R E S E A R C H
Open Access
Toeplitz-type operators in weighted Morrey
spaces
Peizhu Xie
1,2*and Guangfu Cao
1,2*Correspondence:
1School of Mathematics and
Information Science, Guangzhou University, Guangzhou, 510006, China
2Key Laboratory of Mathematics
and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, China
Abstract
LetTj,1andTj,2be singular integrals with non-smooth kernels, which are associated
with an approximation of identity or±I(Iis the identity operator). Denote the Toeplitz-type operator byTb=
N
j=1Tj,1MbTj,2, whereMbf(x) =b(x)f(x). In this paper, the
estimates of the Toeplitz operatorTb(f) related to singular integral operators with
non-smooth kernels andb∈BMO(Rn) in weighted Morrey spaces is established. MSC: 47B35
Keywords: Toeplitz operator; Morrey space;Apweights
1 Introduction
The classical Morrey spaces were introduced by Morrey in [] to investigate the local be-havior of solutions to second-order elliptic partial differential equations. The bounded-ness of the Hardy-Littlewood maximal operator, the singular integral operator, the frac-tional integral operator and the commutator of these operators in Morrey spaces have been studied by many authors; see [–] and the references therein. In [], Komori and Shirai studied the boundedness of these operators in weighted spaces.
It is well known that the commutator [b,T] is defined by [b,T](f) =T(bf) –bT(f), where
T is a Calderón-Zygmund operator andb∈BMO. The commutator generated by the Calderón-Zygmund operators and a locally integrable functionbcan be regarded as a spe-cial case of the Toeplitz operatorTb=
N
j=Tj,MbTj,, whereTj,andTj,are the
Calderón-Zygmund operators or±I(Iis the identity operator),Mbf(x) =b(x)f(x). Whenb∈BMO, Krantz and Li discussed theLp boundedness ofT
b on the homogeneous space, see [, ]. In [], the authors studied the boundedness ofTbin Morrey spaces. In this paper, we study the boundedness of Toeplitz-type operators related to singular integral operators with non-smooth kernels in weighted Morrey spaces.
The singular integral operators with non-smooth kernels previously appeared in []. We say thatT is a singular integral operator with non-smooth kernel if it satisfies the following conditions.
(i) There exists a class of operatorsAtwith kernelsat(x,y), which satisfy the condition
(.) in Section , so that the kernelskt(x,y)of the operators(T–AtT)satisfy the
condition
kt(x,y)≤c
tγ/m
|x–y|n+γ, (.)
when|x–y| ≥ct/mfor someγ,m> .
(ii) There exists a class of operatorsBtwith kernelsbt(x,y), which satisfy the condition
(.), such that(T–TBt)have associated kernelsKt(x,y)and there exist positive
constantsc,csuch that
|x–y|≥ct/m
Kt(x,y)dx≤c for ally∈Rn. (.)
Note that the classes of operatorsAtandBtplay the role of a generalized approximation to the identity. It is not difficult to check that conditions (.) and (.) are consequences of the standard Calderón-Zygmund operator. See Proposition in [].
The paper is organized as follows. In Section , we recall some important estimates on BMO functions, maximal functions and sharp maximal functions. In Section , we prove the main result.
2 Definitions and preliminary results
Let ≤p<∞, <κ< andwbe a weight. The weighted Morrey space is defined by
Lp,κ(w) :=f ∈Lploc(w) :fLp,κ(w)<∞,
where
fLp,κ(w)=sup B
w(B)κ
B
|f|pw dx /p
,
and the supremum is taken over all ballsBinRn. Ifw= andκ=λ/nwith <λ<n, then
Lp,κ(w) =Lp,λ(Rn), the classical Morrey spaces.
The standard Hardy-Littlewood maximal functionMrf, ≤r<∞, is defined by
Mrf(x) = sup B:x∈B
|B|
B
f(y)rdy /r
,
where thesupis taken over all balls containingx. Ifr= ,Mf will be denoted byMf. The Fefferman-Stein sharp maximal function off,f(x), is defined by
f(x) = sup
B:x∈B
|B|
B
f(y) –fBdy,
wherefB=|B| Bf dx. We will sayf ∈BMO(Rn) iff ∈Lloc(Rn) andf(x)∈L∞. Iff ∈BMO,
theBMOsemi-norm off is given by
f∗=sup
x
f(x) =sup
x
sup
x∈B
|B|
B
f(y) –fBdy.
A weightwis a non-negative locally integrable function. We say thatw∈Ap(Rn), <p<
∞, if there exists a constantCsuch that for every ballB⊂Rn,
|B|
B
w dx
|B|
B
w–p dx p–
where p+p = . Forp= , we say thatw∈A(Rn) if there is a constantCsuch that for every ballB⊂Rn,
|B|
B
w dy≤Cw(x) for a.e.x∈B,
or, equivalently,M(w)≤Cwa.e. We denoteA∞(Rn) =≤p<∞Ap(Rn). For the above def-inition, see [].
A family of operatorsAt,t> , is said to be a ‘generalized approximation to the identity’ if, for everyt> ,Atcan be represented by kernelsat(x,y) in the following sense: For every functionf ∈Lp(Rn),p≥,A
tf(x) = Rnat(x,y)f(y)dy, and the following condition holds:
at(x,y)≤ht(x,y) =t–n/ms|x–y|mt–, (.)
in whichmis a positive constant andsis a positive, bounded, decreasing function satisfy-ing
lim
r→∞r
n+N+srm= (.)
for some> .
Note that (.) implies that
at(x,y)≤t–n/m×
+|x–y|
t/m
–(n+)
.
In [], the sharp maximal functionMAf associated with a ‘generalized approximation to the identity’{At,t> }is defined by
MAf(x) =sup
x∈B
|B|
B
f(y) –AtBf(y)dy, (.)
wheretB=rBm, andf ∈Lp(Rn) for somep≥.
The following results are proved in the context of spaces of homogeneous type in [, ] and [].
Lemma .
(i) For everyp∈[,∞),there exists a constantCsuch that for everyf∈Lp(Rn),
Atf(x)≤CMf(x);
(ii) Assume thatb∈BMOandM> .Then,for every ballB(x;r),we have
|bB–bMB| ≤Cb∗logM;
(iii) (John-Nirenberg lemma)Let≤p<∞andB⊂Rn,thenb∈BMOif and only if
|B|
B
Lemma . For <p<∞, <κ< and w∈Ap,we haveMfLp,κ(w)≤CfLp,κ(w).
For the proof of this lemma, see [, Theorem .].
Lemma . Let{At,t> } be a ‘generalized approximation to the identity’ and let b∈
BMO.Then,for every function f∈Lp(Rn),p> ,x∈Rnand <r<∞,we have
sup
x∈B
|B|
B
AtB(b–bB)f(y)dy≤Cb∗Mrf(x), (.)
where tB=rmB.
For the proof of this lemma, see Lemma . in [].
Now, we have the following analogy of the classical Fefferman-Stein inequality [, Chap-ter IV] for the sharp maximal functionMAf. For the proof, see Proposition . in [].
Lemma . Takeλ> ,w∈A∞(Rn),f ∈Llocand a ball Bsuch that there exists x∈B
with Mf(x) <λ.Then,for every <η< ,there exist r,γ > (independent ofλ,B,f,x)
and Cwwhich only depend on w such that
wx∈B:Mf(x) >Dλ,MAf(x)≤γ λ≤Cwηrw(B),
where D> is a fixed constant which depends only on the ‘generalized approximation to the identity’{At,t> }.
3 The main results
In this section, we consider the Toeplitz operator related to a singular integral with non-smooth kernelTb=
M
j=Tj,MbTj,, whereTj, andTj,are singular integrals with
non-smooth kernels, which are associated with an approximation of identity or ±I. Fori= , . . . ,M,j= , , we assume that ifTi,j=±I, then:
(a) Ti,jare bounded operators onL(Rn).
(b) There exist ‘generalized approximations of the identity’{Bijt,t> }such that (Ti,j–Ti,jBijt)have associated kernelsK
ij
t(x,y)and there exist positive constantsC,
Csuch that
|x–y|>Ct/m
Ktij(x,y)dx≤C for ally∈Rn.
(c) There exists a ‘generalized approximation to the identity’{At,t> }such that the
kernelskijt(x,y)of the operators(Ti,j–AtTi,j)satisfy
ktij≤C
tα/m
|x–y|n+α tα/m
d(x,y)α, (.)
It is proved in [] that ifTis an operator satisfying (a) and (b) above, thenTis of weak (, ) and of strong type (p,p) for <p≤. In addition, if (c) is also satisfied, the operator
Tis bounded onLp(Rn) for all <p<∞. Moreover, ifw∈A
p, thenTis bounded onLp(w) (see []).
In order to study the boundedness ofTbin weighted Morrey spaces, we need the fol-lowing result.
Lemma . Let w∈A∞, <κ< and <p<∞.Then,for every f ∈L
locwith Mf ∈Lp,κ(w), there exists a constant Cw,which only depends on w,such that
MfLp,κ(w)≤CwM
AfLp,κ(w). (.)
Proof LetBbe a ball inRn. Set Eλ={x∈B:Mf(x) >λ}. Then from the Whitney de-composition theorem, we know that there exist mutually disjoint cubes Qk such that
Eλ=kQk and Qk∩B\Eλ=∅. DenoteBkto be the ball with the same center asQk andrBk = diameterQk. LetBk= Bk. Then there exists anxk∈Bk∩B\Eλ, that is,
Mf(xk)≤λ. Let us use Lemma .. There areCw;r> andD> such that, if <η< (to be chosen later), we can findγ > in such a way that
wx∈Bk:Mf(x) >Dλ,MAf(x)≤γ λ
≤Cwηrw(Bk).
SetUλ={x∈B:Mf(x) >Dλ,MAf(x)≤γ λ}and soUλ⊂Eλ=kQk⊂
kBksinceD> . Then
w(Uλ)≤ k
wx∈Bk:Mf(x) >Dλ,M
Af(x)≤γ λ
≤Cwηr
k
w(Bk)
≤Cηr k
w(Qk) =Cηrw(Eλ)
=Cηrwx∈B:Mf(x) >λ,
where we used the fact thatA∞ weights are doubling measures andCis a constant that
only depends on the weight. One can prove that
B
|Mf|pw dx=Dp
∞
pλp–wx∈B:Mf(x) >Dλdλ
≤Dp
∞
pλp–w(Uλ) +wx∈B:MAf(x) >γ λdλ
≤CDpηr
B
|Mf|pw dx+Dp γp
B
MAfpw dx.
Let us chooseηsuch thatCDpηr= /. The former inequality turns out to be
B
|Mf|pw dx≤Dp γp
B
This implies that
MfLp,κ(w)≤CM
AfLp,κ(w).
The proof of this lemma is completed.
The aim of this section is to prove the following theorem.
Theorem . Let Ti,jbe operators satisfying the above conditions(a), (b)and(c)or±I.Let
<p<∞, <κ< and w∈Ap.Suppose that T(f) = when f ∈Lp,κ(w).If b∈BMO(Rn), then there exists a constant C such that
Tb(f)Lp,κ(w)≤C
M
j=
Tj,L(Rn)
M
j=
Tj,L(Rn)
b∗fLp,κ(w) (.)
for all f ∈Lp,κ(w).
Proof Without loss of generality, we may assume that
Tj,iL(Rn)≤ for all ≤j≤M,i= , .
Forw∈Ap, it is well known that there existst> such thatw∈Ap/t. Then we can choose two real numbersrandslarger than such that <rs<p<∞andw∈Ap/(rs). We will
prove that there exists a constantCsuch that
MA(Tbf)(x)≤ M
j=
Cb∗Mrs(Tj,f)(x) (.)
for allx∈Rn.
We now prove (.). For an arbitrary fixedx∈Rn, choose a ballB(x;r) ={y∈Rn:|x–
y|<r}which containsx. We have thatT(f) = , and soTbB(f) =bBT(f) = . Thus
Tb(f) =T(b–bB)χB(f) +T(b–bB)χ(B)c(f)
and
AtB(Tbf) =AtB(T(b–bB)χBf) +AtB(T(b–bB)χ(B)cf),
wheretB=rBm. Then
|B|
B
Tb(f)(y) –AtB(Tbf)(y)dy
≤
|B|
B
T(b–bB)χB(f)(y)dy+
|B|
B
AtB(T(b–bB)χBf)(y)dy
+
|B|
B
T(b–bB)χ(B)c(f)(y) –AtB(T(b–bB)χ(B)cf)(y)dy
Letr be the dual ofrsuch that /r+ /r = . By Lemma . and the boundedness ofTj,,
we have
I≤
|B|
Rn
T(b–bB)χB(f)(y) s dy /s ≤ M j= C |B| B
b(y) –bB
Tj,(f)(y)sdy /s ≤ M j= C |B| B
b(y) –bBsr dy
/(sr)
|B|
B
Tj,(f)(y)srdy /(sr)
≤
M
j=
Cb∗Mrs(Tj,f)(x).
Similarly, by Lemma . and the boundedness ofTj,, we obtain
II≤ |B|
B
M(T(b–bB)χBf)(y)dy≤
|B|
Rn
M(T(b–bB)χBf)(y) s dy /s ≤ |B| Rn
T(b–bB)χB(f)(y) s dy /s ≤ M j=
Cb∗Mrs(Tj,f)(x).
We now consider the termIII. There are two cases:
() Suppose thatTj,=±I(j= , . . . ,M), then using the assumption (c), we have
III≤ M j= |B| B
(B)c
kj,
tB(y,z)b(z) –bB
Tj,(f)(z)dz dy
≤C M j= ∞ k=
krB≤|x–z|<k+rB
rα
B
|x–z|n+αb(z) –bB
Tj,(f)(z)dz
≤C M j= ∞ k=
–kα |B(x; kr
B)|
|x–z|<k+rB
b(z) –bB
Tj,(f)(z)dz
≤C M j= ∞ k=
–kα
|B(x; kr B)|
|x–z|<k+rB
b(z) –bk+BTj,(f)(z)dz
+C M j= ∞ k=
–kαbk+B–bB
|B(x; krB)|
|x–z|<k+rB
Tj,(f)(z)dz
≤Cb∗
M j= ∞ k=
–kαMrs(Tj,f)(x) +Cb∗
M j= ∞ k=
–kα(k+ )M(Tj,f)(x)
≤Cb∗
M
j=
() Suppose that there areiidentity or –Ioperators in{Tj,}. Without loss of generality, we assume thatT,, . . . ,Ti,are identity operators, then
III≤
i
j=
|B|
B
b(z) –bB
χ(B)cTj,(f)(z)dz
+ i
j=
|B|
B
AtB
(b–bB)χ(B)cTj,(f)(z)dz
+ M
j=i+
|B|
B
Tj,M(b–bB)χ(B)cTj,(f)(z) –AtB
Tj,M(b–bB)χ(B)cTj,(f)
(z)dz
=III+III+III.
It is obvious thatIII= .
By (.) and Lemma ., we obtain
III≤ i
j=
|B|
B
(B)c
htB(z,y)
b(y) –bB
Tj,(f)(y)dy dz
= i
j=
∞
k=
|B|
B
(k+B)\(kB)
htB(z,y)
b(y) –bB
Tj,(f)(y)dy dz
≤C
i
j=
∞
k=
kns(k–)m
k+B
b(y) –bB
Tj,(f)(y)dy dz
≤Cb∗
i
j=
∞
k=
(k+ )kns(k–)mMrs(Tj,f)(x)
≤Cb∗
i
j=
Mrs(Tj,f)(x).
Moreover, from case () it follows that
III≤Cb∗ M
j=i+
Mrs(Tj,f)(x).
So,III≤Cb∗Mj=Mrs(Tj,f)(x).
Combining the above estimates ofI,IIandIII, we obtain (.).
From (.), we know that ifT is an operator satisfying (a), (b) and (c), then there exists <s<psuch thatw∈Ap/sand
MA(Tf)(x)≤CMsf(x). (.)
For the proof of (.), one can also see [, Proposition .]. Then combining (.), Lem-mas . and ., we have
TfLp,κ(w)≤CM
Combining this, (.), Lemmas . and ., we have
TbfLp,κ(w)≤CM
A(Tbf)Lp,κ(w)≤ M
j=
Cb∗Mrs(Tj,f)Lp,κ(w)≤Cb∗fLp,κ(w)
for allf ∈Lp,κ(w). The proof of this theorem is completed.
Corollary . Let T be operators satisfying the above conditions(a), (b)and(c).Let w∈Ap,
<p<∞.If b∈BMO(Rn),then there exists a constant C such that
[b,T]fLp,κ(w)≤Cb∗fLp,κ(w) (.)
for all f ∈Lp,κ(w).
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.
Acknowledgements
The authors would like to thank the referee for carefully reading the manuscript and for making several useful suggestions. This research was supported by the National Natural Science Foundation of China (Grant No. 11271092), Natural Science Foundation of Guangdong Province (Grant No. s2011010005367), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20114410110001, 20124410120002) and SRF of Guangzhou Education Bureau (Grant No. 2012A088).
Received: 4 July 2012 Accepted: 30 April 2013 Published: 17 May 2013 References
1. Morrey, CB: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc.43, 126-166 (1938)
2. Chiarenza, F, Frasca, M: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl.7, 273-279 (1987) 3. Nakai, E: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized
Morrey spaces. Math. Nachr.166, 95-103 (1994)
4. Palagachev, DK, Softova, LG: Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s. Potential Anal.20, 237-263 (2004)
5. Softova, L: Singular integrals and commutators in generalized Morrey spaces. Acta Math. Sin. Engl. Ser.22, 757-766 (2006) (English summary)
6. Komori, Y, Shirai, S: Weighted Morrey spaces and a singular integral operator. Math. Nachr.282, 219-231 (2009) 7. Krantz, S, Li, S: Boundedness and compactness of integral operators on spaces of homogeneous type and
applications, I. J. Math. Anal. Appl.258, 629-641 (2001)
8. Lu, SZ, Mo, HX: Toeplitz-type operators on Lebesgue spaces. Acta Math. Sci., Ser. B, Engl. Ed.29, 140-150 (2009) 9. Xie, PZ, Cao, GF: Toeplitz-type operators on homogeneous type spaces. Chin. Ann. Math., Ser. A32, 219-228 (2011)
(Chinese)
10. Duong, XT, McIntosh, A: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam.15, 233-265 (1999)
11. Stein, EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993)
12. Martell, JM: Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications. Stud. Math.161, 113-145 (2004)
13. Burger, N: Espace des fonctions à variation moyenne bornée sur un espace de nature homogène. C. R. Acad. Sci. Paris, Sér. A-B286, 139-141 (1978)
14. Bramanti, M, Cerutti, M: Commutators of singular integrals on homogeneous spaces. Boll. Unione Mat. Ital.10, 843-883 (1996)
15. Duong, XT, Yan, LX: Commutators of BMO functions and singular integral operators with non-smooth kernels. Bull. Aust. Math. Soc.67, 187-200 (2003)
doi:10.1186/1029-242X-2013-253
