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

Journal of Inequalities and Applications20142014:369

https://doi.org/10.1186/1029-242X-2014-369

Received: 20 April 2014

Accepted: 27 August 2014

Published: 25 September 2014

Abstract

In this paper, the boundedness for some Toeplitz type operator related to some singular integral operator satisfying a variant of Hörmander’s condition on L p spaces with variable exponent is obtained by using a sharp estimate of the operator.

MSC:42B20, 42B25.

Keywords

Toeplitz type operatorsingular integral operatorvariable L p spaceBMO

1 Introduction

As the development of singular integral operators (see [1, 2]), their commutators have been well studied (see [3, 4]). In [5, 6], some singular integral operators satisfying a variant of Hörmander’s condition and the boundedness for the operators and their commutators are obtained (see [6]). In [79], some Toeplitz type operators related to singular integral operators and strongly singular integral operators are introduced and the boundedness for the operators generated by B M O and Lipschitz functions are obtained. In the last years, the theory of L p spaces with variable exponent has been developed because of its connections with some questions in fluid dynamics, calculus of variations, differential equations and elasticity (see [1013] and their references). Karlovich and Lerner study the boundedness of the commutators of singular integral operators on L p spaces with variable exponent (see [14]). 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 on L p 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, Q will denote a cube of R n with sides parallel to the axes. For any locally integrable function f and δ > 0 , the sharp function of f is defined by
f δ # ( x ) = sup Q x ( 1 | Q | Q | f ( y ) f Q | δ d y ) 1 / δ ,
where, and in what follows, f Q = | Q | 1 Q f ( x ) d x . It is well known that (see [1, 2])
f δ # ( x ) sup Q x inf c C ( 1 | Q | Q | f ( y ) c | δ d y ) 1 / δ .
We write that f δ # = f # if δ = 1 . We say that f belongs to B M O ( R n ) if f # belongs to L ( R n ) and define f B M O = f # L . Let M be the Hardy-Littlewood maximal operator defined by
M ( f ) ( x ) = sup Q x | Q | 1 Q | f ( y ) | d y .
For k N , we denote by M k the operator M iterated k times, i.e., M 1 ( f ) ( x ) = M ( f ) ( x ) and
M k ( f ) ( x ) = M ( M k 1 ( f ) ) ( x ) when  k 2 .
Let Ψ be a Young function and Ψ ˜ be the complementary associated to Ψ. We denote the Ψ-average by, for a function f,
f Ψ , Q = inf { λ > 0 : 1 | Q | Q Ψ ( | f ( y ) | λ ) d y 1 }
and the maximal function associated to Ψ by
M Ψ ( f ) ( x ) = sup Q x f Ψ , Q .
The Young functions to be used in this paper are Ψ ( t ) = t ( 1 + log t ) r and Ψ ˜ ( t ) = exp ( t 1 / r ) , the corresponding average and maximal functions are denoted by L ( log L ) r , Q , M L ( log L ) r and exp L 1 / r , Q , M exp L 1 / r . Following [4], we know the generalized Hölder inequality,
1 | Q | Q | f ( y ) g ( y ) | d y f Ψ , Q g Ψ ˜ , Q ,
and the following inequality: for r , r j 1 , j = 1 , , l with 1 / r = 1 / r 1 + + 1 / r l and any x R n , b B M O ( R n ) ,
f L ( log L ) 1 / r , Q M L ( log L ) 1 / r ( f ) C M L ( log L ) l ( f ) C M l + 1 ( f ) , f f Q exp L r , Q C f B M O , | f 2 k + 1 Q f 2 Q | C k f B M O .
The non-increasing rearrangement of a measurable function f on R n is defined by
f ( t ) = inf { λ > 0 : | { x R n : | f ( x ) | > λ } | t } ( 0 < t < ) .
For λ ( 0 , 1 ) and a measurable function f on R n , the local sharp maximal function of f is defined by
M λ # ( f ) ( x ) = sup Q x inf c C ( ( f c ) χ Q ) ( λ | Q | ) .
Let p : R n [ 1 , ) be a measurable function. Denote by L p ( ) ( R n ) the sets of all Lebesgue measurable functions f on R n such that m ( λ f , p ) < for some λ = λ ( f ) > 0 , where
m ( f , p ) = R n | f ( x ) | p ( x ) d x .
The sets become Banach spaces with respect to the following norm:
f L p ( ) = inf { λ > 0 : m ( f / λ , p ) 1 } .
Denote by M ( R n ) the sets of all measurable functions p : R n [ 1 , ) such that the Hardy-Littlewood maximal operator M is bounded on L p ( ) ( R n ) and the following holds:
1 < p = ess inf x R n p ( x ) , ess sup x R n p ( x ) = p + < .
(1)

In recent years, the boundedness of classical operators on spaces L p ( ) ( R n ) has attracted a great deal of attention (see [1014] and their references).

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 # 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 .

In this paper, we study some singular integral operator as follows (see [5]).

Definition 3 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 a singular integral operator related to the kernel K by
T ( f ) ( x ) = R n K ( x y ) f ( y ) d y .
Let b be a locally integrable function on R n and T be a singular integral operator with variable Calderón-Zygmund kernels. The Toeplitz type operator associated to T is defined by
T b = j = 1 m T j , 1 M b T j , 2 ,

where T j , 1 are the singular integral operators T with variable Calderón-Zygmund kernels or ±I (the identity operator), T j , 2 are the linear operators for j = 1 , , m and M b ( f ) = b f .

Remark Note that the classical Calderón-Zygmund singular integral operator satisfies Definition 3 (see [2, 4]).

We shall prove the following theorems.

Theorem 1 Let T be a singular integral operator as in Definition  3, 0 < δ < 1 and b B M O ( 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 L 0 ( R n ) and x ˜ R n ,
M Φ , δ # ( T b ( f ) ) ( x ˜ ) C b B M O j = 1 m M 2 ( T j , 2 ( f ) ) ( x ˜ ) .
Theorem 2 Let T be a singular integral operator as in Definition  3, p ( ) M ( R n ) and b B M O ( R n ) . If T 1 ( g ) = 0 for any g L u ( R n ) ( 1 < u < ) and T j , 2 are the bounded linear operators on L p ( ) ( R n ) for k = 1 , , m , then T b is bounded on L p ( ) ( R n ) , that is,
T b ( f ) L p ( ) C b B M O f L p ( ) .

Corollary Let [ b , T ] ( f ) = b T ( f ) T ( b f ) be a commutator generated by the singular integral operators T and b. Then Theorems  1 and 2 hold for [ b , T ] .

3 Proofs of theorems

To prove the theorems, we need the following lemmas.

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

Let 0 < p < q < . We define that for any function f 0 and 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 ([4])

Let r j 1 for j = 1 , , l , we denote that 1 / r = 1 / r 1 + + 1 / r l . Then
1 | Q | Q | f 1 ( x ) f l ( x ) g ( x ) | d x f exp L r 1 , Q f exp L r l , Q g L ( log L ) 1 / r , Q .

Lemma 3 (see [5])

Let T be a singular integral operator as in Definition  3. Then T is weak bounded of ( L 1 , L 1 ) .

Lemma 4 ([13])

Let p : R n [ 1 , ) be a measurable function satisfying (1). Then L 0 ( R n ) is dense in L p ( ) ( R n ) .

Lemma 5 ([14])

Let f L loc 1 ( R n ) and g be a measurable function satisfying
| { x R n : | g ( x ) | > α } | < for all  α > 0 .
Then
R n | f ( x ) g ( x ) | d x C n R n M λ n # ( f ) ( x ) M ( g ) ( x ) d x .

Lemma 6 ([5, 14])

Let δ > 0 , 0 < λ < 1 , f L loc δ ( R n ) and Φ = { ϕ 1 , , ϕ m } L ( R n ) such that | det [ ϕ j ( y i ) ] | 2 R H ( R n m ) . Then
M λ # ( f ) ( x ) ( 1 / λ ) 1 / δ M Φ , δ # ( f ) ( x ) .

Lemma 7 ([13, 14])

Let p : R n [ 1 , ) be a measurable function satisfying (1). If f L p ( ) ( R n ) and g L p ( ) ( R n ) with p ( x ) = p ( x ) / ( p ( x ) 1 ) , then fg is integrable on R n and
R n | f ( x ) g ( x ) | d x C f L p ( ) g L p ( ) .

Lemma 8 ([14])

Let p : R n [ 1 , ) be a measurable function satisfying (1). Set
f L p ( ) = sup { R n | f ( x ) g ( x ) | d x : f L p ( ) ( R n ) , g L p ( ) ( R n ) } .

Then f L p ( ) f L p ( ) C f L p ( ) .

Proof of Theorem 1 It suffices to prove for f L 0 ( R n ) and some constant C 0 that the following inequality holds:
( 1 | Q | Q | T b ( f ) ( x ) C 0 | δ d x ) 1 / δ C b B M O 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 2 Q ) χ ( 2 Q ) c T j , 2 ( f ) ( y ) d y . Without loss of generality, we may assume that 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 | δ d x ) 1 / δ C ( 1 | Q | Q | f 1 ( x ) | δ d x ) 1 / δ + C ( 1 | Q | Q | f 2 ( x ) C 0 | δ d x ) 1 / δ = I + I I .
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 ) | δ d x ) 1 / δ | Q | 1 T j , 1 M ( b b 2 Q ) χ 2 Q T j , 2 ( f ) χ Q L δ | Q | 1 / δ 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 B M O 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 ) | δ d x ) 1 / δ C b B M O j = 1 m M 2 ( T j , 2 ( f ) ) ( x ˜ ) .
For II, we get, for x Q ,
| T j , 1 M ( b b 2 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 B M O M 2 ( T j , 2 ( f ) ) ( x ˜ ) k = 1 k 2 k δ C b B M O M 2 ( T j , 2 ( f ) ) ( x ˜ ) ,
thus
I I C | Q | Q j = 1 m | T j , 1 M ( b b 2 Q ) χ ( 2 Q ) c T j , 2 ( f ) ( x ) i = 1 l g j i ϕ i ( x 0 x ) | d x C b B M O j = 1 m M 2 ( T j , 2 ( f ) ) ( x ˜ ) .

This completes the proof of Theorem 1. □

Proof of Theorem 2 By Lemmas 4-7, we get, for f L 0 ( R n ) and g L p ( ) ( R n ) ,
R n | T b ( f ) ( x ) g ( x ) | d x C R n M λ n # ( T b ( f ) ) ( x ) M ( g ) ( x ) d x C R n M Φ , δ # ( T b ( f ) ) ( x ) M ( g ) ( x ) d x C b B M O j = 1 m R n M 2 ( T j , 2 ( f ) ) ( x ) M ( g ) ( x ) d x C b B M O j = 1 m M 2 ( T j , 2 ( f ) ) L p ( ) M ( g ) L p ( ) C b B M O j = 1 m T j , 2 ( f ) L p ( ) M ( g ) L p ( ) C b B M O f L p ( ) g L p ( ) ,
thus, by Lemma 8,
T b ( f ) L p ( ) b B M O f L p ( ) .

This completes the proof of Theorem 2. □

Declarations

Authors’ Affiliations

(1)
Changsha Commerce and Tourism College

References

  1. Garcia-Cuerva J, Rubio de Francia JL North-Holland Math. Stud. 116. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.Google Scholar
  2. Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.MATHGoogle Scholar
  3. Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611-635. 10.2307/1970954MathSciNetView ArticleMATHGoogle Scholar
  4. Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672-692. 10.1112/S0024610702003174View ArticleMATHMathSciNetGoogle Scholar
  5. Grubb DJ, Moore CN: A variant of Hörmander’s condition for singular integrals. Colloq. Math. 1997, 73: 165-172.MathSciNetMATHGoogle Scholar
  6. 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.MathSciNetMATHGoogle Scholar
  7. Krantz S, Li S: Boundedness and compactness of integral operators on spaces of homogeneous type and applications. J. Math. Anal. Appl. 2001, 258: 629-641. 10.1006/jmaa.2000.7402MathSciNetView ArticleMATHGoogle Scholar
  8. Lin Y, Lu SZ: Toeplitz type operators associated to strongly singular integral operator. Sci. China Ser. A 2006, 36: 615-630.Google Scholar
  9. Lu SZ, Mo HX: Toeplitz type operators on Lebesgue spaces. Acta Math. Sci. 2009,29(1):140-150. 10.1016/S0252-9602(09)60014-XMathSciNetView ArticleMATHGoogle Scholar
  10. Cruz-Uribe D, Fiorenza A, Neugebauer CJ:The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn., Math. 2003, 28: 223-238.MathSciNetMATHGoogle Scholar
  11. Diening L:Maximal function on generalized Lebesgue spaces L p ( ) . Math. Inequal. Appl. 2004, 7: 245-253.MathSciNetMATHGoogle Scholar
  12. Diening L, Ruzicka M:Calderón-Zygmund operators on generalized Lebesgue spaces L p ( ) and problems related to fluid dynamics. J. Reine Angew. Math. 2003, 563: 197-220.MathSciNetMATHGoogle Scholar
  13. Nekvinda A:Hardy-Littlewood maximal operator on L p ( x ) ( R n ) . Math. Inequal. Appl. 2004, 7: 255-265.MathSciNetMATHGoogle Scholar
  14. Karlovich AY, Lerner AK:Commutators of singular integral on generalized L p spaces with variable exponent. Publ. Mat. 2005, 49: 111-125.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Feng; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.