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# $M 2$-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

## Abstract

In this paper, we prove the $M 2$-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.

## 1 Introduction

As the development of singular integral operators (see [1, 2]), their commutators have been well studied. In [3, 4], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on $L p ( R n )$ for $1. Chanillo (see ) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [6, 7], some Toeplitz-type operators related to the singular integral operators and strongly singular integral operators are introduced, and the boundedness for the operators generated by BMO and Lipschitz functions is obtained. In , some singular integral operators satisfying a variant of Hörmander’s condition are introduced, 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 application, 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, Q will denote a cube of $R n$ with sides parallel to the axes. For any locally integrable function f, the sharp maximal function of f is defined by

$f # (x)= sup Q ∋ x 1 | Q | ∫ Q | f ( y ) − f Q | dy,$

where, and in what follows, $f Q = | Q | − 1 ∫ Q f(x)dx$. We say that f belongs to $BMO( R n )$ if $f #$ belongs to $L ∞ ( R n )$ and define $∥ f ∥ BMO = ∥ f # ∥ L ∞$. It has been known that (see )

$∥ f − f 2 k Q ∥ BMO ≤Ck ∥ f ∥ BMO .$

Let M be the Hardy-Littlewood maximal operator defined by

$M(f)(x)= sup Q ∋ x 1 | Q | ∫ Q | f ( y ) | dy.$

For $η>0$, let $M η (f)=M ( | f | η ) 1 / η$. For $k∈N$, we denote by $M k$ the operator M iterated k times, i.e., $M 1 (f)=M(f)$ and

Let Φ be a Young function and $Φ ˜$ be the complementary associated to Φ. We denote the Φ-average by, for a function f,

$∥ f ∥ Φ , Q =inf { λ > 0 : 1 | Q | ∫ Q Φ ( | f ( y ) | λ ) d y ≤ 1 }$

and the maximal function associated to Φ by

$M Φ (f)(x)= sup x ∈ Q ∥ f ∥ Φ , Q .$

The Young functions to be used in this paper are $Φ(t)=t(1+logt)$ and $Φ ˜ (t)=exp(t)$, the corresponding average and maximal functions denoted by $∥ ⋅ ∥ L ( log L ) , Q$, $M L ( log L )$ and $∥ ⋅ ∥ exp L , Q$, $M exp L$. Following , we know that the generalized Hölder inequality and the following inequalities hold:

$1 | Q | ∫ Q | f ( y ) g ( y ) | d y ≤ ∥ f ∥ Φ , Q ∥ g ∥ Φ ˜ , Q , ∥ f ∥ L ( log L ) , Q ≤ M L ( log L ) ( f ) ≤ C M 2 ( f ) , ∥ f − f Q ∥ exp L , Q ≤ C ∥ f ∥ BMO$

and

$∥ f − f Q ∥ exp L , 2 k Q ≤Ck ∥ f ∥ BMO .$

The $A p$ weight is defined by (see )

$A p = { w ∈ L loc 1 ( R n ) : sup Q ( 1 | Q | ∫ Q w ( x ) d x ) ( 1 | Q | ∫ Q w ( x ) − 1 / ( p − 1 ) d x ) p − 1 < ∞ } , 1 < p < ∞ ,$

and

$A 1 = { w ∈ L loc p ( R n ) : M ( w ) ( x ) ≤ C w ( x ) , a.e. } .$

Given a weight function w, for $1≤p<∞$, the weighted Lebesgue space $L p (w)$ is the space of functions f such that

$∥ f ∥ L p ( w ) = ( ∫ R n | f ( x ) | p w ( x ) d x ) 1 / p <∞.$

Definition 1 Let $Φ={ ϕ 1 ,…, ϕ l }$ be a finite family of bounded functions in $R n$. For any locally integrable function f, the Φ sharp maximal function of f is defined by

$M Φ # (f)(x)= sup Q ∋ x inf { c 1 , … , c l } 1 | Q | ∫ Q | f ( y ) − ∑ i = 1 l c i ϕ i ( x Q − y ) | dy,$

where the infimum is taken over all m-tuples ${ c 1 ,…, c l }$ of complex numbers and $x Q$ is the center of Q. For $η>0$, let

$M Φ , η # (f)(x)= sup Q ∋ x inf { c 1 , … , c l } ( 1 | Q | ∫ Q | f ( y ) − ∑ i = 1 l c j ϕ i ( x Q − y ) | η d y ) 1 / η .$

Remark We note that $M Φ # ≈ f #$ if $l=1$ and $ϕ 1 =1$.

Definition 2 Given a positive and locally integrable function f in $R n$, we say that f satisfies the reverse Hölder condition (write this as $f∈R H ∞ ( R n )$) if for any cube Q centered at the origin, we have

$0< sup x ∈ Q f(x)≤C 1 | Q | ∫ Q f(y)dy.$

Definition 3 Let φ be a positive, increasing function on $R +$, and there exists a constant $D>0$ such that

Let w be a weight function and f be a locally integrable function on $R n$. Set, for $1≤p<∞$,

$∥ f ∥ L p , φ ( w ) = sup x ∈ R n , d > 0 ( 1 φ ( d ) ∫ Q ( x , d ) | f ( y ) | p w ( y ) d y ) 1 / p ,$

where $Q(x,d)={y∈ R n :|x−y|. The generalized Morrey space is defined by

$L p , φ ( R n , w ) = { f ∈ L loc 1 ( R n ) : ∥ f ∥ L p , φ ( w ) < ∞ } .$

If $φ(d)= d η$, $η>0$, then $L p , φ ( R n ,w)= L p , η ( R n ,w)$, which is the classical weighted Morrey spaces (see [8, 9]). If $φ(d)=1$, then $L p , φ ( R n ,w)= L p ( R n ,w)$, which is the weighted Lebesgue spaces (see ).

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 [5, 811]).

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

Definition 4 Let $K∈ L 2 ( R n )$ and satisfy

$∥ K ∥ L ∞ ≤ C , | K ( x ) | ≤ C | x | − n ,$

there exist functions $B 1 ,…, B l ∈ L loc 1 ( R n −{0})$ and $Φ={ ϕ 1 ,…, ϕ l }⊂ L ∞ ( R n )$ such that $| det [ ϕ j ( y i ) ] | 2 ∈R H ∞ ( R n l )$, and for a fixed $δ>0$ and any $|x|>2|y|>0$,

$| K ( x − y ) − ∑ i = 1 l B i ( x ) ϕ i ( y ) | ≤C | y | δ | x − y | n + δ .$

For $f∈ C 0 ∞$, we define the singular integral operator related to the kernel K by

$T(f)(x)= ∫ R n K(x−y)f(y)dy.$

Moreover, let b be a locally integrable function on $R n$. The Toeplitz-type operator related to T is defined by

$T b = ∑ j = 1 m T j , 1 M b T j , 2 ,$

where $T j , 1$ are T or ±I (the identity operator), $T j , 2$ are the bounded linear operators on $L p (w)$ for $1 and $w∈ A 1$, $j=1,…,m$, $M b (f)=bf$.

Remark Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 4 (see , ). Also note that the commutator $[b,T](f)=bT(f)−T(bf)$ is a particular operator of the Toeplitz-type operators $T b$. The Toeplitz-type operators $T b$ 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 [12, 14]). The main purpose of this paper is to prove the sharp maximal inequalities for the Toeplitz-type operator $T b$. As the application, we obtain the weighted $L p$-norm inequality and Morrey space boundedness for the Toeplitz-type operators $T b$.

## 3 Theorems and lemmas

We shall prove the following theorems.

Theorem 1 Let T be the singular integral operator as Definition 4, $0 and $b∈BMO( R n )$. If $T 1 (g)=0$ for any $g∈ L u ( R n )$ ($1), then there exists a constant $C>0$ such that, for any $f∈ C 0 ∞ ( R n )$ and $x ˜ ∈ R n$,

$M Φ , r # ( T b ( f ) ) ( x ˜ )≤C ∥ b ∥ BMO ∑ j = 1 m M 2 ( T j , 2 ( f ) ) ( x ˜ ).$

Theorem 2 Let T be the singular integral operator as Definition 4, $1, $w∈ A 1$ and $b∈BMO( R n )$. If $T 1 (g)=0$ for any $g∈ L u ( R n )$ ($1), then $T b$ is bounded on $L p (w)$.

Theorem 3 Let T be the singular integral operator as Definition 4, $0, $1, $w∈ A 1$ and $b∈BMO( R n )$. If $T 1 (g)=0$ for any $g∈ L u ( R n )$ ($1), then $T b$ is bounded on $L p , φ ( R n ,w)$.

To prove the theorems, we need the following lemmas.

Lemma 1 ([, p.485])

Let $0 and for any function $f≥0$. We define that for $1/r=1/p−1/q$,

$∥ f ∥ W L q = sup λ > 0 λ | { x ∈ R n : f ( x ) > λ } | 1 / q , N p , q ( f ) = sup E ∥ f χ E ∥ L p / ∥ χ E ∥ L r ,$

where the sup is taken for all measurable sets E with $0<|E|<∞$. Then

$∥ f ∥ W L q ≤ N p , q (f)≤ ( q / ( q − p ) ) 1 / p ∥ f ∥ W L q .$

Lemma 2 (see )

We have

$1 | Q | ∫ Q | f ( x ) g ( x ) | dx≤ ∥ f ∥ exp L , Q ∥ g ∥ L ( log L ) , Q .$

Lemma 3 (see )

Let T be the singular integral operator as Definition 4. Then T is bounded on $L p (w)$ for $1, $w∈ A 1$ and weak $( L 1 , L 1 )$ bounded.

Lemma 4 (see )

Let $1, $0<η<∞$, $w∈ A ∞$ and $Φ={ ϕ 1 ,…, ϕ l }⊂ L ∞ ( R n )$ such that $| det [ ϕ j ( y i ) ] | 2 ∈R H ∞ ( R n l )$. Then, for any smooth function f for which the left-hand side is finite,

$∫ R n M η (f) ( x ) p w(x)dx≤C ∫ R n M Φ , η # (f) ( x ) p w(x)dx.$

Lemma 5 (see [5, 11])

Let $1, $w∈ A 1$ and $0. Then, for any smooth function f for which the left-hand side is finite,

$∥ M ( f ) ∥ L p , φ ( w ) ≤C ∥ f ∥ L p , φ ( w ) .$

Lemma 6 Let $1, $0<η<∞$, $w∈ A 1$, $0 and $Φ={ ϕ 1 ,…, ϕ l }⊂ L ∞ ( R n )$ such that $| det [ ϕ j ( y i ) ] | 2 ∈R H ∞ ( R n l )$. Then, for any smooth function f for which the left-hand side is finite,

$∥ M η ( f ) ∥ L p , φ ( w ) ≤C ∥ M Φ , η # ( f ) ∥ L p , φ ( w ) .$

Proof For any cube $Q=Q( x 0 ,d)$ in $R n$, we know $M(w χ Q )∈ A 1$ for any cube $Q=Q(x,d)$ by . By Lemma 4, we have, for $f∈ L p , φ ( R n ,w)$,

$∫ Q | M η ( f ) ( y ) | p w ( y ) d y = ∫ R n | M η ( f ) ( y ) | p w ( y ) χ Q ( y ) d y ≤ ∫ R n | M η ( f ) ( y ) | p M ( w χ Q ) ( y ) d y ≤ C ∫ R n | M Φ , η # ( f ) ( y ) | p M ( w χ Q ) ( y ) d y = C ( ∫ Q | M Φ , η # ( f ) ( y ) | p M ( w χ Q ) ( y ) d y + ∑ k = 0 ∞ ∫ 2 k + 1 Q ∖ 2 k Q | M Φ , η # ( f ) ( y ) | p M ( w χ Q ) ( y ) d y ) ≤ C ( ∫ Q | M Φ , η # ( f ) ( y ) | p w ( y ) d y + ∑ k = 0 ∞ ∫ 2 k + 1 Q ∖ 2 k Q | M Φ , η # ( f ) ( y ) | p w ( Q ) | 2 k + 1 Q | d y ) ≤ C ( ∫ Q | M Φ , η # ( f ) ( y ) | p w ( y ) d y + ∑ k = 0 ∞ ∫ 2 k + 1 Q | M Φ , η # ( f ) ( y ) | p M ( w ) ( y ) 2 n ( k + 1 ) d y ) ≤ C ( ∫ Q | M Φ , η # ( f ) ( y ) | p w ( y ) d y + ∑ k = 0 ∞ ∫ 2 k + 1 Q | M Φ , η # ( f ) ( y ) | p w ( y ) 2 n k d y ) ≤ C ∥ M Φ , η # ( f ) ∥ L p , φ ( w ) p ∑ k = 0 ∞ 2 − n k φ ( 2 k + 1 d ) ≤ C ∥ M Φ , η # ( f ) ∥ L p , φ ( w ) p ∑ k = 0 ∞ ( 2 − n D ) k φ ( d ) ≤ C ∥ M Φ , η # ( f ) ∥ L p , φ ( w ) p φ ( d ) ,$

thus

$( 1 φ ( d ) ∫ Q M η ( f ) ( x ) p w ( x ) d x ) 1 / p ≤C ( 1 φ ( d ) ∫ Q M Φ , η # ( f ) ( x ) p w ( x ) d x ) 1 / p$

and

$∥ M η ( f ) ∥ L p , φ ( w ) ≤C ∥ M Φ , η # ( f ) ∥ L p , φ ( w ) .$

This finishes the proof. □

Lemma 7 Let T be the singular integral operator as Definition 3 or the bounded linear operator on $L r (w)$ for any $1 and $w∈ A 1$, $1, $w∈ A 1$ and $0. Then

$∥ T ( f ) ∥ L p , φ ( w ) ≤C ∥ f ∥ L p , φ ( w ) .$

The proof of the lemma is similar to that of Lemma 6 by Lemma 3, we omit the details.

## 4 Proofs of theorems

Proof of Theorem 1 It suffices to prove that for $f∈ C 0 ∞ ( R n )$ and some constant $C 0$, the following inequality holds:

$( 1 | Q | ∫ Q | T b ( f ) ( x ) − C 0 | r d x ) 1 / r ≤C ∥ b ∥ BMO ∑ j = 1 m M 2 ( T j , 2 ( f ) ) ( x ˜ ),$

where Q is any cube centered at $x 0$, $C 0 = ∑ j = 1 m ∑ i = 1 l g j i ϕ i ( x 0 −x)$ and $g j i = ∫ R n B i ( x 0 −y) M ( b − b Q ) χ ( 2 Q ) c T j , 2 (f)(y)dy$. Without loss of generality, we may assume $T j , 1$ are T ($j=1,…,m$). Let $x ˜ ∈Q$. Fix a cube $Q=Q( x 0 ,d)$ and $x ˜ ∈Q$. Write

$T b (f)(x)= T b − b 2 Q (f)(x)= T ( b − b 2 Q ) χ 2 Q (f)(x)+ T ( b − b 2 Q ) χ ( 2 Q ) c (f)(x)= f 1 (x)+ f 2 (x).$

Then

$( 1 | Q | ∫ Q | T b ( f ) ( x ) − C 0 | r d x ) 1 / r ≤ C ( 1 | Q | ∫ Q | f 1 ( x ) | r d x ) 1 / r + C ( 1 | Q | ∫ Q | f 2 ( x ) − C 0 | r d x ) 1 / r = I + II .$

For I, by Lemmas 1, 2 and 3, we obtain

$( 1 | Q | ∫ Q | T j , 1 M ( b − b 2 Q ) χ 2 Q T j , 2 ( f ) ( x ) | r d x ) 1 / r ≤ | Q | − 1 ∥ T j , 1 M ( b − b 2 Q ) χ 2 Q T j , 2 ( f ) χ Q ∥ L r | Q | 1 / r − 1 ≤ C | Q | − 1 ∥ T j , 1 M ( b − b 2 Q ) χ 2 Q T j , 2 ( f ) ∥ W L 1 ≤ C | Q | − 1 ∥ M ( b − b 2 Q ) χ 2 Q T j , 2 ( f ) ∥ L 1 ≤ C | Q | − 1 ∫ 2 Q | b ( x ) − b 2 Q | | T j , 2 ( f ) ( x ) | d x ≤ C ∥ b − b 2 Q ∥ exp L , 2 Q ∥ T j , 2 ( f ) ∥ L ( log L ) , 2 Q ≤ C ∥ b ∥ BMO M 2 ( T j , 2 ( f ) ) ( x ˜ ) ,$

thus

$I≤C ∑ j = 1 m ( 1 | Q | ∫ Q | T j , 1 M ( b − b 2 Q ) χ 2 Q T j , 2 ( f ) ( x ) | r d x ) 1 / r ≤C ∥ b ∥ BMO ∑ j = 1 m M 2 ( T j , 2 ( f ) ) ( x ˜ ).$

For II, we get, for $x∈Q$,

$| T j , 1 M ( b − b Q ) χ ( 2 Q ) c T j , 2 ( f ) ( x ) − ∑ i = 1 l g j i ϕ i ( x 0 − x ) | ≤ | ∫ R n ( K ( x − y ) − ∑ i = 1 l B i ( x 0 − y ) ϕ i ( x 0 − x ) ) ( b ( y ) − b 2 Q ) χ ( 2 Q ) c ( y ) T j , 2 ( f ) ( y ) d y | ≤ ∑ k = 1 ∞ ∫ 2 k d ≤ | y − x 0 | < 2 k + 1 d | K ( x − y ) − ∑ i = 1 l B i ( x 0 − y ) ϕ i ( x 0 − x ) | | b ( y ) − b 2 Q | | T j , 2 ( f ) ( y ) | d y ≤ C ∑ k = 1 ∞ ∫ 2 k d ≤ | y − x 0 | < 2 k + 1 d | x − x 0 | δ | y − x 0 | n + δ | b ( y ) − b 2 Q | | T j , 2 ( f ) ( y ) | d y ≤ C ∑ k = 1 ∞ d δ ( 2 k d ) n + δ ( 2 k d ) n ∥ b − b 2 Q ∥ exp L , 2 k + 1 Q ∥ T j , 2 ( f ) ∥ L ( log L ) , 2 k + 1 Q ≤ C ∥ b ∥ BMO M 2 ( T j , 2 ( f ) ) ( x ˜ ) ∑ k = 1 ∞ k 2 − k δ ≤ C ∥ b ∥ BMO M 2 ( T j , 2 ( f ) ) ( x ˜ ) ,$

thus

$II≤ 1 | Q | ∫ Q ∑ j = 1 m | T j , 1 M ( b − b Q ) χ ( 2 Q ) c T j , 2 ( f ) ( x ) − C 0 | dx≤C ∥ b ∥ BMO ∑ j = 1 m M 2 ( T j , 2 ( f ) ) ( x ˜ ).$

This completes the proof of Theorem 1. □

Proof of Theorem 2 By Theorem 1 and Lemmas 3-4, we have

$∥ T b ( f ) ∥ L p ( w ) ≤ ∥ M r ( ( T b ( f ) ) ∥ L p ( w ) ≤ C ∥ M Φ , r # ( T b ( f ) ) ∥ L p ( w ) ≤ C ∥ b ∥ BMO ∑ j = 1 m ∥ M 2 ( T j , 2 ( f ) ) ∥ L p ( w ) ≤ C ∥ b ∥ BMO ∑ j = 1 m ∥ T j , 2 ( f ) ∥ L p ( w ) ≤ C ∥ b ∥ BMO ∥ f ∥ L p ( w ) .$

This completes the proof. □

Proof of Theorem 3 By Theorem 1 and Lemmas 5-7, we have

$∥ T b ( f ) ∥ L p , φ ( w ) ≤ ∥ M r ( T b ( f ) ) ∥ L p , φ ( w ) ≤ C ∥ M Φ , r # ( T b ( f ) ) ∥ L p , φ ( w ) ≤ C ∥ b ∥ BMO ∑ j = 1 m ∥ M 2 ( T j , 2 ( f ) ) ∥ L p , φ ( w ) ≤ C ∥ b ∥ BMO ∑ j = 1 m ∥ T j , 2 ( f ) ∥ L p , φ ( w ) ≤ C ∥ b ∥ BMO ∥ f ∥ L p , φ ( w ) .$

This completes the proof. □

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Correspondence to Qiufen Feng.

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Feng, Q. $M 2$-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. J Inequal Appl 2013, 589 (2013). https://doi.org/10.1186/1029-242X-2013-589

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### Keywords

• Toeplitz-type operator
• singular integral operator
• sharp maximal function
• BMO
• Morrey space 