M2 Type sharp estimates and boundedness on a Morrey space for Toeplitz type operators associated to singular integral operators satisfying a variant of Hörmander’s condition

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R E S E A R C H

Open Access

M



-Type sharp estimates and boundedness

on a Morrey space for Toeplitz-type operators

associated to singular integral operators

satisfying a variant of Hörmander’s condition

Qiufen Feng

*

*_{Correspondence:}

[email protected]

Changsha Commerce and Tourism College, Yuhua District, White Road No. 16, Changsha, Hunan, China

Abstract

In this paper, we prove theM2_{-type sharp maximal function estimates for the}

Toeplitz-type operators associated to certain singular integral operators satisfying a variant of Hörmander’s condition. As an application, we obtain the weighted boundedness of the operators on the Lebesgue and Morrey spaces.

MSC: 42B20; 42B25

Keywords: Toeplitz-type operator; singular integral operator; sharp maximal function;BMO; Morrey space

1 Introduction

As the development of singular integral operators (see [, ]), their commutators have been well studied. In [, ], the authors prove that the commutators generated by the sin-gular integral operators andBMOfunctions are bounded onLp₍_Rn_{) for  <}_p_<_∞_{. Chanillo}

(see []) proves a similar result when singular integral operators are replaced by the frac-tional integral operators. In [, ], some Toeplitz-type operators related to the singular integral operators and strongly singular integral operators are introduced, and the bound-edness for the operators generated byBMOand Lipschitz functions is obtained. In [], some singular integral operators satisfying a variant of Hörmander’s condition are intro-duced, and the boundedness for the operators is obtained (see [], []). In this paper, we prove the sharp maximal function inequalities for the Toeplitz-type operator related to some singular integral operators satisfying a variant of Hörmander’s condition. As an ap-plication, we obtain the weighted boundedness of the Toeplitz-type operator on Lebesgue and Morrey spaces.

2 Preliminaries

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, the sharp maximal function off is deﬁned by

f#(x) =sup

Qx

 |Q|

Q

f(y) –fQdy,

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where, and in what follows,fQ=|Q|–

Qf(x)dx. We say thatf belongs toBMO(Rn) iff #

belongs toL∞(Rn_{) and deﬁne}_f

BMO=f#L∞. It has been known that (see [])

f–f_k_QBMO≤CkfBMO.

LetMbe the Hardy-Littlewood maximal operator deﬁned by

M(f)(x) =sup

Qx

 |Q|

Q

f(y)dy.

Forη> , letMη(f) =M(|f|η)/η. Fork∈N, we denote byMkthe operatorMiteratedk times,i.e.,M(f) =M(f) and

Mk(f) =MMk–(f) 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

x∈Q f,Q.

The Young functions to be used in this paper are(t) =t( +logt) and˜(t) =exp(t), the corresponding average and maximal functions denoted by · L(logL),Q,ML(logL) and

· expL,Q,MexpL. Following [], we know that the generalized Hölder inequality and the

following inequalities hold:

 |Q|

Q

f(y)g(y)dy≤ f,Qg˜,Q,

fL(logL),Q≤ML(logL)(f)≤CM(f),

f–fQexpL,Q≤CfBMO

and

f–fQexpL,k_Q≤CkfBMO.

TheApweight is deﬁned by (see [])

Ap=

w∈L_locRn:sup

Q

 |Q|

Q

w(x)dx 

|Q|

Q

w(x)–/(p–)dx p–

<∞

,

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and

A=

w∈Lp_locRn:M(w)(x)≤Cw(x), a.e..

Given a weight functionw, for ≤p<∞, the weighted Lebesgue spaceLp₍_w_{) is the}

space of functionsf such that

fLp₍_w₎=

Rn

f(x)pw(x)dx

/p

<∞.

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#

≈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.

Deﬁnition  Letϕbe a positive, increasing function onR+, and there exists a constant

D>  such that

ϕ(t)≤Dϕ(t) fort≥.

Letwbe a weight function andf be a locally integrable function onRn_{. Set, for }_≤_p_<_∞_,

fLp,ϕ₍_w₎= sup x∈Rn_,_d_>



ϕ(d)

Q(x,d)

f(y)pw(y)dy

/p

,

whereQ(x,d) ={y∈Rn_:_|_x_–_y_|_<_d_}_{. The generalized Morrey space is deﬁned by}

Lp,ϕRn,w=f ∈L_locRn:fLp,ϕ₍_w₎<∞.

Ifϕ(d) =dη_,_η_{> , then}_Lp,ϕ₍_Rn_,_w_{) =}_Lp,η₍_Rn_,_w_{), which is the classical weighted Morrey}

spaces (see [, ]). Ifϕ(d) = , thenLp,ϕ₍_Rn_,_w_{) =}_Lp₍_Rn_,_w_{), which is the weighted Lebesgue}

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As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [, –]).

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

Deﬁnition  LetK∈L₍_Rn_{) and satisfy}

KL∞≤C, K(x)≤C|x|–n,

there exist functions B, . . . ,Bl∈Lloc(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 the singular integral operator related to the kernelKby

T(f)(x) =

RnK(x–y)f(y)dy.

Moreover, letbbe a locally integrable function onRn_{. The Toeplitz-type operator related}

toT is deﬁned by

Tb=

m

j=

Tj,MbTj,,

whereTj,_are_T _or_±_I_{(the identity operator),}_Tj,_{are the bounded linear operators on}

Lp(w) for  <p<∞andw∈A,j= , . . . ,m,Mb(f) =bf.

Remark Note that the classical Calderón-Zygmund singular integral operator satisﬁes Deﬁnition  (see [], []). Also note that the commutator [b,T](f) =bT(f) –T(bf) is a particular operator of the Toeplitz-type operatorsTb_{. The Toeplitz-type operators}_Tb

are the non-trivial generalizations of the commutator. It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [, ]). The main purpose of this paper is to prove the sharp maximal inequalities for the Toeplitz-type operatorTb_{. As the application, we obtain the weighted}_Lp_-norm

inequality and Morrey space boundedness for the Toeplitz-type operatorsTb_.

3 Theorems and lemmas

We shall prove the following theorems.

Theorem  Let T be the singular integral operator as Deﬁnition,  <r< and b∈ BMO(Rn_)._{If T}₍_g_{) = }_{for any g}_∈_Lu₍_Rn_{) ( <}_u_<_∞_),_{then there exists a constant C}_{> } such that,for any f∈C_∞(Rn₎_and_x_˜_∈_Rn_,

M#_,_rTb(f)(x˜)≤CbBMO m

j=

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Theorem  Let T be the singular integral operator as Deﬁnition,  <p<∞,w∈Aand

b∈BMO(Rn).If T(g) = for any g∈Lu(Rn) ( <u<∞),then Tbis bounded on Lp(w).

Theorem  Let T be the singular integral operator as Deﬁnition,  <D< n_{,  <}_p_<_∞_,

w∈Aand b∈BMO(Rn).If T(g) = for any g∈Lu(Rn) ( <u<∞),then Tbis bounded

on Lp,ϕ₍_Rn_,_w_).

To prove the theorems, we need the following lemmas.

Lemma ([, p.]) Let <p<q<∞and for any function f ≥.We deﬁne that for

/r= /p– /q,

fWLq=sup λ>

λx∈Rn:f(x) >λ/q,

Np,q(f) =sup

E

fχELp/χELr,

where the sup is taken for all measurable sets E with <|E|<∞.Then

fWLq≤N_p_,_q(f)≤q/(q–p)/pf_WLq.

Lemma (see []) We have

 |Q|

Q

f(x)g(x)dx≤ fexpL,QgL(logL),Q.

Lemma  (see []) Let T be the singular integral operator as Deﬁnition. Then T is bounded on Lp₍_w₎_for_{ <}_p_<_∞_,_w_∈_A

and weak(L,L)bounded.

Lemma (see []) Let <p<∞,  <η<∞,w∈A∞and={φ, . . . ,φl} ⊂L∞(Rn)such that|det[φj(yi)]|∈RH∞(Rnl).Then,for any smooth function f for which the left-hand side is ﬁnite,

RnMη(f)(x)

p_w₍_x₎_dx_≤_C

RnM

#

,η(f)(x)pw(x)dx.

Lemma (see [, ]) Let <p<∞,w∈Aand <D< n.Then,for any smooth function

f for which the left-hand side is ﬁnite,

M(f)_Lp,ϕ₍_w₎≤CfLp,ϕ₍_w₎.

Lemma  Let <p<∞,  <η<∞,w∈A,  <D< nand={φ, . . . ,φl} ⊂L∞(Rn)such that|det[φj(yi)]|∈RH∞(Rnl).Then,for any smooth function f for which the left-hand side is ﬁnite,

Mη(f)_Lp,ϕ₍_w₎≤CM

#

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Proof For any cubeQ=Q(x,d) inRn, we knowM(wχQ)∈A for any cubeQ=Q(x,d)

by []. By Lemma , we have, forf ∈Lp,ϕ₍_Rn_,_w_),

Q

Mη(f)(y)

p w(y)dy

=

Rn

Mη(f)(y)

p

w(y)χQ(y)dy

≤

Rn

Mη(f)(y)

p

M(wχQ)(y)dy

≤C

Rn

M#,η(f)(y)

p

M(wχQ)(y)dy

=C

Q

M#,η(f)(y)

p

M(wχQ)(y)dy+

∞

k=

k+_Q_\_k_Q

M#,η(f)(y)

p

M(wχQ)(y)dy

≤C

Q

M#,η(f)(y)

p

w(y)dy+

∞

k=

k+_Q_\_k_Q

M#,η(f)(y)

p w(Q)

|k+_Q_|dy

≤C

Q _M#

,η(f)(y)

p

w(y)dy+

∞

k=

k+_Q

_M#

,η(f)(y)

pM(w)(y)

n(k+) dy

≤C

Q

M#_,_η(f)(y)pw(y)dy+

∞

k=

k+_Q

M#_,_η(f)(y)pw(y) nk dy

≤CM#_,_η(f)p_Lp,ϕ₍_w₎

∞

k=

–nkϕk+d

≤CM#,η(f)

p Lp,ϕ₍_w₎

∞

k=

–nDkϕ(d)

≤CM#,η(f)

p

Lp,ϕ₍_w₎ϕ(d),

thus



ϕ(d)

Q

Mη(f)(x)pw(x)dx

/p

≤C



ϕ(d)

Q

M#,η(f)(x)pw(x)dx

/p

and

Mη(f)_Lp,ϕ₍_w₎≤CM

#

,η(f)Lp,ϕ₍_w₎.

This ﬁnishes the proof.

Lemma  Let T be the singular integral operator as Deﬁnitionor the bounded linear operator on Lr₍_w₎_{for any}_{ <}_r_<_∞_{and w}_∈_A

,  <p<∞,w∈Aand <D< n.Then

T(f)_Lp,ϕ₍_w₎≤CfLp,ϕ₍_w₎.

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4 Proofs of theorems

Proof of Theorem It suﬃces to prove that forf ∈C_∞(Rn) and some constantC, the

following inequality holds:

 |Q|

Q

Tb(f)(x) –C

r dx

/r

≤CbBMO m

j=

MTj,(f)(x˜),

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} _Tj, _are _T ₍_j₌

, . . . ,m). Letx˜∈Q. Fix a cubeQ=Q(x,d) andx˜∈Q. Write

Tb(f)(x) =Tb–bQ₍_f₎₍_x_{) =}_T(b–bQ)χQ₍_f₎₍_x_{) +}_T(b–bQ)χ(Q)c₍_f₎₍_x_{) =}_f

(x) +f(x).

Then

 |Q|

Q

Tb(f)(x) –Crdx /r

≤C

 |Q|

Q

f(x)rdx /r

+C

 |Q|

Q

_f_₍_x_{) –}_C_r dx

/r

=I+II.

ForI, by Lemmas ,  and , we obtain

 |Q|

Q _Tj,_M

(b–bQ)χQT

j,₍_f₎₍_x₎r_dx /r

≤ |Q|–T j,_M

(b–bQ)χQTj,(f)χQLr |Q|/r–

≤C|Q|–_Tj,_M

(b–bQ)χQT

j,₍_f₎

WL

≤C|Q|–M(b–bQ)χQT

j,₍_f₎

L

≤C|Q|–

Q

b(x) –bQTj,(f)(x)dx

≤Cb–bQexpL,QTj,(f)_L₍logL),Q

≤CbBMOM

Tj,(f)(x˜),

thus

I≤C m

j=

 |Q|

Q _Tj,_M

(b–bQ)χQT

j,₍_f₎₍_x₎r_dx /r

≤CbBMO m

j=

MTj,(f)(x˜).

ForII, we get, forx∈Q,

Tj,M(b–bQ)χ(Q)cTj,(f)(x) –

l

i=

g_jiφi(x–x)

≤ Rn

K(x–y) –

l

i=

Bi(x–y)φi(x–x)

b(y) –bQ

χ(Q)c(y)Tj,(f)(y)dy

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≤

∞

k=

k_d_≤|_y_–_x |<k+d

K(x–y) – l

i=

Bi(x–y)φi(x–x)

b(y) –bQTj,(f)(y)dy

≤C ∞

k=

k_d_≤|_y_–_x |<k+d

|x–x|δ

|y–x|n+δ

b(y) –bQTj,(f)(y)dy

≤C ∞

k=

dδ (k_d₎n+δ

kdnb–bQexpL,k+_QTj,(f)

L(logL),k+_Q

≤CbBMOM

Tj,(f)(x˜)

∞

k=

k–kδ

≤CbBMOM

Tj,(f)(x˜),

thus

II≤ 

|Q|

Q m

j=

Tj,M(b–bQ)χ(Q)cTj,(f)(x) –Cdx≤CbBMO m

j=

MTj,(f)(x˜).

This completes the proof of Theorem .

Proof of Theorem By Theorem  and Lemmas -, we have

Tb(f)_Lp₍_w₎≤Mr(

Tb(f)_Lp₍_w₎≤CM#,r

Tb(f)_Lp₍_w₎

≤CbBMO m

j=

MTj,(f)_Lp₍_w₎≤CbBMO m

j=

Tj,(f)_Lp₍_w₎

≤CbBMOfLp₍_w₎.

This completes the proof.

Proof of Theorem By Theorem  and Lemmas -, we have

Tb(f)_Lp,ϕ₍_w₎≤Mr

Tb(f)_Lp,ϕ₍_w₎≤CM

#

,r

Tb(f)_Lp,ϕ₍_w₎

≤CbBMO m

j=

MTj,(f)_Lp,ϕ₍_w₎≤CbBMO m

j=

Tj,(f)_Lp,ϕ₍_w₎

≤CbBMOfLp,ϕ₍_w₎.

This completes the proof.

Competing interests

The author declares that they have no competing interests.

Received: 4 June 2013 Accepted: 15 November 2013 Published:17 Dec 2013

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