Boundedness of Toeplitz type operator associated to general integral operator on Lp spaces with variable exponent

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R E S E A R C H

Open Access

Boundedness of Toeplitz-type operator

associated to general integral operator on

L

p

spaces with variable exponent

Xin-Sheng Yuan and Zhi-Gang Wang

*

*_{Correspondence:}

[email protected] School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455002, P.R. China

Abstract

In this paper, the boundedness for some Toeplitz-type operator related to some general integral operator onLpspaces with variable exponent is obtained by using a sharp estimate of the operator. The operator includes Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

MSC: 42B20; 42B25

Keywords: Toeplitz-type operator; Littlewood-Paley operator; Marcinkiewicz operator; Bochner-Riesz operator; variableLp_{space; BMO}

1 Introduction

As the development of singular integral operators (see [, ]), their commutators have been well studied. In [–], the authors proved that the commutators generated by the singular integral operators andBMOfunctions are bounded onLp(Rn) for  <p<∞. In [–], some Toeplitz-type operators associated to the singular integral operators and strongly singular integral operators were introduced, and the boundedness for the operators was obtained. In the last years, the theory ofLpspaces with variable exponent was 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 studied the boundedness of the commutators of singular integral operators onLp_spaces

with variable exponent (see []). Motivated by these papers, in this paper, we have the purpose to introduce some Toeplitz-type operator related to some integral operator and

BMOfunctions, and prove the boundedness for the operator onLp_{spaces with variable}

exponent by using a sharp estimate of the operator. The operators include Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz 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

f_δ#(x) =sup

Qx

 |Q|

Q

f(y) –fQ

δ

dy

/δ ,

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where, and in what follows,fQ=|Q|–

Qf(x)dx. It is well known that (see [, ])

f_δ#(x)≈sup

Qx

inf

c∈C

 |Q|

Q

f(y) –cδdy

/δ .

We write thatf#₌_f#

δ 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 byMk_{the operator}_M_iterated_k_times,_i.e._,_M₍_f₎₍_x_{) =}_M₍_f₎₍_x_{) and}

Mk(f)(x) =MMk–(f)(x) whenk≥.

Letbe a Young function and let˜ be the complementary associated to. For a func-tionf, we denote the-average by

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)r _and₍˜ _t_{) =}_exp(_t/r_),

the corresponding average and maximal functions are denoted by · L(logL)r_,_Q,M_L₍_log_L₎r

and · _expL/r_,_Q,M_expL/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+· · ·+ /rl, and any x∈Rn_,_b_∈_BMO₍_Rn_),

fL(logL)/r_,_Q≤M_L₍_logL₎/r(f)≤CM_L₍_log_L₎l(f)≤CMl+(f),

f–fQexpLr_,_Q≤Cf_BMO,

|f_k+_Q–f__Q| ≤Ckf_BMO.

The non-increasing rearrangement of a measurable functionf onRn_{is deﬁned by}

f∗(t) =infλ>  :x∈Rn:f(x)>λ≤t ( <t<∞).

Forλ∈(, ) and the measurable functionf onRn, the local sharp maximal function off

is deﬁned by

M#λ(f)(x) =sup

Qx

inf

c∈C

(f–c)χQ

∗

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Letp:Rn_→_[,_∞_{) be a measurable function. Denote by}_Lp(·)₍_Rn_{) the sets of all Lebesgue}

measurable functionsf onRn_{such that}_m_(λ_f_,_p_{) <}_∞_{for some}_λ₌_λ(_f_{) > , where}

m(f,p) =

Rn

f(x)p(x)dx.

The sets become Banach spaces with respect to the following norm:

f_Lp(·)=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 hold:

 <p–=ess inf

x∈Rnp(x), ess sup

x∈Rnp(x) =p+<∞. ()

In this paper, we study some integral operators as follows (see [, ]).

Deﬁnition LetFt(x,y) be deﬁned onRn×Rn×[, +∞), set

Ft(f)(x) =

Rn

Ft(x,y)f(y)dy

for every bounded and compactly supported functionf.Ftsatisﬁes: for ﬁxedδ> ,

Ft(x–y)≤C|x–y|–n

and

Ft(y–x) –Ft(z–x)≤C|y–z|δ|x–z|–n–δ

if |y–z| ≤ |x–z|. We deﬁne thatT(f)(x) =Ft(f)(x).

LetHbe the Banach spaceH={h:h<∞}. For each ﬁxedx∈Rn_{, we view}_F

t(f)(x) and Fb

t(f)(x) as the mappings from [, +∞) toH. Set

T(f)(x) =Ft(f)(x).

Moreover, letbbe a locally integrable function onRn. The Toeplitz-type operator related toT is deﬁned by

Tb(f) =F_tb(f),

where

F_tb(f) =

m

k=

F_tk,MbFtk,(f),

Ftk,(f) areFt(f) or±I(the identity operator),Tk,(f) =Ftk,(f)are the operators fork=

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Note that the commutator is a particular operator of the Toeplitz-type operator Tb_.

The Toeplitz-type operatorsTb_{are the non-trivial generalizations of the commutator. It}

is well known that commutators are of great interest in harmonic analysis, and they have been widely studied by many authors (see [, ]). In recent years, the boundedness of classical operators on spacesLp(·)₍_Rn_{) have attracted great attention (see [–] and their}

references). The main purpose of this paper is twofold. First, we establish a sharp estimate for the operatorTb_{; and second, we prove the boundedness for the operator on}_Lp_spaces

with variable exponent by using the sharp estimate. In Section , we give some applications of the theorems in this paper.

We shall prove the following theorems.

Theorem  Let T be the integral operator as deﬁned in Deﬁnition,  <δ< and b∈

BMO(Rn_)._{If F}

t(g) = for any g∈Lu(Rn) ( <u<∞),then there exists a constant C>  such that for any f ∈L∞_(Rn₎_and_x_˜_∈_Rn_,

Tb(f)#_δ(x˜)≤CbBMO m

k=

MTk,(f)(x˜).

Theorem  Let T be the integral operator as deﬁned in Deﬁnition,p(·)∈M(Rn₎_{and b}_∈

BMO(Rn_)._{If F}

t(g) = for any g∈Lu(Rn) ( <u<∞)and Tk,are the bounded 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 the commutator generated by the integral

oper-ator T and b.Then Theoremsandhold 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

fχELp/χ_E_Lr,

where thesupis taken for all measurable sets E with <|E|<∞.Then

fWLq≤N_p_,_q(f)≤q/(q–p)/pf_WLq.

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 [] Let T be the integral operator as deﬁned in Deﬁnition.Then T is bounded

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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

RnM #

λn(f)(x)M(g)(x)dx.

Lemma [, ] Letδ> ,  <λ< and f ∈Lδ

loc(Rn).Then

M#_λ(f)(x)≤(/λ)/δf_δ#(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 R}n_and

Rn

f(x)g(x)dx≤Cf_Lp(·)g_Lp(·).

Lemma [] Let p:Rn_→_[,_∞₎_{be a measurable function satisfying}_()._Set

f_Lp(·)=sup

Rn

f(x)g(x)dx:f ∈Lp(·)Rn,g∈Lp(·)Rn.

Thenf_Lp(·)≤ f

Lp(·)≤CfLp(·).

Proof of Theorem It suﬃces to prove, forf ∈C_∞(Rn_{) and some constant}_C_{, that the}

following inequality holds:

 |Q|

Q

Tb(f)(x) –C δ

dx

/δ

≤CbBMO m

k=

MTk,(f)(x˜).

Without loss of generality, we may assume thatTk,_are_T ₍_k_{= , . . . ,}_m_{). Fix a cube}_Q₌

Q(x,d) andx˜∈Q. We write, byF

t(g) = ,

F_tb(f)(x) =Ftb–bQ(f)(x) =F

(b–bQ)χQ

t (f)(x) +F

(b–bQ)χ(Q)c

t (f)(x) =f(x) +f(x).

Then

 |Q|

Q

_Tb₍_f₎₍_x_{) –}_f_(_x_)δ_dx

/δ

=

 |Q|

Q

F_tb(f)(x)–f(x)δdx

/δ ≤

 |Q|

Q

F_tb(f)(x) –f(x)δdx

/δ

≤C

 |Q|

Q

f(x)δdx

/δ +C

 |Q|

Q

f(x) –f(x)δdx

/δ

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ForI, by the weak (L_,_L_{) boundedness of}_T _{(see Lemma ) and Kolmogorov’s inequality} (see Lemma ), we obtain

 |Q|

Q

F_tk,M(b–bQ)χQF k,

t (f)(x)

δ dx /δ =  |Q|

Q

Tk,M(b–bQ)χQT

k,₍_f₎₍_x₎δ

dx

/δ

≤|Q_| |/δ–

Q|/δ

Tk,M(b–bQ)χQT k,₍_f_)χ

QLδ

χQLδ/(–δ)

≤_|C

Q|T k,_M

(b–bQ)χQT k,₍_f₎

WL

≤ C

|Q|

Rn

M(b–bQ)χQT

k,₍_f₎₍_x₎_dx

≤C|Q|–M(b–bQ)χQT k,₍_f₎

L

≤C|Q|–

Q

b(x) –bQTk,(f)(x)dx

≤Cb–bQexpL,QTk,(f)_L₍_logL_),_Q

≤CbBMOM

Tk,(f)(x˜).

Thus

I≤C

m

k=

 |Q|

Q

F_tk,M(b–bQ)χQF k,

t (f)(x)

δ

dx

/δ

≤CbBMO m

k=

MTk,(f)(x˜).

ForI, we get, forx∈Q,

F_tk,M(b–bQ)χ(Q)cFtk,(f)(x) –Ftk,M(b–bQ)χ(Q)cFtk,(f)(x)

≤

(Q)c

b(y) –bQFt(x,y) –Ft(x,y)Tk,(f)(y)dy

≤C ∞

j=

jd≤|y–x|<j+d

b(y) –bQ

|x–x|δ |x–y|n+δ

Tk,(f)(y)dy

≤C ∞

j=

–jδ  |j+_Q_|

j+_Q

b(y) –bQTk,(f)(y)dy

≤C ∞

j=

–jδb–bQexpL,j+_QTk,(f)

L(logL),j+_Q

≤C ∞

j=

j–jδbBMOM

Tk,(f)(x˜)

≤CbBMOM

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Thus

I≤

C

|Q|

Q m

k=

F_tk,M(b–bQ)χ(Q)cFtk,(f)(x) –Ftk,M(b–bQ)χ(Q)cFtk,(f)(x)dx

≤CbBMO m

k=

MTk,(f)(x˜).

These complete the proof of Theorem .

Proof of Theorem By Lemmas -, we get, forf ∈L∞_ (Rn_{) and}_g_∈_Lp(·)₍_Rn_),

Rn

Tb(f)(x)g(x)dx≤C

RnM #

λn

Tb(f)(x)M(g)(x)dx

≤C

Rn

Tb(f)#_δ(x)M(g)(x)dx

≤CbBMO m

k=

RnM

_Tk,₍_f₎₍_x₎_M₍_g₎₍_x₎_dx

≤CbBMO m

k=

MTk,(f)_Lp(·)M(g)_Lp(·)

≤CbBMO m

k=

Tk,(f)_Lp(·)M(g)_Lp(·)

≤CbBMOfLp(·)g_Lp(·).

Thus, by Lemma ,

Tb(f)_Lp(·)≤ fLp(·).

This completes the proof of Theorem .

4 Applications

In this section we shall apply Theorems  and  of the paper to some particular operators such as Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Application  Littlewood-Paley operator.

Fixedε> . Letψbe a ﬁxed function which satisﬁes:

() _Rnψ(x)dx= , () |ψ(x)| ≤C( +|x|)–(n+)_,

() |ψ(x+y) –ψ(x)| ≤C|y|ε_{( +}_|_x_|₎–(n++ε)_when__|_y_|_<_|_x_|_.

Letψt(x) =t–nψ(x/t) fort>  andFt(f)(x) =

Rnf(y)ψt(x–y)dy. The Littlewood-Paley

operator is deﬁned by (see [])

gψ(f)(x) =

∞



Ft(f)(x)

dt

t

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SetHto be the space

H= h:h=

∞



h(t)dt/t

/ <∞

.

Letb be a locally integrable function onRn. The Toeplitz-type operator related to the Littlewood-Paley operator is deﬁned by

gψb(f)(x) =

∞



F_tb(f)(x)dt

t

/ ,

where

Fb t =

m

k=

Fk,

t MbFtk,,

F_tk,areFtor±I(the identity operator),Tk,=Ftk,are the bounded linear operators on Lp(Rn) for  <p<∞andk= , . . . ,m,Mb(f) =bf. Then, for each ﬁxedx∈Rn,Ftb(f)(x) may

be viewed as the mapping from [, +∞) toH, and it is clear that

g_ψb(f)(x) =F_tb(f)(x), gψ(f)(x) =Ft(f)(x).

It is easy to see thatgψb satisﬁes the conditions of Theorems  and  (see [, , ]), thus Theorems  and  hold forgψb.

Application  Marcinkiewicz operator.

Fixed  <γ ≤. Letbe homogeneous of degree zero onRn_with

Sn–(x)dσ(x) = .

Assume that∈Lip_γ(Sn–_{). Set}_F

t(f)(x) =

|x–y|≤t

(x–y)

|x–y|n–f(y)dy. The Marcinkiewicz

oper-ator is deﬁned by (see [])

μ(f)(x) =

∞



Ft(f)(x)

dt

t

/ .

SetHto be the space

H= h:h=

∞



h(t)dt/t

/ <∞

.

Letb be a locally integrable function onRn_{. The Toeplitz-type operator related to the}

Marcinkiewicz operator is deﬁned by

μb(f)(x) =

∞



F_tb(f)(x)dt

t

/ ,

where

F_tb=

m

k=

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Ftk,areFtor±I(the identity operator),Tk,=Ftk,are the bounded linear operators on Lp(Rn) for  <p<∞andk= , . . . ,m,Mb(f) =bf. Then it is clear that

μb(f)(x) =F_tb(f)(x), μ(f)(x) =Ft(f)(x).

It is easy to see thatμbsatisﬁes the conditions of Theorems  and  (see [, , ]), thus Theorems  and  hold forμb.

Application  Bochner-Riesz operator.

Letδ> (n– )/,Fδ

t(f)(ξˆ ) = ( –t|ξ|)δ+fˆ(ξ) andBδt(z) =t–nBδ(z/t) fort> . The maximal

Bochner-Riesz operator is deﬁned by (see [])

Bδ,∗(f)(x) =sup t>

F_tδ(f)(x).

SetHto be the spaceH={h:h=sup_t_>|h(t)|<∞}. Letbbe a locally integrable func-tion onRn_{. The Toeplitz-type operator related to the maximal Bochner-Riesz operator is}

deﬁned by

Bb_δ,_∗(f)(x) =sup

t>

Bb_δ,_t(f)(x),

where

Bb_δ,_t=

m

k=

F_tk,MbFtk,,

Ftk,areFtor±I(the identity operator),Tk,=Ftk,are the bounded linear operators on Lp₍_Rn_{) for  <}_p_<_∞_and_k_{= , . . . ,}_m_,_M

b(f) =bf. Then

Bb_δ,_∗(f)(x) =Bb_δ,_t(f)(x), Bδ

∗(f)(x) =Bδt(f)(x).

It is easy to see thatBb

δ,∗satisﬁes the conditions of Theorems  and  (see [, ]), thus

Theorems  and  hold forBb_δ,_∗.

Competing interests

The authors declare that they have no competing interests.

Authors’ information

All authors read and approved the ﬁnal manuscript.

Acknowledgements

The present investigation was supported by theNational Natural Science Foundationunder Grants 11226088, 11301008 and 11101053, theOpen Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universitiesunder Grants 11FEFM02 and 12FEFM02, and theKey Project of Natural Science Foundation of Educational Committee of Henan

Provinceunder Grant 12A110002 of the People’s Republic of China.

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