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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

Journal of Inequalities and Applications20132013:589

https://doi.org/10.1186/1029-242X-2013-589

Received: 4 June 2013

Accepted: 15 November 2013

Published: 17 December 2013

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.

Keywords

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

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 < p < . Chanillo (see [5]) 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 [6], some singular integral operators satisfying a variant of Hörmander’s condition are introduced, and the boundedness for the operators is obtained (see [6], [20]). 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 | d y ,
where, and in what follows, f Q = | Q | 1 Q f ( x ) d x . 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 [2])
f f 2 k Q BMO C k f BMO .
Let M be the Hardy-Littlewood maximal operator defined by
M ( f ) ( x ) = sup Q x 1 | Q | Q | f ( y ) | d y .
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
M k ( f ) = M ( M k 1 ( f ) ) when  k 2 .
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 + log t ) 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 [2], 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 C k f BMO .
The A p weight is defined by (see [1])
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 ) | d y ,
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 ) d y .
Definition 3 Let φ be a positive, increasing function on R + , and there exists a constant D > 0 such that
φ ( 2 t ) D φ ( t ) for  t 0 .
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 | < d } . 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 [6]).

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 [12]).

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 ) d y .
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 < p < and w A 1 , j = 1 , , m , M b ( f ) = b f .

Remark Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 4 (see [13], [19]). Also note that the commutator [ b , T ] ( f ) = b T ( f ) T ( b f ) 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 < r < 1 and b BMO ( R n ) . If T 1 ( g ) = 0 for any g L u ( R n ) ( 1 < u < ), 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 < p < , w A 1 and b BMO ( R n ) . If T 1 ( g ) = 0 for any g L u ( R n ) ( 1 < u < ), then T b is bounded on L p ( w ) .

Theorem 3 Let T be the singular integral operator as Definition 4, 0 < D < 2 n , 1 < p < , w A 1 and b BMO ( R n ) . If T 1 ( g ) = 0 for any g L u ( R n ) ( 1 < u < ), then T b is bounded on L p , φ ( R n , w ) .

To prove the theorems, we need the following lemmas.

Lemma 1 ([[1], p.485])

Let 0 < p < q < 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 [2])

We have
1 | Q | Q | f ( x ) g ( x ) | d x f exp L , Q g L ( log L ) , Q .

Lemma 3 (see [15])

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

Lemma 4 (see [12])

Let 1 < p < , 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 ) d x C R n M Φ , η # ( f ) ( x ) p w ( x ) d x .

Lemma 5 (see [5, 11])

Let 1 < p < , w A 1 and 0 < D < 2 n . 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 < p < , 0 < η < , w A 1 , 0 < D < 2 n 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 [3]. 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 < r < and w A 1 , 1 < p < , w A 1 and 0 < D < 2 n . 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 ) d y . 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 | d x 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. □

Declarations

Authors’ Affiliations

(1)
Changsha Commerce and Tourism College

References

  1. Di Fazio G, Ragusa MA: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 1993, 112: 241–256. 10.1006/jfan.1993.1032MathSciNetView ArticleGoogle Scholar
  2. Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672–692. 10.1112/S0024610702003174View ArticleGoogle Scholar
  3. Coifman R, Rochberg R: Another characterization of BMO. Proc. Am. Math. Soc. 1980, 79: 249–254. 10.1090/S0002-9939-1980-0565349-8MathSciNetView ArticleGoogle Scholar
  4. Peetre J:On the theory of L p , λ -spaces. J. Funct. Anal. 1969, 4: 71–87. 10.1016/0022-1236(69)90022-6MathSciNetView ArticleGoogle Scholar
  5. Chiarenza F, Frasca M: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. 1987, 7: 273–279.MathSciNetGoogle Scholar
  6. Garcia-Cuerva J, Rubio de Francia JL North-Holland Math. Stud. 16. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.Google Scholar
  7. Grubb DJ, Moore CN: A variant of Hörmander’s condition for singular integrals. Colloq. Math. 1997, 73: 165–172.MathSciNetGoogle Scholar
  8. Liu LZ: Interior estimates in Morrey spaces for solutions of elliptic equations and weighted boundedness for commutators of singular integral operators. Acta Math. Sci. 2005, 25(1):89–94.MathSciNetGoogle Scholar
  9. Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. ICM-90 Satell. Conf. Proc. Harmonic Analysis 1990, 183–189. Sendai, 1990Google Scholar
  10. Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611–635. 10.2307/1970954MathSciNetView ArticleGoogle Scholar
  11. Di Fazio G, Ragusa MA: Commutators and Morrey spaces. Boll. Unione Mat. Ital., A 1991, 5(7):323–332.MathSciNetGoogle Scholar
  12. Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.Google Scholar
  13. Trujillo-Gonzalez R: Weighted norm inequalities for singular integral operators satisfying a variant of Hörmander’s condition. Comment. Math. Univ. Carol. 2003, 44: 137–152.MathSciNetGoogle Scholar
  14. Torchinsky A Pure and Applied Math. 123. In Real Variable Methods in Harmonic Analysis. Academic Press, New York; 1986.Google Scholar
  15. Chanillo S: A note on commutators. Indiana Univ. Math. J. 1982, 31: 7–16. 10.1512/iumj.1982.31.31002MathSciNetView ArticleGoogle Scholar

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© Feng; licensee Springer. 2013

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