R E S E A R C H
Open Access
Inequalities and boundedness for
commutators related to integral operator
with general kernel
Jiasheng Zeng
**Correspondence: [email protected] College of Mathematics, Hunan University of Commerce, Changsha, 410205, P.R. China
Abstract
In this paper, we establish the sharp maximal function inequalities for the
commutators related to some integral operator with general kernel and theBMOand Lipschitz functions. As an application, we obtain the boundedness of the
commutators on Lebesgue, Morrey and Triebel-Lizorkin space. The operator includes Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.
MSC: 42B20; 42B25
Keywords: commutator; Littlewood-Paley operator; Marcinkiewicz operator; Bochner-Riesz operator; sharp maximal function; Morrey space; Triebel-Lizorkin space;
BMO; Lipschitz function
1 Introduction and preliminaries
As the development of singular integral operators (see [–]), their commutators have been well studied (see [–]). In [–], the authors prove that the commutators generated by the singular integral operators andBMOfunctions are bounded onLp(Rn) for <p<∞. Chanillo (see []) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [–], the boundedness for the commutators gener-ated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and Lp(Rn) ( <p<∞) spaces are obtained. In [], some singular integral operators with gen-eral kernel are introduced, and the boundedness for the operators and their commutators generated byBMOand Lipschitz functions are obtained (see [, ]). The purpose of this paper is to prove the sharp maximal function inequalities for the commutator associated
with some integral operator with general kernel and theBMOand Lipschitz functions. As
an application, we obtain the boundedness of the commutator on Lebesgue, Morrey and Triebel-Lizorkin space. The operator includes Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.
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 maximal function off is defined by
M#(f)(x) =sup
Qx |Q|
Q
f(y) –fQdy;
here, and in what follows,fQ=|Q|–
Qf(x)dx. It is well known that (see [, ])
M#(f)(x)≈sup
Qx
inf
c∈C |Q|
Q
f(y) –cdy.
We say that f belongs to BMO(Rn) if M#(f) belongs to L∞(Rn) and define fBMO =
M#(f)L∞. It has been known that (see [])
f–fkQBMO≤CkfBMO.
Let
M(f)(x) =sup
Qx |Q|
Q
f(y)dy.
Forη> , letMη(f)(x) =M(|f|η)/η(x).
For <η< and ≤r<∞, set
Mη,r(f)(x) =sup Qx
|Q|–rη/n
Q
f(y)rdy /r
.
TheApweight is defined by (see [])
Ap=
w∈LlocRn :sup
Q
|Q|
Q w(x)dx
|Q|
Q
w(x)–/(p–)dx p–
<∞
,
<p<∞,
and
A=w∈LplocRn :M(w)(x)≤Cw(x), a.e..
Forβ> andp> , letF˙pβ,∞(Rn) be the weighted homogeneous Triebel-Lizorkin space (see []).
Forβ> , the Lipschitz spaceLipβ(Rn) is the space of functionsf such that
fLipβ= sup
x,y∈Rn x=y
|f(x) –f(y)| |x–y|β <∞.
In this paper, we will study some integral operators as follows (see []).
Definition LetFt(x,y) be defined onRn×Rn×[, +∞) andbbe a locally integrable function onRn, set
Ft(f)(x) =
RnFt(x,y)f(y)dy
and
Ftb(f)(x) =
Rn
b(x) –b(y) Ft(x,y)f(y)dy
LetHbe the Banach spaceH={h:h<∞}. For each fixedx∈Rn, we viewF
t(f)(x) and Fb
t(f)(x) as the mappings from [, +∞) toH. Set
T(f)(x) =Ft(f)(x),
whichTis bounded onL(Rn). The commutator related toFb
t is defined by
Tb(f)(x) =Ftb(f)(x)
and forFtwe find that there is a sequence of positive constant numbers{Ck}such that for anyk≥,
|y–z|<|x–y|
Ft(x,y) –Ft(x,z)+Ft(y,x) –Ft(z,x) dx≤C
and
k|z–y|≤|x–y|<k+|z–y|
Ft(x,y) –Ft(x,z)+Ft(y,x) –Ft(z,x) qdy /q
≤Ck
k|z–y| –n/q,
where <q< and /q+ /q= .
Definition Letϕ be a positive, increasing function onR+ and there exists a constant D> such that
ϕ(t)≤Dϕ(t) fort≥.
Letf be a locally integrable function onRn. Set, for ≤η<nand ≤p<n/η,
fLp,η,ϕ= sup
x∈Rn,d>
ϕ(d)–pη/n
Q(x,d)
f(y)pdy /p
,
whereQ(x,d) ={y∈Rn:|x–y|<d}. The generalized fractional Morrey space is defined by
Lp,η,ϕRn =f∈LlocRn :fLp,η,ϕ<∞.
We writeLp,η,ϕ(Rn) =Lp,ϕ(Rn) ifη= , which is the generalized Morrey space. Ifϕ(d) =dδ,
δ> , thenLp,ϕ(Rn) =Lp,δ(Rn), which is the classical Morrey spaces (see [, ]). Ifϕ(d) = , thenLp,ϕ(Rn,w) =Lp(Rn), which is the Lebesgue spaces.
It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [, ]). In [], Pérez and Trujillo-Gonzalez prove a sharp estimate for the multilinear commutator. The main purpose of this paper is to prove the sharp maximal inequalities for the commutator. As the application, we obtain
theLp-norm inequality, Morrey and Triebel-Lizorkin spaces boundedness for the
com-mutator.
2 Theorems
We shall prove the following theorems.
Theorem Let T be the integral operator as Definition,the sequence{Ck} ∈l, <β< , q≤s<∞and b∈Lipβ(Rn).Then there exists a constant C> such that,for any f ∈ C∞(Rn)andx˜∈Rn,
M#Tb(f) (x)˜ ≤CbLipβ
Mβ,s(f)(x) +˜ Mβ,s
T(f) (x)˜ .
Theorem Let T be the integral operator as Definition,the sequence{kβCk} ∈l, <
β< ,q≤s<∞and b∈Lipβ(Rn).Then there exists a constant C> such that,for any f ∈C∞(Rn)andx˜∈Rn,
sup
Q˜x
inf
c∈Rn |Q|+β/n
Q
Tb(f)(x) –cdx≤CbLipβ
Ms(f)(x) +˜ Ms
T(f) (x)˜ .
Theorem Let T be the integral operator as Definition,the sequence{kCk} ∈l,q≤s< ∞and b∈BMO(Rn).Then there exists a constant C> such that,for any f∈C∞
(Rn)and ˜
x∈Rn,
M#Tb(f) (x)˜ ≤CbBMOMs(f)(x) +˜ Ms
T(f) (x)˜ .
Theorem Let T be the integral operator as Definition,the sequence{Ck} ∈l, <β<
min(,n/q),q<p<n/β, /r= /p–β/n and b∈Lipβ(Rn).Then Tbis bounded from Lp(Rn) to Lr(Rn).
Theorem Let T be the integral operator as Definition,the sequence{Ck} ∈l, <D< n, <β<min(,n/q),q<p<n/β, /r= /p–β/n and b∈Lip
β(Rn).Then Tbis bounded
from Lp,β,ϕ(Rn)to Lr,ϕ(Rn).
Theorem Let T be the integral operator as Definition,the sequence{kβCk} ∈l, <
β<min(,n/q),q<p<n/β, /r= /p–β/n and b∈Lipβ(Rn).Then Tbis bounded from Lp(Rn)toF˙β,∞
r (Rn).
Theorem Let T be the integral operator as Definition,the sequence{kCk} ∈l and b∈BMO(Rn).Then Tbis bounded on Lp(Rn)for q≤p<∞.
3 Proofs of theorems
Lemma (see []) Let T be the integral operator as Definition,the sequence{Ck} ∈l. Then T is bounded on Lp(Rn)for <p<∞.
Lemma (see []) For <β< and <p<∞,we have
fF˙β,∞
p ≈
sup
Q·
|Q|+β/n
Q
f(x) –fQdx Lp
≈sup
Q·
inf
c |Q|+β/n
Q
f(x) –cdx Lp
.
Lemma (see []) Let <p<∞and w∈≤r<∞Ar.Then,for any smooth function f for which the left-hand side is finite,
RnM(f)(x)
pw(x)dx≤C
RnM
#
(f)(x)pw(x)dx.
Lemma (see []) Suppose that <η<n, <s<p<n/ηand/q= /p–η/n.Then
Mη,s(f)Lq≤CfLp.
Lemma Let <p<∞, <D< n.Then,for any smooth function f for which the left-hand side is finite,
M(f)Lp,ϕ≤CM #
(f)Lp,ϕ.
Proof For any cubeQ=Q(x,d) inRn, we knowM(χQ)∈Afor any cubeQ=Q(x,d) by []. Noticing thatM(χQ)≤ andM(χQ)(x)≤dn/(|x–x|–d)nifx∈Qc, by Lemma , we have, forf ∈Lp,ϕ(Rn),
Q
M(f)(x)pdx=
Rn
M(f)(x)pχQ(x)dx
≤
RnM(f)(x) pM(χ
Q)(x)dx
≤C
RnM
#(f)(x)pM(χ Q)(x)dx
=C
Q
M#(f)(x)pM(χQ)(x)dx
+
∞
k=
k+Q\kQ
M#(f)(x)pM(χQ)(x)dx
≤C
Q
M#(f)(x)pdx+
∞
k=
k+Q\kQM
#
(f)(x)p |Q| |k+Q|dx
≤C
Q
M#(f)(x)pdx+
∞
k=
k+QM #
(f)(x)p–kndy
≤CM#(f)pLp,ϕ ∞
k=
≤CM#(f)pLp,ϕ ∞
k=
–nD kϕ(d)
≤CM#(f)pLp,ϕϕ(d),
thus
ϕ(d)
Q
M(f)(x)pdx /p
≤C
ϕ(d)
Q
M#(f)(x)pdx /p
and
M(f)Lp,ϕ≤CM#(f)Lp,ϕ.
This finishes the proof.
Lemma Let T be the integral operator as Definition , ≤η<n, < D< n and ≤p<∞.Then
T(f)
Lp,η,ϕ≤CfLp,η,ϕ.
Lemma Let <D< n, <η<n, ≤s<p<n/ηand/q= /p–η/n.Then
Mη,s(f)Lq,ϕ≤CfLp,η,ϕ.
The proofs of the two lemmas are similar to that of Lemma by Lemmas and , we omit the details.
Proof of Theorem It suffices to prove forf ∈C∞(Rn) and some constantC, the following inequality holds:
|Q|
Q
Tb(f)(x) –Cdx≤CbLipβ
Mβ,s(f)(x) +˜ Mβ,s
T(f) (x)˜ .
Fix a cubeQ=Q(x,d) andx˜∈Q. Write, forf=fχQandf=fχ(Q)c,
Ftb(f)(x) =b(x) –bQ Ft(f)(x) –Ft
(b–bQ)f (x) –Ft
(b–bQ)f (x).
Then
|Q|
Q
Ftb(f)(x) –Ft
(bQ–b)f (x)dx
≤| Q|
Q
b(x) –bQ Ft(f)(x)dx+ |Q|
Q Ft
(b–bQ)f (x)dx
+
|Q|
Q Ft
(b–bQ)f (x) –Ft
(b–bQ)f (x)dx
ForI, by Hölder’s inequality and Lemma , we obtain
I≤ C |Q|xsup∈Q
b(x) –bQ|Q|–/s
Q
T(f)(x)sdx /s
≤C|Q|–/sbLipβ|Q|β/n|Q|/s–β/n
|Q|–sβ/n
Q
T(f)(x)sdx /s
≤CbLipβMβ,s
T(f) (x).˜
ForI, by the boundedness ofT, we get
I≤
|Q|
Rn
T(b–bQ)f (x) s dx /s ≤C |Q|
Rn
b(x) –bQ f(x)sdx /s
≤C|Q|–/s|Q|β/n||Q|/s–β/n
|Q|–sβ/n
Q
f(x)sdx /s
≤CbLipβMβ,s(f)(x).˜
ForI, recalling thats>q, we have
I≤
|Q|
Q
(Q)c
b(y) –bQf(y)Ft(x,y) –Ft(x,y)dy dx
≤
|Q| Q ∞ k=
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y)b(y) –bk+Qf(y)dy dx
+
|Q| Q ∞ k=
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y)bk+Q–bQf(y)dy dx
≤ C
|Q| Q ∞ k=
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y)qdy /q
× sup
y∈k+Q
b(y) –bk+Q
k+Q
f(y)qdy /q
dx
+ C
|Q|
Q
∞
k=
|bk+Q–bQ|
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y)qdy /q
×
k+Q
f(y)qdy /q dx ≤C ∞ k= Ck
kd –n/qk+Qβ/nbLipβk+Q/q
–/s
k+Q/s–β/n
×
|k+Q|–sβ/n
k+Q
f(y)sdy /s
+C
∞
k=
bLipβkQ
β/n Ck
×
|k+Q|–sβ/n
k+Q
f(y)sdy /s
≤CbLipβMβ,s(f)(x)˜
∞
k= Ck
≤CbLipβMβ,s(f)(x).˜
This completes the proof of Theorem .
Proof of Theorem It suffices to prove forf ∈C∞ (Rn) and some constantC, the following inequality holds:
|Q|+β/n
Q
Tb(f)(x) –Cdx≤CbLipβ
Ms(f)(x) +˜ Ms
T(f) (x)˜ .
Fix a cubeQ=Q(x,d) andx˜∈Q. Write, forf=fχQandf=fχ(Q)c,
Ftb(f)(x) =b(x) –bQ Ft(f)(x) –Ft
(b–bQ)f (x) –Ft
(b–bQ)f (x).
Then
|Q|+β/n
Q
Ftb(f)(x) –Ft
(bQ–b)f (x)dx
≤
|Q|+β/n
Q
b(x) –bQ Ft(f)(x)dx+ |Q|+β/n
Q Ft
(b–bQ)f (x)dx
+
|Q|+β/n
Q Ft
(b–bQ)f (x) –Ft
(b–bQ)f (x)dx
=II+II+II.
By using the same argument as in the proof of Theorem , we get
II≤ C
|Q|+β/n sup x∈Q
b(x) –bQ|Q|–/s
Q
T(f)(x)sdx /s
≤CbLipβ|Q|β/n|Q|–/s|Q|/s–β/n
|Q|
Q
T(f)(x)sdx /s
≤CbLipβMs
T(f) (x),˜
II≤
|Q|+β/n|Q| –/s
Rn
T(b–bQ)f (x)sdx /s
≤ | C
Q|+β/n|Q| –/s
Rn
b(x) –bQ f(x) s
dx /s
≤ | C
Q|+β/n|Q|
–/s|Q|β/n|Q|/s
|Q|
Q
f(x)sdx /s
≤CbLipβMs(f)(x),˜
II≤ |Q|+β/n
Q
(Q)c
≤ |Q|+β/n
Q ∞ k=
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y)b(y) –bk+Qf(y)dy dx
+
|Q|+β/n Q ∞ k=
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y)bk+Q–bQf(y)dy dx
≤ C
|Q|+β/n Q ∞ k=
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y) q
dy /q
× sup
y∈k+Q
b(y) –bk+Q
k+Q
f(y)qdy /q
dx
+ C
|Q|+β/n
Q
∞
k=
|bk+Q–bQ|
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y) q
dy /q
×
k+Q
f(y)qdy /q
dx
≤C|Q|–β/n
∞
k= Ck
kd –n/qk+Qβ/nbLipβk+Q
/q
×
|k+Q|
k+Q
f(y)sdy /s
+C|Q|–β/n
∞
k=
bLipβkQ
β/n Ck
kd –n/qk+Q/q
×
|k+Q|
k+Q
f(x)sdx /s
≤CbLipβMs(f)(x)˜
∞
k= kβCk
≤CbLipβMs(f)(x).˜
This completes the proof of Theorem .
Proof of Theorem It suffices to prove forf ∈C∞ (Rn) and some constantC, the following inequality holds:
|Q|
Q
Tb(f)(x) –Cdx≤CbBMOMs(f)(x) +˜ Ms
T(f) (x)˜ .
Fix a cubeQ=Q(x,d) andx˜∈Q. Write, forf=fχQandf=fχ(Q)c,
Ftb(f)(x) =b(x) –bQ Ft(f)(x) –Ft
(b–bQ)f (x) –Ft
(b–bQ)f (x).
Then
|Q|
Q
Ftb(f)(x) –Ft
(bQ–b)f (x)dx
≤| Q|
Q
b(x) –bQ Ft(f)(x)dx+ |Q|
Q Ft
+ |Q|
Q Ft
(b–bQ)f (x) –Ft
(b–bQ)f (x)dx
=III+III+III.
ForIII, by Hölder’s inequality, we get
III≤
|Q|
Q
b(x) –bQ s
dx /s
|Q|
Q
T(f)(x)sdx /s
≤CbBMOMs
T(f) (x).˜
ForIII, choose <r<s, by Hölder’s inequality and the boundedness ofT, we obtain
III≤
|Q|
Rn
T(b–bQ)f (x) r
dx /r
≤C|Q|–/r
Rn
(b–bQ)f(x)rdx /r
≤C|Q|–/r
Q
b(x) –bQsr/(s–r)dx
(s–r)/sr
Q
f(x)sdx /s
≤CbBMO
|Q|
Q
f(x)sdx /s
≤CbBMOMs(f)(x).˜
ForIII, recalling thats>q, taking <p<∞, <r<swith /p+ /q+ /r= , by Hölder’s inequality, we obtain
III≤ |Q|
Q
(Q)c
b(y) –bQf(y)Ft(x,y) –Ft(x,y)dy dx
≤ | Q| Q ∞ k=
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y)b(y) –bQf(y)dy dx
≤ | Q| Q ∞ k=
kd≤|y–x|<k+d
Ft(x,y) –Ft(x,y) q
dy /q
×
k+Q
b(x) –bQ p
dx
/p
k+Q
f(y)rdy /r dx ≤C ∞ k=
kkd n/q
kd≤|y–x|<k+d
K(x,y) –K(x,y)qdy /q
× bBMO
|k+Q|
k+Q
f(y)sdy /s
≤CbBMOMs(f)(x)˜
∞
k= kCk
≤CbBMOMs(f)(x).˜
Proof of Theorem Chooseq<s<pin Theorem , we have, by Lemmas , , and ,
Tb(f)
Lr ≤M
Tb(f) Lr≤CM#
Tb(f) Lr
≤CbLipβMβ,s
T(f) Lr+Mβ,s(f)Lr
≤CbLipβT(f)Lp+fLp ≤CbLipβfLp.
This completes the proof of Theorem .
Proof of Theorem Chooseq<s<pin Theorem , we have, by Lemmas -,
Tb(f)Lr,ϕ≤M
Tb(f) Lr,ϕ≤CM #
Tb(f) Lr,ϕ
≤CbLipβMβ,s
T(f) Lr,ϕ+Mβ,s(f)Lr,ϕ
≤CbLipβT(f)Lp,β,ϕ+fLp,β,ϕ ≤CbLipβfLp,ϕ.
This completes the proof of Theorem .
Proof Theorem Chooseq<s<pin Theorem . By using Lemma , we obtain
Tb(f)F˙β,∞
r ≤C
sup
Q·
|Q|+β/n
Q
Tb(f)(x) –T(bQ–b)f (x)dx Lr
≤CbLipβMs
T(f) Lr+Ms(f)Lr
≤CbLipβT(f)Lp+fLp ≤CbLipβfLp.
This completes the proof of the theorem.
Proof of Theorem Chooseq≤s<pin Theorem , we have
Tb(f)Lp≤M
Tb(f) Lp≤CM
#
Tb(f) Lp
≤CbBMOMs
T(f) Lp+Ms(f)Lp
≤CbBMOT(f)Lp+fLp ≤CbBMOfLp.
This completes the proof of the theorem.
4 Applications
In this section we shall apply Theorems - of the paper to some particular operators such as the Littlewood-Paley operators, Marcinkiewicz operator and Bochner-Riesz operator.
Application Littlewood-Paley operators.
() Rnψ(x)dx= ,
() |ψ(x)| ≤C( +|x|)–(n+),
() |ψ(x+y) –ψ(x)| ≤C|y|ε( +|x|)–(n++ε)when|y|<|x|. We denote (x) ={(y,t)∈Rn+
+ :|x–y|<t}and the characteristic function of (x) by
χ (x). The Littlewood-Paley commutators are defined by
gψb(f)(x) =
∞
Fb
t(f)(x) dt
t /
,
Sbψ(f)(x) =
(x)
Ftb(f)(x,y)dy dt tn+
/ ,
and
gμb(f)(x) =
Rn++
t t+|x–y|
nμ
Ftb(f)(x,y)dy dt tn+
/ ,
where
Ftb(f)(x) =
Rn
b(x) –b(y) ψt(x–y)f(y)dy,
Ftb(f)(x,y) =
Rn
b(x) –b(z) f(z)ψt(y–z)dz
andψt(x) =t–nψ(x/t) fort> . SetFt(f)(y) =f∗ψt(y). We also define
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,Fb
t(f)(x) andFtb(f)(x,y) may be viewed as the mapping from [, +∞) toH, and it is clear that
Sbψ(f)(x) =χ (x)Ftb(f)(x,y), Sψ(f)(x) =χ (x)Ft(f)(y)
and
gμb(f)(x) =
t+|xt–y|nμ/Ftb(f)(x,y), gμ(f)(x) =
t+|xt–y|nμ/Ft(f)(y) .
It is easily to see thatgψb,Sbψ, andgμbsatisfy the conditions of Theorems - (see [–]),
thus Theorems - hold forgb
ψ,Sbψ, andgμb.
Application Marcinkiewicz operators.
Fixedλ>max(, n/(n+ )) and <γ ≤. Letbe homogeneous of degree zero onRn withSn–(x)dσ(x) = . Assume that∈Lipγ(Sn–). The Marcinkiewicz commutators
are defined by
μb(f)(x) =
∞
Ftb(f)(x)dt t
/ ,
μbS(f)(x) =
(x)
Ftb(f)(x,y)dy dt tn+
/ ,
and
μbλ(f)(x) =
Rn++
t t+|x–y|
nλ
Ftb(f)(x,y)dy dt tn+
/ ,
where
Ftb(f)(x) =
|x–y|≤t
b(x) –b(y) (x–y) |x–y|n–f(y)dy
and
Ftb(f)(x,y) =
|y–z|≤t
b(x) –b(z) (y–z) |y–z|n–f(z)dz.
Set
Ft(f)(x) =
|x–y|≤t
(x–y) |x–y|n–f(y)dy.
We also define
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
μb(f)(x) =Fb
t(f)(x), μ(f)(x) =Ft(f)(x),
μbS(f)(x) =χ (x)Ftb(f)(x,y), μS(f)(x) =χ (x)Ft(f)(y),
and
μbλ(f)(x) =
t t+|x–y|
nλ/
Ftb(f)(x,y), μλ(f)(x) =
t+|xt–y|nλ/Ft(f)(y) .
It is easy to see thatμb,μbS, andμbλsatisfy the conditions of Theorems - (see [–]), thus Theorems - hold forμb,μbS, andμbλ.
Application Bochner-Riesz operator.
Letδ> (n– )/,Bδ
t(fˆ)(ξ) = ( –t|ξ|)δ+ˆf(ξ) andBδt(z) =t–nBδ(z/t) fort> . Set
Fδb,t(f)(x) =
Rn
b(x) –b(y) Bδt(x–y)f(y)dy.
The maximal Bochner-Riesz commutator is defined by
Bbδ,∗(f)(x) =sup
t>
Bbδ,t(f)(x).
We also define
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
Bbδ,∗(f)(x) =Bbδ,t(f)(x), Bδ
∗(f)(x) =Bδt(f)(x).
It is easy to see thatBb
δ,∗ satisfies the conditions of Theorems - (see []), thus
Theo-rems - hold forBb
δ,∗.
Competing interests
The author declares that they have no competing interests.
References
1. Garcia-Cuerva, J, Rubio de Francia, JL: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 16. North-Holland, Amsterdam (1985)
2. Stein, EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993)
3. Torchinsky, A: Real Variable Methods in Harmonic Analysis. Pure and Applied Math., vol. 123. Academic Press, New York (1986)
4. Coifman, RR, Rochberg, R, Weiss, G: Factorization theorems for Hardy spaces in several variables. Ann. Math.103, 611-635 (1976)
5. Pérez, C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal.128, 163-185 (1995) 6. Pérez, C, Trujillo-Gonzalez, R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc.65, 672-692
(2002)
7. Di Fazio, G, Ragusa, MA: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal.112, 241-256 (1993)
8. Chanillo, S: A note on commutators. Indiana Univ. Math. J.31, 7-16 (1982)
9. Chen, WG: Besov estimates for a class of multilinear singular integrals. Acta Math. Sin.16, 613-626 (2000) 10. Janson, S: Mean oscillation and commutators of singular integral operators. Ark. Math.16, 263-270 (1978) 11. Paluszynski, M: Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss.
Indiana Univ. Math. J.44, 1-17 (1995)
12. Chang, DC, Li, JF, Xiao, J: Weighted scale estimates for Calderón-Zygmund type operators. Contemp. Math.446, 61-70 (2007)
13. Lin, Y: Sharp maximal function estimates for Calderón-Zygmund type operators and commutators. Acta Math. Sci. Ser. A Chin. Ed.31, 206-215 (2011)
14. Peetre, J: On convolution operators leavingLp,λ-spaces invariant. Ann. Mat. Pura Appl.72, 295-304 (1966) 15. Peetre, J: On the theory ofLp,λ-spaces. J. Funct. Anal.4, 71-87 (1969)
16. Chiarenza, F, Frasca, M: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat.7, 273-279 (1987) 17. Di Fazio, G, Ragusa, MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital., A (7)5, 323-332 (1991) 18. Liu, LZ: Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for
commutators of singular integral operators. Acta Math. Sci.25(1), 89-94 (2005)
19. Mizuhara, T: Boundedness of some classical operators on generalized Morrey spaces. In: Harmonic Analysis. ICM-90 Satellite Conference Proceedings (Sendai, 1990), pp. 183-189 (1991)
20. Coifman, RR, Rochberg, R: Another characterization of BMO. Proc. Am. Math. Soc.79, 249-254 (1980) 21. Liu, LZ: The continuity of commutators on Triebel-Lizorkin spaces. Integral Equ. Oper. Theory49, 65-76 (2004) 22. Liu, LZ: Triebel-Lizorkin space estimates for multilinear operators of sublinear operators. Proc. Indian Acad. Sci. Math.
Sci.113, 379-393 (2003)
23. Liu, LZ: Boundedness for multilinear Littlewood-Paley operators on Triebel-Lizorkin spaces. Methods Appl. Anal.10(4), 603-614 (2004)
24. Torchinsky, A, Wang, S: A note on the Marcinkiewicz integral. Colloq. Math.60(61), 235-243 (1990) 25. Lu, SZ: Four Lectures on RealHpSpaces. World Scientific, River Edge (1995)
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