#belongs toL∞(Rn) andfBMO=f#L∞. For < r<∞, we denotef#
r by
fr#(x) =|f|r#(x)/r.
Let M be the Hardy-Littlewood maximal operator, that is, M(f)(x) =supx∈Q|Q|–×
Q|f(y)|dy. Fork∈N, we denote byM
kthe operatorMiteratedktimes,i.e.,M(f)(x) =
Letbe a Young function and˜ be the complementary associated to, we denote the
-average by, for a functionf,
f,Q=inf
λ> :
|Q|
Q
|f(y)|
λ
dy≤
and the maximal function associated toby
M(f)(x) =sup
x∈Q
f,Q.
The Young functions to be used in this paper are(t) =exp(tr) – and(t) =tlogr(t+e), the corresponding-average and maximal functions are denoted by · expLr,Q,MexpLr and · L(logL)r,Q,ML(logL)r. We have the following inequality, for anyr> andm∈N(see [])
M(f)≤ML(logL)r(f), ML(logL)m(f)≈Mm+(f).
Forr≥, we denote that
boscexpLr =sup
Q
b–bQexpLr,Q,
the spaceOscexpLr is defined by
OscexpLr=b∈Llog Rn:bosc
expLr <∞
.
It has been known that (see [])
b–bQexpLr,kQ≤CkbOscexpLr.
It is obvious thatOscexpLr coincides with theBMOspace ifr= , andOscexpLr⊂BMOif r> . We denote the Muckenhoupt weights byApfor ≤p<∞(see []).
Now we state our main results as follows.
Theorem Let <s<∞,rj≥and DαAj∈Osc
expLrjfor allαwith|α|=mjand j= , . . . ,l. Define/r= /r+· · ·+ /rl.Then,for any <p< ,there exists a constant C> such that
for any f ={fi} ∈C∞(Rn)and x∈Rn,
TA(f) s
# p(x)≤C
l
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
(x).
Theorem Let <s<∞,rj≥and DαAj∈Osc
expLrj for allα with |α|=mj and j= , . . . ,l.
() If <p<∞andw∈Ap,then
TA(f)
sLp(w)≤C l
j= |αj|=mj
DαjAj Osc
expLrj
() Ifw∈A.Define/r= /r+· · ·+ /rland(t) =tlog/r(t+e).Then there exists a constantC> such that for allλ> ,
wx∈Rn:TA(f)(x)s>λ
≤C
Rn
λ–
l
j= |αj|=mj
DαjAj Osc
expLrj
f(x)s
w(x)dx.
Remark The conditions in Theorems and are satisfied by many operators.
Now we give some examples.
Example Littlewood-Paley operators.
Fixε> andμ> (n+ )/n. Let ψ be a fixed function which satisfies the following properties:
() Rnψ(x)dx= ,
() |ψ(x)| ≤C( +|x|)–(n+),
() |ψ(x+y) –ψ(x)| ≤C|y|ε( +|x|)–(n++ε)when|y|<|x|.
We denote that(x) ={(y,t)∈R+n+:|x–y|<t}and the characteristic function of(x) byχ(x). The Littlewood-Paley multilinear operators are defined by
gψA(f)(x) =
∞
FtA(f)(x)dt t
/
,
SAψ(f)(x) =
(x)
FtA(f)(x,y)dy dt tn+
/
and
gμA(f)(x) =
Rn++
t t+|x–y|
nμ
FtA(f)(x,y)dy dt tn+
/
,
where
FtA(f)(x) =
Rn
l
j=Rmj+(Aj;x,y)
|x–y|m ψt(x–y)f(y)dy,
FtA(f)(x,y) =
Rn
l
j=Rmj+(Aj;x,z)
|x–z|m f(z)ψt(y–z)dz
andψt(x) =t–nψ(x/t) fort> . SetFt(f)(y) =f∗ψt(y). We also define that
gψ(f)(x) =
∞
Ft(f)(x)dt t
/
,
Sψ(f)(x) =
(x)
Ft(f)(y)
dy dt
tn+ /
and
gμ(f)(x) =
Rn++
t t+|x–y|
nμ
Ft(f)(y)dy dt tn+
/
which are the Littlewood-Paley operators (see []). LetHbe the space
H=
h:h=
∞
h(t)dt/t
/
<∞
or
H=
h:h=
Rn++
h(y,t)dy dt/tn+
/
<∞
.
Then, for each fixedx∈Rn,FtA(f)(x) andFtA(f)(x,y) may be viewed as the mapping from [, +∞) toH, and it is clear that
gψA(f)(x) =FtA(f)(x), gψ(f)(x) =Ft(f)(x),
SAψ(f)(x) =χ(x)FtA(f)(x,y), Sψ(f)(x) =χ(x)Ft(f)(y)
and
gμA(f)(x) =
t+|xt–y|nμ/FtA(f)(x,y), gμ(f)(x) =
t+|xt–y|nμ/Ft(f)(y).
It is easy to see thatgψ,Sψandgμsatisfy the conditions of Theorems and (see [–]),
thus Theorems and hold forgA
ψ,SψAandgμA.
Example Marcinkiewicz operators.
Fixλ>max(, n/(n+ )) and <γ ≤. Letbe homogeneous of degree zero onRn with Sn–(x)dσ(x) = . Assume that∈Lipγ(Sn–). The Marcinkiewicz multilinear
operators are defined by
μA(f)(x) =
∞
FtA(f)(x)dt t
/
,
μAS(f)(x) =
(x)
FtA(f)(x,y)dy dt tn+
/
and
μAλ(f)(x) =
Rn++
t t+|x–y|
nλ
FtA(f)(x,y)dy dt tn+
/
,
where
FtA(f)(x) =
|x–y|≤t
l
j=Rmj+(Aj;x,y)
|x–y|m
(x–y)
|x–y|n–f(y)dy
and
FtA(f)(x,y) =
|y–z|≤t
l
j=Rmj+(Aj;y,z)
|y–z|m
(y–z)
Set
Ft(f)(x) =
|x–y|≤t
(x–y)
|x–y|n–f(y)dy.
We also define that
μ(f)(x) =
∞
Ft(f)(x)dt t
/
,
μS(f)(x) =
(x)
Ft(f)(y)
dy dt
tn+ /
and
μλ(f)(x) =
Rn++
t t+|x–y|
nλ
Ft(f)(y)
dy dt
tn+ /
,
which are the Marcinkiewicz operators (see []). LetHbe the space
H=
h:h=
∞
h(t)dt/t
/
<∞
or
H=
h:h=
Rn++
h(y,t)dy dt/tn+
/
<∞
.
Then it is clear that
μA(f)(x) =FtA(f)(x), μ(f)(x) =Ft(f)(x),
μAS(f)(x) =χ(x)FtA(f)(x,y), μS(f)(x) =χ(x)Ft(f)(y)
and
μAλ(f)(x) =
t t+|x–y|
nλ/
FtA(f)(x,y), μλ(f)(x) =
t+|xt–y|nλ/Ft(f)(y).
It is easy to see thatμ,μSandμλsatisfy the conditions of Theorems and (see [,
]), thus Theorems and hold forμA
,μAS andμAλ.
Example Bochner-Riesz operators. Letδ> (n– )/,Bδ
t(fˆ)(ξ) = ( –t|ξ|)δ+ˆf(ξ) andBδt(z) =t–nBδ(z/t) fort> . Set
FδA,t(f)(x) =
Rn
l
j=Rmj+(Aj;x,y)
|x–y|m B
δ
t(x–y)f(y)dy.
The maximal Bochner-Riesz multilinear operators are defined by
BAδ,∗(f)(x) =sup
t>
We also define that
Bδ,∗(f)(x) =sup
t>
Bδt(f)(x),
which is the maximal Bochner-Riesz operator (see []). LetHbe the spaceH={h:h= supt>|h(t)|<∞}, then
BAδ,∗(f)(x) =BAδ,t(f)(x), B
δ
∗(f)(x) =Bδt(f)(x). It is easy to see thatBA
δ,∗satisfies the conditions of Theorems and (see []), thus
The-orems and hold forBA
δ,∗.
2 Some lemmas
We give some preliminary lemmas.
Lemma ([]) Let A be a function on Rnand DαA∈Lq(Rn)for allαwith|α|=m and some q>n.Then
Rm(A;x,y)≤C|x–y|m
|α|=m
| ˜Q(x,y)|
˜
Q(x,y)
DαA(z)qdz
/q ,
whereQ is the cube centered at x and having side length˜ √n|x–y|.
Lemma ([, p.]) Let <p<q<∞and for any function f ≥,we define that,for
/r= /p– /q,
fWLq=sup
λ>
λx∈Rn:f(x) >λ/q, Np,q(f) =sup
E
fχELp/χELr,
where the sup is taken for all measurable sets E with <|E|<∞.Then
fWLq≤Np,q(f)≤q/(q–p)/pfWLq.
Lemma ([]) Let rj≥for j= , . . . ,m,we denote that/r= /r+· · ·+ /rm.Then
|Q|
Q
f(x)· · ·fm(x)g(x)dx≤ fexpLr,Q· · · fexpLrm,QgL(logL)/r,Q.
3 Proof of the theorem It is only to prove Theorem .
Proof of Theorem It suffices to prove forf ∈C∞(Rn) and some constantC that the
following inequality holds:
|Q|
Q
TA(f)(x)s–Cpdx /p
≤C l
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
Without loss of generality, we may assumel= . Fix a cubeQ=Q(x,d) andx˜∈Q. LetQ˜ =
√nQandAj˜ (x) =Aj(x) –|α|=mα!(D αAj)
˜
Qxα, thenRm(Aj;x,y) =Rm(Aj˜ ;x,y) andDαAj˜ = DαA
j– (DαAj)Q˜ for|α|=mj. We splitf =g+h={gi}+{hi}forgi=fiχQ˜ andhi=fiχRn\ ˜Q. Write
FtA(fi)(x) =
Rn
j=Rmj+(Aj˜ ;x,y)
|x–y|m Ft(x,y)fi(y)dy
=
Rn
j=Rmj+(A˜j;x,y)
|x–y|m Ft(x,y)hi(y)dy+
Rn
j=Rmj(A˜j;x,y)
|x–y|m Ft(x,y)gi(y)dy
–
|α|=m
α!
Rn
Rm(A˜;x,y)(x–y) α
|x–y|m D
αA˜
(y)Ft(x,y)gi(y)dy
–
|α|=m
α!
Rn
Rm(A˜;x,y)(x–y)α |x–y|m D
αA˜
(y)Ft(x,y)gi(y)dy
+
|α|=m,|α|=m
α!α!
Rn
(x–y)α+αDαA˜
(y)DαA˜(y)
|x–y|m Ft(x,y)gi(y)dy.
Then, by Minkowski’s inequality, we have
|Q|
Q
TA(f)(x)s–TA˜(h)(x)s
p dx /p ≤
|Q|
Q
FtA(f)(x)s–FtA˜(h)(x)s
p dx /p ≤
|Q|
Q
∞
i=
FtA(fi)(x) –F
˜
A
t (hi)(x)
s p/s
dx
/p
≤
C
|Q|
Q ∞ i= Rn
j=Rmj(Aj˜ ;x,y)
|x–y|m Ft(x,y)gi(y)dy
s
p/s
dx
/p
+
C
|Q|
Q ∞ i=
|α|=m
α!
×
Rn
Rm(A˜;x,y)(x–y) α
|x–y|m D
αA˜
(y)Ft(x,y)gi(y)dy s
p/s
dx
/p
+
C
|Q|
Q ∞ i=
|α|=m
α!
×
Rn
Rm(A˜;x,y)(x–y) α
|x–y|m D
αA˜
(y)Ft(x,y)gi(y)dy
s
p/s
dx
/p
+
C
|Q|
Q ∞ i=
|α|=m,|α|=m
α!α!
×
Rn
(x–y)α+αDαA˜
(y)DαA˜(y)
|x–y|m Ft(x,y)gi(y)dy
s
p/s
dx
+
C
|Q|
Q
∞
i= FA˜
t (hi)(x) –FtA˜(hi)(x)
s p/s
dx
/p
:=I+I+I+I+I.
Now, let us estimateI,I,I,IandI, respectively. First, forx∈Qandy∈ ˜Q, by Lemma ,
we get
Rmj(Aj˜ ;x,y)≤C|x–y|
mj
|αj|=mj
DαjAj Osc
expLrj .
Thus, by Lemma and the weak type (, ) of|T|s, we obtain
I≤C
j= |αj|=mj
DαjA jOsc
expLrj
|Q|
Q
T(g)(x)psdx
/p
=C
j= |αj|=mj
DαjAj Osc
expLrj
|Q|–|T(g)|sχQLp
|Q|/p–
≤C
j= |αj|=mj
DαjAj Osc
expLrj
|Q|–T(g)sWL
≤C
j= |αj|=mj
DαjAj Osc
expLrj
|Q|–|g|sL
≤C
j= |αj|=mj
DαjAj Osc
expLrj
M|f|s
(x˜)
≤C
j= |αj|=mj
DαjA jOsc
expLrj
ML(logL)/r
|f|s
(x˜).
ForI, note thatχQexpLr,Q≤C, similar to the proof ofIand by using Lemma , we get
I≤C
|α|=m DαA
OscexpLr
|α|=m
|Q|
Rn
TDαA˜ g
(x)psdx
/p
≤C
|α|=m DαA
OscexpLr
|α|=m
|Q|–TDαA˜ g
(x)sχQWL
≤C
|α|=m DαA
Osc
expLr
|α|=m
|Q|
Rn
DαA˜
(x)g(x)sdx
≤C
|α|=m DαA
Osc
expLrχQexpL
r,Q
×
|α|=m DαA
–
DαA
˜
QexpLr,Q˜|f|sL(logL)/r,Q˜
≤C
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
ForI, similar to the proof ofI, we get
I≤C
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
(x˜).
Similarly, forI, by using Lemma , we get
I≤C
|α|=m,|α|=m
|Q|
Rn
TDαA˜
DαA˜f
(x)psdx
/p
≤C
|α|=m,|α|=m
|Q|–TDαA˜
DαA˜gsχQWL
≤C
|α|=m,|α|=m
|Q|
Rn
DαA˜
(x)DαA˜(x)g(x)sdx
≤C
|α|=m DαA
–
DαA
˜
QexpLr,Q˜
×
|α|=m DαA
–
DαA
˜
QexpLr,Q˜|f|sL(logL)/r,Q˜
≤C
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
(x˜).
ForI, we write
FtA˜(hi)(x) –F
˜
A t (hi)(x)
=
Rn
Ft(x,y)
|x–y|m –
Ft(x,y) |x–y|m
j=
Rmj(Aj˜ ;x,y)hi(y)dy
+
Rn
Rm(A˜;x,y) –Rm(A˜;x,y)
Rm(A˜;x,y) |x–y|m
Ft(x,y)hi(y)dy
+
Rn
Rm(A˜;x,y) –Rm(A˜;x,y)
Rm(A˜;x,y) |x–y|m
Ft(x,y)hi(y)dy
–
|α|=m
α!
Rn
Rm(A˜;x,y)(x–y) α
|x–y|m Ft(x,y) –
Rm(A˜;x,y)(x–y) α
|x–y|m
Ft(x,y)
×DαA˜
(y)hi(y)dy
–
|α|=m
α!
Rn
Rm(A˜;x,y)(x–y) α
|x–y|m Ft(x,y) –
Rm(A˜;x,y)(x–y) α
|x–y|m
Ft(x,y)
×DαA˜
(y)hi(y)dy
+
|α|=m,|α|=m
α!α!
Rn
(x–y)α+α
|x–y|m Ft(x,y) –
(x–y)α+α |x–y|m
Ft(x,y)
×DαA˜
(y)DαA˜(y)hi(y)dy
By Lemma , we know that, forx∈Qandy∈k+Q˜ \kQ˜,
Rmj(Aj˜ ;x,y)≤C|x–y|
mj
|αj|=mj
DαjA Osc
expLrj
+DαjA
˜
Q(x,y)–
DαjA
˜
Q
≤Ck|x–y|mj
|αj|=mj
DαjA Osc
expLrj .
Note that|x–y| ∼ |x–y|forx∈Qandy∈Rn\ ˜Q, we obtain, by the condition ofFt,
I()≤C
Rn
|
x–x|
|x–y|m+n+
+ |x–x|
ε
|x–y|m+n+ε
j=
Rmj(Aj˜;x,y)hi(y)dy
≤C
j= |αj|=mj
DαjAj Osc
expLrj
×
∞
k=
k+Q˜\kQ˜ k
|
x–x|
|x–y|n+
+ |x–x|
ε
|x–y|n+ε
fi(y)dy
≤C
j= |αj|=mj
DαjAj Osc
expLrj
∞
k=
k–k+ –εk
|kQ˜|
kQ˜
fi(y)dy.
Thus, by Minkowski’s inequality, we get
∞
i= I()s
/s
≤C
j= |α|=mj
DαA
jOsc
expLrj
×
∞
k=
k–k+ –εk
|kQ˜|
kQ˜
f(y)sdy
≤C
j= |α|=mj
DαAjOsc
expLrj
M|f|s
(x˜)
≤C
j= |αj|=mj
DαjA jOsc
expLrj
ML(logL)/r
|f|s
(x˜).
ForI(), by the formula (see [])
Rmj(A˜;x,y) –Rmj(A˜;x,y) =
|β|<mj
β!Rmj–|β|
DβA˜;x,x
(x–y)β
and Lemma , we have
Rmj(A˜;x,y) –Rmj(A˜;x,y)≤C
|β|<mj
|α|=mj
|x–x|mj–|β||x–y||β|DαA Osc
thus
∞
i= I()s
/s
≤C
j= |αj|=mj
DαjAj Osc
expLrj
∞
k=
k+Q˜\kQ˜k
|x–x|
|x–y|n+
f(y)sdy
≤C
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
(x˜).
Similarly,
∞
i= I()s
/s
≤C
j= |αj|=mj
DαjA jOsc
expLrj
ML(logL)/r
|f|s
(x˜).
ForI(), similar to the proof ofI(),I()andI, we get
∞
i= I()s
/s
≤C
|α|=m
Rn\ ˜Q
(x–|yx)–αyFt|m(x,y)–
(x–y)αFt(x,y) |x–y|m
×Rm(A˜;x,y)DαA˜(y)f(y)sdy
+C
|α|=m
Rn\ ˜Q
Rm(A˜;x,y) –Rm(A˜;x,y)
× (x–y)αFt(x,y) |x–y|m
DαA˜
(y)f(y)sdy
≤C
|α|=m DαA
Osc
expLr
×
|α|=m
∞
k=
k–k+ –εk
|kQ˜|
kQ˜
DαA˜
(y)f(y)sdy
≤C
|α|=m DαA
Osc
expLr
|α|=m
∞
k=
k–k+ –εk
×DαA –
DαA
˜
QexpLr,kQ˜|f|sL(logL)/r,kQ˜
≤C
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
(x˜).
Similarly,
∞
i= I()s
/s
≤C
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
ForI(), by using Lemma , we obtain
∞
i= I()s
/s
≤C
|α|=m,|α|=m
Rn\ ˜Q
(x–y)α+αF
t(x,y)
|x–y|m –
(x–y)α+αFt(x,y) |x–y|m
×DαA˜
(y)DαA˜(y)f(y)sdy
≤C
|α|=m,|α|=m
∞
k=
–k+ –εk
|kQ˜|
kQ˜
DαA˜
(y)DαA˜(y)f(y)sdy
≤C
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
(x˜).
Thus
I≤C
j= |αj|=mj
DαjAj Osc
expLrj
ML(logL)/r
|f|s
(x˜).
This completes the proof of Theorem .
By Theorem and theLp-boundedness ofML(logL)/r, we may obtain the conclusions (), () of Theorem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper together. They also read and approved the final manuscript.
Received: 15 January 2014 Accepted: 11 July 2014 Published:21 Aug 2014
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10.1186/1029-242X-2014-315
