R E S E A R C H
Open Access
Estimates of singular integrals and multilinear
commutators in weighted Morrey spaces
Xiao Feng Ye
*and Xu Sheng Zhu
*Correspondence: [email protected]; [email protected]
Department of Mathematics and Information Sciences, East China Jiao Tong University, Nanchang, 330013, P.R. China
Abstract
SupposeTis a singular integral operator whose kernel is a variable kernel with mixed homogeneity; the purpose of this paper is to study the continuity of the operator in weighted Morrey spacesLp,κ(
ω
), 1≤p<∞, 0 <κ
< 1. A special attention is paid alsoto the multilinear commutator of this operator withBMOfunction. MSC: 42B20; 42B35
Keywords: mixed homogeneity; multilinear commutators; weighted Morrey spaces;
BMO; Orlicz maximal operator
1 Introduction
LetK(x,ξ) :Rn×Rn\{} →Rbe a variable kernel. The singular integral operator is
de-fined by
Tf(x) =p.v.
RnK(x,x–y)f(y)dy (.)
and its multilinear commutator with theBMOfunction
[b,T]f(x) =
Rn
N
i=
bi(x) –bi(y)
K(x,x–y)f(y)dy, (.)
whereb= (b, . . . ,bn),bi∈BMO, ≤i≤N. The variable kernelK(x,ξ) depends on some
parameterxand possesses ‘good’ properties with respect to the second variableξ, which was firstly introduced by Fabes and Rieviève in []. They generalized the classical Calderón and Zygmund kernel and the parabolic kernel studied by Jones in []. By introducing a new metricρ, Fabes and Rieviève studied (.) inLp(Rn), whereRnwas endowed with the
topology induced byρand defined by ellipsoids.
By using this metricρ, Softova in [] obtained that the integral operator (.) and com-mutator were continuous in generalized Morrey spacesLp,ω(Rn), <p<∞,ωsatisfying
suitable conditions.
The multilinear commutator was introduced by Pérez and González [] who proved the weighted Lebesgue estimates. Xu in [] also showed that the multilinear commutators (.) were continuous inLp,ω(Rn), <p<∞.
The weighted Morrey spacesLp,κ(w) were introduced by Komori and Shirai [].
More-over, they showed some classical integral operators and corresponding commutators were
bounded in weighted Morrey spaces. Recently, Wang [–] obtained that some other kind of integral operators (e.g., Bochner-Riesz operator, Marcinkiewicz operatorsetc.) and commutators were also bounded in weighted Morrey spaces. He Sha [] showed that mul-tilinear operators were bounded on weighted Morrey spaces with the symbol ofb∈Lip(β). The main purpose of this paper is to discuss the continuity of the singular integral operator whose kernel is a variable kernel with mixed homogeneity and its multilinear commutator in the weighted Morrey spacesLp,κ(ω), <p<∞, <κ< , where the weight functionωis Apweight. Furthermore, we shall give the weighted weak type estimate of theses operators
in the weighted Morrey spacesL,κ(ω), <κ< . Our main results are stated as follows.
Theorem . Let <p<∞, <κ< .If w∈Ap,then there exists a constant C> such that
TfLp,κ(w)≤CfLp,κ(w).
When p= ,for anyλ> and ellipsoidE,there exists a constant C> such that
λwx∈E:Tf(x)>λ≤CfL,κ(w).
IfK(x,ξ) is a constant kernel and a metricρis Euclidean one, this result is just Theo-rem . in [].
Theorem . Let <p<∞, <κ < .If bi∈BMO(Rn), ≤i≤N,w∈Ap,then there exists a constant C> such that
[b,T]f Lp,κ(w)≤CbfLp,κ(w),
whereb=Ni=bi∗.When p= ,for anyλ> and ellipsoidE,then there exists a con-stant C> such that
λwx∈E:[b,T]f(x)>λ≤CbfL,κ(w), where(t) =tlogN(e+t)andfL,κ(w)=(|f|)L,κ(w).
In what follows, we denote byCpositive constants which are independent of the main parameters but may vary from line to line.
2 Some notations and lemmas
In this section, we introduce some basic definitions and lemmas needed for the proof of the main results.
Letα, . . . ,αnbe real numbers,αi≥ and|α|=
n
i=αi. Following Fabes and Riviève
[], there exists a functionρsuch thatρ(x–y) defines a distance between any two points
x,y∈Rn. ThusRnendowed with the metric ρ results in a homogeneous metric space [, ]. The balls with respect toρ(x) centered at the origin and of radius r are the ellipsoids
Er() =
x∈Rn: x
rα +· · ·+ x
n rαn <
with Lebesgue measure|Er|=C(n)r|α|. It is easy to see that the unit sphere with respect to
this metric coincides with the unit spherenwith respect to the Euclidean one.
Definition . The functionK(x,ξ) :Rn×Rn\{} →Ris called a variable kernel with
mixed homogeneity if
(i) for every fixedx, the functionK(x,·)is a constant kernel satisfying () K(x,·)∈C∞(Rn\{});
() for anyμ> ,αi≥,|α|=
n i=αi Kx,μαξ
, . . . ,μαnξn
=μ–|α|K(x,ξ);
()
nK(x,ξ)dσξ = and
n|K(x,ξ)|dσξ <∞;
(ii) for every multiindexβ,supξ∈n|DβξK(x,ξ)| ≤C(β)independent ofx.
In the caseαi= , ≤i≤n, Definition . gives rise to the classical Calderón-Zygmund
kernel. On the other hand, when αi= , ≤i≤n– andαn≥, we obtain the kernel
studied by Jones in [] and discussed in [].
Definition . Let ≤p<∞, <κ< andwbe a weight function. Then a weighted Morrey space is defined by
Lp,κ(w) :=f ∈Lloc(w) :fLp,κ(w)<∞,
where
fLp,κ(w)=sup
E
w(E)κ
E
f(x)pw(x)dx
/p
,
the supremum is taken over all ellipsoidEinRn.
Definition . For the functionb∈Lloc(Rn) and any ellipsoidE,bis called aBMO
func-tion if
b∗=sup
E
|E|
E
b(x) –bEdx<∞,
wherebE =|E|Eb(y)dy. The quantityb∗is a norm in theBMOmodulo constant func-tion under whichBMOresults in a Banach space (see []).
Definition . Let <p<∞. For any locally integrable functionwand ellipsoidE, if
|E|
Ew(x)dx
|E|
Ew(x)
–pdx
p–
<∞
holds, thenwbelongs to the Muckenhoupt classAp. We denoteA∞=
<p<∞Ap.
Whenp= ,w∈Aif there existsC> such that
Mw(x)≤Cw(x)
Remark . Given a weight functionw∈Ap, ≤p≤ ∞, it also satisfies the doubling
condition: for any ellipsoidE, there exists a constantC> such thatw(E)≤Cw(E).
In fact,w∈, we have the following inequality.
Lemma A[, ] Suppose w∈,there exists a constant D> such that w(E)≥Dw(E)
for any ellipsoidE.
Lemma B[] Suppose w∈A∞,then the norm of BMO(w)is equivalent to the norm of BMO(Rn),where
BMO(w) =
b:b∗,w=sup
E
w(E)
E
b(x) –bE,ww(x)dx<∞
,
where bE,w=w(E)
Eb(x)w(x)dx.
Lemma C[] Let the ellipsoidE=E(x,r)centered at xwith side length of r.For any positive integer i, iE denotes the ellipsoid centered at x
with side length ofir,we have the inequality
|biE–bE| ≤Cib∗,
where bE,w=|E|
Eb(x)w(x)dx.
Lemma D[] Suppose <p<∞, <κ< and w∈Ap,ifT is the classical Calderón-¯ Zygmund operator with a constant kernel,then the operatorT is bounded on L¯ p,κ(w).
If p= , <κ< and w∈A,then there exists a constant C> such that
λwx∈E:Tf¯ (x)>λ≤CfL,κ(w)w(E)κ for allλ> and any ellipsoidE.
Definition . Let(t) =tlogN(t+e). The Orlicz maximal operatorMis given by
Mf(x) =sup
x∈Ef,E=
sup
x∈E
|E|
E
|f|(x)dx.
From the above definition, observe thatMf(x)≤Mf(x)≤M((|f|))(x). This inequality will be relevant in our work.
Aside from the properties of anApweight function and aBMOfunction, we need some
estimates of multilinear commutators. The following results were proved by Pérez and González [].
Lemma E Let <p<∞and w∈Ap.Suppose bj∈BMO(Rn), ≤j≤N,then there exists a constant C> such that
Rn
[b,T¯](f)(x)pw(x)dx≤Cbp
Rn
Although the commutators with aBMOfunction are not of weak type (, ), we have the following inequality.
Lemma F Let w∈A∞.There exists a constant C> such that
sup
t>
(t)w
x∈Rn:[b,T¯](f)(x)>t
≤Csup
t>
(
t)
wx∈Rn:Mbf(x) >t,
where(t) =tlogN(e+t).
By the above inequality, we have the following result.
Lemma G Let w∈A.There exists a constant C> such that,for allλ> ,
wx∈Rn:[b,T¯](f)(x)>λ≤C
Rn
|f|(x)w(x)dx,
where(t) =tlogN(e+t).
Finally, we need the spherical harmonics and their properties (see more detail in [, , ]). Recall that any homogeneous polynomialP:Rn→Rof degreemthat
sat-isfiesP= is called ann-dimensional solid harmonic of degreem. Its restriction to the unit spherenwill be called ann-dimensional spherical harmonic of degree m. Denote by Hmthe space of alln-dimensional spherical harmonics of degreem. In general, it results
in a finite-dimensional linear space withgm=dimHmsuch thatg= ,g=nand
gm=Cmn–+n––Cnm–+n–≤C(n)mn–, m≥. (.)
Furthermore, let{Ysm}gs=m be an orthonormal basis ofHm. Then{Ysm}gsm∞=m=is a complete
orthonormal system inL(
n) and
sup
x∈n Dβ
xYsm(x)≤C(n)m|β|+(n–)/, m= , , . . . . (.)
If, for instance,φ∈C∞(n), thens,mbsmYsm(x) is the Fourier series expansion ofφ(x)
with respect to{Ysm}s,m(s,msubstitutes∞m=
gm
s=) and
bsm=
n
φ(x)Ysm(x)dσ, |bsm| ≤C(n,l)m–l sup
|β|=l y∈n
Dβyφ(y), (.)
for any integer l. In particular, the expansion ofφ into spherical harmonics converges uniformly toφ. For more detail, we can see [].
3 Proof of the theorems
In this section, we shall use the complete orthonormal system inL(n) and some lemmas
Proof of Theorem. In order to ensure the existence of the operator (.) inLp,κ(w), ≤ p<∞, we restrict our consideration to the functionf ∈Lp,κ(w), for which the norm of Lp(w) is finite. For the sake of convenience, we still denote these spaces byLp,κ(w). Let x,y∈Rnandy¯=y/ρ(y)∈
n. In view of the properties of the kernelKwith respect to the
second variable and the complete of{Ysm(x)}inL(n), we get K(x,x–y) =ρ(x–y)–|α|K(x,x–y)
=ρ(x–y)–|α| s,m
bsm(x)Ysm(x–y).
Replacing the kernel with its series expansion, (.) can be written as
Tf(x) =lim
→Tf(x)
=lim
→
ρ(x–y)>
s,m
bsm(x)ρ(x–y)–|α|Ysm(x–y)f(y)dy.
From the properties of (.)-(.), the series expansion s,m|bsm(x)Ysm(x–y)| ≤ C(n,α)m(n–)/–l, where the integer l is preliminarily chosen greater than (n– )/.
Along with theρ(x–y)–|α|f(y)∈L(Rn) for a.a.x∈Rn, by the Fubini dominated
conver-gence theorem, we have
Tf(x) =
s,m
bsm(x)lim →
ρ(x–y)>
Hsm(x–y)f(y)dy=
s,m
bsm(x)Tsmf(x),
where Hsm(x–y) =ρ(x–y)–|α|Ysm(x–y). Instead of the operatorsTf(x), we shall study
the existence and boundedness inLp,κ(ω) of the operatorsT
smf(x) with a kernelHsm(·).
Observe thatHsm(·) is a constant kernel and satisfies
Hsm(x)≤C(n,α)m
n–
ρ–|α|; ∇Hsm(x)≤C(n,α)mnρ–|α|–.
From Lemma D, it follows
Tsmf(x) Lp,κ(ω)≤C(n,α)m
n
f(x) Lp,κ(ω)
for <p<∞. Consequently, by the above inequality and (.)-(.), we show
Tf(x) Lp,κ(ω)≤C
s,m
bsm(x) L∞ Tsmf(x) Lp,κ(ω)
≤C
s,m
m–l+n f(x) Lp,κ(ω)
≤CfLp,κ(ω),
where the integerlis preliminary chosen greater thatl>n. Forp= , by Lemma D, we have
λwx∈E:Tsmf(x)>λ
for anyλ> and ellipsoidE. Therefore, one gets
λwx∈E:Tf(x)>λ≤C s,m
bsm(x) L∞λw
x∈E:Tsmf(x)>λ
≤C
s,m
m–l+nf L,κ(w)
≤C f(x) L,κ(w),
thus we complete the proof of Theorem ..
Next we begin with the second theorem, for which further discussion is needed.
Proof of Theorem. As above, we use the series expansion of a kernelK(x,y), the operator [b,T]f(x) is divided into
[b,T]f(x) =
s,m
bsm(x)[b,Tsm]f(x).
Instead of the operator [b,T]f(x), we only consider the existence and boundedness in
Lp,κ(w) of the operators [b,T sm]f(x).
Let <p<∞. For any ellipsoidE, we only need to obtain the inequality
E
[b,Tsm]f(x) p
w(x)dx≤Cmmp bpw(E)kfp Lp,κ(w).
In fact, by the series expansion of a kernelK(x,y), we have
[b,T]f(x) Lp,κ(ω)≤C
s,m
bsm(x) L∞ [b,Tsm]f(x) Lp,κ(ω)
≤C
s,m
m–l+nf
Lp,κ(ω)≤CfLp,κ(ω),
where the integerlis chosen greater thanl>n
. Next, fix the above ellipsoidE=E(x,r)
and decomposef=f+f, wheref=fχE,χE denotes the characteristic function of E,
then we have
E
[b,Tsm](f)(x) p
w(x)dx≤C
E
[b,Tsm](f)(x)
p
+[b,Tsm](f)(x)
p w(x)dx
=C{I+II}. (.)
By using Lemma E, we get
I≤
Rn
[b,Tsm](f)(x)
p w(x)dx
≤Cmnp bpw(E)κfp
For the termII, without loss of generality, we can assumeN= . Thus, the operator [b,Tsm]
can be divided into four parts,
[b,Tsm]f(x) =
b(x) –λ
b(x) –λ
RnHsm(x–y)f(y)dy
+
Rn
Hsm(x–y)
b(y) –λ
b(y) –λ
f(y)dy
–b(x) –λ
RnHsm(x–y)
b(y) –λ
f(y)dy
–b(x) –λ
Rn
Hsm(x–y)
b(y) –λ
f(y)dy
=II(x) +II(x) +II(x) +II(x), (.)
whereλi= (bi)E,w=w(E)
Ebi(x)w(x)dx,i= , . For the termII(x), observing thatx∈E
andy∈Rn\E, we haveρ(x
–y)≤Cρ(x–y). Thus, it yields
E
II(x)
p
w(x)dx≤Cmnp
E
b(x) – (b)E,w
b(x) – (b)E,w p
w(x)dx
×
Rn\E
|f(y)|
ρ(x–y)|α| dy
p
≤Cmnp w(E)
w(E)
E
b(x) – (b)E,wpw(x)dx
× w(E) E
b(x) – (b)E,w
p w(x)dx
× ∞ j=
j+E\jE
|f(y)|
ρ(x–y)|α| dy
p
≤Cmnp b
p∗bp∗w(E) ∞
j=
|jE|
j+E
f(y)pw(y)dy
p
×
j+Ew(y) –pp
dx
p p
,
sincew∈Ap, and by the definition of a weighted Morrey space, we get
E
II(x)
p
w(x)dx≤Cmnp bpw(E)
∞
j=
wj+E–p
j+E
f(y)pw(y)dy
pp
≤Cmnp bpw(E)
∞
j=
wj+E
κ–
p f
Lp,κ(w)
p
≤Cmnp bpw(E)
∞
j=
Djκ–p w(E)κ–p f
Lp,κ(w)
p
≤Cmnp bpw(E)κfp
Lp,κ(w). (.)
ForII(x), note thatλi= (bi)E,w=w(E)
Ebi(x)w(x)dx,i= , . By Hölder’s inequality and
ρ(x–y)≤Cρ(x–y), we get
E
II(x)pw(x)dx≤Cmnp w(E)
Rn\E
|(b(y) – (b)E,w)(b(y) – (b)E,w)|
ρ(x–y)|α|
f(y)dy
p
≤Cmnp w(E)
∞
j=
|jE|
j+E\jE
b(y) – (b)E,w
×b(y) – (b)E,wf(y)dy
p
≤Cmnp w(E)
∞
j=
|jE|
j+E
f(y)pw(y)dy
p
×
j+E
b(y) – (b)E,wp
w(y)–p pdy p ×
j+E
b(y) – (b)E,w
p w(y)–p
p dy p p .
Indeed, by Lemma B we knowBMO(Rn) is equivalent toBMO(w),w∈A
∞. LetW=w– p
p ∈
Ap⊂A∞,bi∈BMO(Rn),i= , . For any ellipsoidE, by using Lemma B and Lemma C,
we show
W(j+E)
j+E
bi(y) – (bi)E,w
p W(y)dy
p
≤Cjbi∗.
Thus, sincew∈Ap, it yields
E
II(x)
p
w(x)dx≤Cmnp w(E)
∞
j= j
|jE|b∗b∗W
j+E
p
×
j+E
f(y)pw(y)dy
p
p
≤Cmnp w(E)bpfp Lp,κ(w)
∞
j= j
w(j+E)–κp p
≤Cmnp w(E)κbpfp
Lp,κ(w). (.)
The last inequality is obtained by Lemma A and the D’Alembert judge method of positive series.
ForII(x), by the inequalityρ(x–y)≤Cρ(x–y) sincew∈Ap⊂A∞, by Lemma B, we have
E
II(x)
p w(x)dx
≤Cmnp
E
b(x) –λ
Rn\E
|b(y) –λ|
ρ(x–y)|α| f(y)dy
≤Cmnp
E
b(x) –λpw(x)dx
Rn\E
|b(y) –λ|
ρ(x–y)|α|
f(y)dy
p
≤Cmnpw(E)bp∗
Rn\E
|b(y) –λ|
ρ(x–y)|α|
f(y)dy
p
.
By Hölder’s inequality, Lemma B and Lemma C, we get
Rn\E
|b(y) –λ|
ρ(x–y)|α|
f(y)dy≤C∞
j=
|jE|
j+E
b(y) –λf(y)dy
≤C
∞
j=
|jE|
j+E
f(y)p w(y)dy
p
×
j+E
b(y) –λ
p w(y)–
p p dy
p
≤Cb∗fLp,κ(w)
∞
j= jw(
j+E)κp
|jE|
j+Ew(y) –pp
dy
p
≤Cb∗fLp,κ(w)
∞
j= j
w(j+E)–κp ,
indeed for <κ< , by using Lemma A, we have that ∞
j= j
w(j+E)–κp
≤
∞
j= j
D(j+)–κp
w(E)κ–p ≤Cw(E)κ–p .
Thus, we conclude
E
II(x)
p
w(x)dx≤Cmnp w(E)κbpfp
Lp,κ(w). (.)
In the same way, we shall get the result ofII(x)
E
II(x)
p
w(x)dx≤Cmnpw(E)κbpfp
Lp,κ(w). (.)
Which together with (.)-(.), for <p<∞, the proof of Theorem . is finished. Now, we are in a position to consider the casep= . In general, the singularity of the commutator is stronger than the singular integral, and the endpoint casep= of the com-mutator is not even obtained. Thus, the result for the casep= of the multilinear com-mutator is interesting. We splitf as above byf =f+f, which yields
λwx∈E:[b,T]f(x)>λ≤C s,m
bsmL∞λw
x∈E:[b,Tsm]f(x)>λ
≤C
s,m
m–lλwx∈E:[b,Tsm]f(x)>λ/
+λwx∈E:[b,Tsm]f(x)>λ/
=C s,m
for any ellipsoidE,λ> and integerl> . For the termIII, we use Lemma G. It follows that
III≤C
Rn
|f|
(x)w(x)dx
≤Cw(E)κfL,κ(w). (.)
For the last termIV, without loss of generality, we still supposeN= . By homogeneity, it is enough to assumeλ/ =b∗=b∗= , and hence we only need to prove
wx∈E:[b,Tsm]f(x)>
≤Cw(E)κfL,κ(w).
In fact, by Lemma F, we get
wx∈E:[b,Tsm]f(x)>
≤sup
t>
(t)w
x∈E:[b,Tsm]f(x)>t
≤Csup
t>
(t)w
x∈E:Mf(x) >t
=Csup
t>
(
t)
wx∈E:M|f|
(x) >t, (.)
where(t) =tlogN(e+t). We use the Fefferman-Stein maximal inequality
x:Mf(x)>t
φ(t)dx≤C t
Rn
f(x)Mφ(x)dx,
for any functionsf andφ≥. This yields
wx∈E:M|f|
(x) >t≤ t
{x∈Rn:M(|f
|)(x)>t}
χE(x)w(x)dx
≤C
t
Rn
|f|
(x)M(wχE)(x)dx
=C
t
E
+
Rn\E
|f|
(x)M(wχE)(x)dx
=C
t(IV+IV). (.)
ForIV, sincew∈A, it follows that
IV≤C
E
|f|(x)w(x)dx ≤Cw(E)κ |f|
L,κ(w)
≤Cw(E)κfL,κ. (.)
To estimate the termIV, we first consider the form
|F|
for anyx∈Rn\E,x∈FandF∩E=∅. By simple geometric observation, we have
|F|
E∩Fw(y)dy≤C
ρ(x–x)|α|
Ew(y)dy
= C
ρ(x–x)|α| w(E).
Therefore, we obtain
M(wχE)(x)≤ C
ρ(x–x)|α| w(E).
Sincew∈Asatisfies the doubling condition and Lemma A, we estimate the termIVas
follows:
IV≤C
Rn\E
(|f|)(x)
ρ(x–x)|α| w(E)dx
≤Cw(E)
∞
j=
|jE|
j+E
|f|(x)dx
≤Cw(E) |f| L,κ(w)
∞
j=
w(jE)–κ
≤Cw(E)κfL,κ. (.)
The last inequality is similar to (.). Noting thatt(t) > , from (.)-(.), we conclude
wx∈E:[b,Tsm]f(x)>
≤Cw(E)κfL,κ(w).
Thus, the proof of Theorem . is completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
XFY conceived of the study and drafted the manuscript. XSZ participated in the discussion. All authors read and approved the final manuscript.
Acknowledgements
XFY research was partially supported by the National Natural Sciences Foundation of China (10961015; 11161021). XSZ research was supported by the National Natural Sciences Foundation of China (11161021).
Received: 24 August 2012 Accepted: 5 November 2012 Published: 17 December 2012 References
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