R E S E A R C H
Open Access
Boundedness of Toeplitz type operator
associated to singular integral operator
satisfying a variant of Hörmander’s condition
on
L
p
spaces with variable exponent
Qiufen Feng
**Correspondence:
Changsha Commerce and Tourism College, Yuhua District, White Road No. 16, Changsha, Hunan, P.R. China
Abstract
In this paper, the boundedness for some Toeplitz type operator related to some singular integral operator satisfying a variant of Hörmander’s condition onLpspaces
with variable exponent is obtained by using a sharp estimate of the operator.
MSC: 42B20; 42B25
Keywords: Toeplitz type operator; singular integral operator; variableLpspace; BMO
1 Introduction
As the development of singular integral operators (see [, ]), their commutators have been well studied (see [, ]). In [, ], some singular integral operators satisfying a variant of Hörmander’s condition and the boundedness for the operators and their commutators are obtained (see []). In [–], some Toeplitz type operators related to singular integral operators and strongly singular integral operators are introduced and the boundedness for the operators generated byBMOand Lipschitz functions are obtained. In the last years, the theory ofLpspaces with variable exponent has been developed because of its connections with some questions in fluid dynamics, calculus of variations, differential equations and elasticity (see [–] and their references). Karlovich and Lerner study the boundedness of the commutators of singular integral operators onLp spaces with variable exponent (see []). Motivated by these papers, the main purpose of this paper is to introduce some Toeplitz type operator related to some singular integral operator satisfying a variant of Hörmander’s condition and prove the boundedness for the operator onLp spaces with variable exponent by using a sharp estimate of the operator.
2 Preliminaries and results
First, let us introduce some notations. Throughout this paper,Qwill denote a cube ofRn with sides parallel to the axes. For any locally integrable functionf andδ> , the sharp function off is defined by
fδ#(x) =sup
Qx
|Q|
Q
f(y) –fQ
δ dy
/δ
,
where, and in what follows,fQ=|Q|–
Qf(x)dx. It is well known that (see [, ])
fδ#(x)≈sup
Qx
inf
c∈C
|Q|
Q
f(y) –cδdy /δ
.
We write thatf#
δ =f#ifδ= . We say thatf belongs toBMO(Rn) iff#belongs toL∞(Rn)
and definefBMO=f#L∞. LetMbe the Hardy-Littlewood maximal operator defined by
M(f)(x) =sup
Qx|
Q|–
Q
f(y)dy.
Fork∈N, we denote byMkthe operatorMiteratedktimes,i.e.,M(f)(x) =M(f)(x) and
Mk(f)(x) =MMk–(f)(x) whenk≥.
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
Qx
f,Q.
The Young functions to be used in this paper are(t) =t( +logt)rand˜(t) =exp(t/r), the corresponding average and maximal functions are denoted by · L(logL)r,Q,ML(logL)r
and · expL/r,Q,MexpL/r. Following [], we know the generalized Hölder inequality,
|Q|
Q
f(y)g(y)dy≤ f,Qg˜,Q,
and the following inequality: forr,rj≥,j= , . . . ,lwith /r= /r+· · ·+ /rland anyx∈Rn,
b∈BMO(Rn),
fL(logL)/r,Q≤ML(logL)/r(f)≤CML(logL)l(f)≤CMl+(f),
f–fQexpLr,Q≤CfBMO,
|fk+Q–fQ| ≤CkfBMO.
The non-increasing rearrangement of a measurable functionf onRnis defined by
f∗(t) =infλ> :x∈Rn:f(x)>λ≤t ( <t<∞).
Forλ∈(, ) and a measurable functionf onRn, the local sharp maximal function off is defined by
M#λ(f)(x) =sup
Qx
inf
c∈C
(f–c)χQ
Letp:Rn→[,∞) be a measurable function. Denote byLp(·)(Rn) the sets of all Lebesgue measurable functionsf onRnsuch thatm(λf,p) <∞for someλ=λ(f) > , where
m(f,p) =
Rn
f(x)p(x)dx.
The sets become Banach spaces with respect to the following norm:
fLp(·)=inf
λ> :m(f/λ,p)≤.
Denote byM(Rn) the sets of all measurable functionsp:Rn→[,∞) such that the Hardy-Littlewood maximal operatorMis bounded onLp(·)(Rn) and the following holds:
<p–=ess inf
x∈Rnp(x), ess sup
x∈Rnp(x) =p+<∞. ()
In recent years, the boundedness of classical operators on spacesLp(·)(Rn) has attracted a great deal of attention (see [–] and their references).
Definition Let={φ, . . . ,φl}be a finite family of bounded functions inRn. For any locally integrable functionf, thesharp maximal function off is defined by
M#(f)(x) =sup
Qx
inf {c,...,cl}
|Q|
Q
f(y) –
l
i=
ciφi(xQ–y)
dy,
where the infimum is taken over allm-tuples{c, . . . ,cl}of complex numbers andxQis the center ofQ. Forη> , let
M#,η(f)(x) =sup
Qx
inf {c,...,cl}
|Q|
Q
f(y) –
l
i=
cjφi(xQ–y)
η
dy /η
.
Remark We note thatM#≈M
#
(f) ifl= andφ= .
Definition Given a positive and locally integrable functionfinRn, we say thatf satisfies the reverse Hölder condition (write this asf ∈RH∞(Rn)) if, for any cubeQcentered at the
origin, we have
<sup
x∈Q
f(x)≤C
|Q|
Q
f(y)dy.
In this paper, we study some singular integral operator as follows (see []).
Definition LetK∈L(Rn) and satisfy
KL∞≤C,
There exist functionsB, . . . ,Bl∈Lloc(Rn–{}) and={φ, . . . ,φl} ⊂L∞(Rn) such that
|det[φj(yi)]|∈RH∞(Rnl), and for a fixedδ> and any|x|> |y|> ,
K(x–y) –
l
i=
Bi(x)φi(y)
≤C
|y|δ
|x–y|n+δ.
Forf∈C∞, we define a singular integral operator related to the kernelKby
T(f)(x) =
RnK(x–y)f(y)dy.
Letbbe a locally integrable function onRnandT be a singular integral operator with variable Calderón-Zygmund kernels. The Toeplitz type operator associated toTis defined by
Tb= m
j=
Tj,MbTj,,
whereTj,are the singular integral operatorsTwith variable Calderón-Zygmund kernels or±I(the identity operator),Tj,are the linear operators forj= , . . . ,mandMb(f) =bf.
Remark Note that the classical Calderón-Zygmund singular integral operator satisfies Definition (see [, ]).
We shall prove the following theorems.
Theorem Let T be a singular integral operator as in Definition, <δ < and b∈ BMO(Rn).If T(g) = for any g∈Lu(Rn) ( <u<∞),then there exists a constant C>
such that for any f ∈L∞(Rn)andx˜∈Rn,
M#,δ
Tb(f)
(x˜)≤CbBMO m
j=
MTj,(f)(x˜).
Theorem Let T be a singular integral operator as in Definition,p(·)∈M(Rn)and
b∈BMO(Rn).If T(g) = for any g∈Lu(Rn) ( <u<∞)and Tj,are the bounded linear
operators on Lp(·)(Rn)for k= , . . . ,m,then Tbis bounded on Lp(·)(Rn),that is,
Tb(f)Lp(·)≤CbBMOfLp(·).
Corollary Let[b,T](f) =bT(f) –T(bf)be a commutator generated by the singular integral operators T and b.Then Theoremsandhold for[b,T].
3 Proofs of theorems
To prove the theorems, we need the following lemmas.
Lemma ([, p.]) Let <p<q<∞.We define that for any function f ≥and/r= /p– /q,
fWLq=sup
λ>
λx∈Rn:f(x) >λ/q, Np,q(f) =sup E
where thesupis taken for all measurable sets E with <|E|<∞.Then
fWLq≤Np,q(f)≤q/(q–p)/pfWLq.
Lemma ([]) Let rj≥for j= , . . . ,l,we denote that/r= /r+· · ·+ /rl.Then
|Q|
Q
f(x)· · ·fl(x)g(x)dx≤ fexpLr,Q· · · fexpLrl,QgL(logL)/r,Q.
Lemma (see []) Let T be a singular integral operator as in Definition.Then T is weak bounded of(L,L).
Lemma ([]) Let p:Rn→[,∞)be a measurable function satisfying().Then L∞ (Rn)
is dense in Lp(·)(Rn).
Lemma ([]) Let f ∈L
loc(Rn)and g be a measurable function satisfying
x∈Rn:g(x)>α<∞ for allα> .
Then
Rn
f(x)g(x)dx≤Cn
Rn
Mλ#n(f)(x)M(g)(x)dx.
Lemma ([, ]) Letδ> , <λ< ,f ∈Lδ
loc(Rn)and={φ, . . . ,φm} ⊂L∞(Rn)such
that|det[φj(yi)]|∈RH∞(Rnm).Then
M#λ(f)(x)≤(/λ)/ δ
M#,δ(f)(x).
Lemma ([, ]) Let p:Rn→[,∞)be a measurable function satisfying ().If f ∈
Lp(·)(Rn)and g∈Lp(·)(Rn)with p(x) =p(x)/(p(x) – ),then fg is integrable on Rnand
Rn
f(x)g(x)dx≤CfLp(·)gLp(·).
Lemma ([]) Let p:Rn→[,∞)be a measurable function satisfying().Set
fLp(·)=sup
Rn
f(x)g(x)dx:f ∈Lp(·)Rn,g∈Lp(·)Rn.
ThenfLp(·)≤ f
Lp(·)≤CfLp(·).
Proof of Theorem It suffices to prove forf ∈L∞ (Rn) and some constantC
that the fol-lowing inequality holds:
|Q|
Q
Tb(f)(x) –C
δ dx
/δ
≤CbBMO m
j=
where Q is any cube centered at x, C=
m
j=
l
i=gjiφi(x–x) and gji=
RnBi(x–
y)M(b–bQ)χ(Q)cT
j,(f)(y)dy. Without loss of generality, we may assume that Tj, are T
(j= , . . . ,m). Letx˜∈Q. Fix a cubeQ=Q(x,d) andx˜∈Q. Write
Tb(f)(x) =Tb–bQ(f)(x) =T(b–bQ)χQ(f)(x) +T(b–bQ)χ(Q)c(f)(x) =f(x) +f(x).
Then
|Q|
Q
Tb(f)(x) –C
δ dx /δ ≤C
|Q|
Q
f(x)
δ dx /δ +C
|Q|
Q
f(x) –C
δ dx
/δ
=I+II.
ForI, by Lemmas , and , we obtain
|Q|
Q
Tj,M(b–bQ)χQT
j,(f)(x)δ dx
/δ
≤ |Q|–T
j,M
(b–bQ)χQT
j,(f)χ QLδ
|Q|/δ–
≤C|Q|–Tj,M(b–bQ)χQT
j,(f) WL
≤C|Q|–M(b–bQ)χQT
j,(f) L
≤C|Q|–
Q
b(x) –bQTj,(f)(x)dx
≤Cb–bQexpL,QTj,(f)L(logL),Q
≤CbBMOM
Tj,(f)(x˜),
thus
I≤C
m
j=
|Q|
Q
Tj,M(b–bQ)χQT
j,(f)(x)δ dx
/δ
≤CbBMO m
j=
MTj,(f)(x˜).
ForII, we get, forx∈Q,
Tj,M(b–bQ)χ(Q)cTj,(f)(x) –
l
i=
gjiφi(x–x)
≤ Rn
K(x–y) – l
i=
Bi(x–y)φi(x–x)
b(y) –bQ
χ(Q)c(y)Tj,(f)(y)dy
≤ ∞ k=
kd≤|y–x |<k+d
K(x–y) –
l
i=
Bi(x–y)φi(x–x)
b(y) –bQTj,(f)(y)dy
≤C ∞
k=
kd≤|y–x |<k+d
|x–x|δ
|y–x|n+δ
b(y) –bQTj,(f)(y)dy
≤C ∞
k=
dδ
(kd)n+δ
kdnb–bQexpL,k+QTj,(f)
≤CbBMOM
Tj,(f)(x˜)
∞
k=
k–kδ
≤CbBMOM
Tj,(f)(x˜),
thus
II≤ C
|Q|
Q m
j=
Tj,M(b–bQ)χ(Q)cTj,(f)(x) –
l
i=
gijφi(x–x)
dx
≤CbBMO m
j=
MTj,(f)(x˜).
This completes the proof of Theorem .
Proof of Theorem By Lemmas -, we get, forf ∈L∞ (Rn) andg∈Lp(·)(Rn),
Rn
Tb(f)(x)g(x)dx≤C
RnM
#
λn
Tb(f)
(x)M(g)(x)dx
≤C
RnM
#
,δ
Tb(f)
(x)M(g)(x)dx
≤CbBMO m
j=
RnM
Tj,(f)(x)M(g)(x)dx
≤CbBMO m
j=
MTj,(f)Lp(·)M(g)Lp(·)
≤CbBMO m
j=
Tj,(f)
Lp(·)M(g)Lp(·)
≤CbBMOfLp(·)gLp(·),
thus, by Lemma ,
Tb(f)
Lp(·)≤ bBMOfLp(·).
This completes the proof of Theorem .
Competing interests
The author declares that they have no competing interests.
Received: 20 April 2014 Accepted: 27 August 2014 Published:25 Sep 2014
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Cite this article as:Feng:Boundedness of Toeplitz type operator associated to singular integral operator satisfying a variant of Hörmander’s condition onLpspaces with variable exponent.Journal of Inequalities and Applications