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Boundedness of Toeplitz-type operator associated to general integral operator on L p spaces with variable exponent

Abstract

In this paper, the boundedness for some Toeplitz-type operator related to some general integral operator on L p spaces 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.

1 Introduction

As the development of singular integral operators (see [1, 2]), their commutators have been well studied. In [35], the authors proved that the commutators generated by the singular integral operators and BMO functions are bounded on L p ( R n ) for 1<p<. In [68], 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 of L p spaces with variable exponent was developed because of its connections with some questions in fluid dynamics, calculus of variations, differential equations and elasticity (see [913] and their references). Karlovich and Lerner studied the boundedness of the commutators of singular integral operators on L p spaces with variable exponent (see [12]). Motivated by these papers, in this paper, we have the purpose to introduce some Toeplitz-type operator related to some integral operator and BMO functions, and prove the boundedness for the operator on L p 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, 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)dx. 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 # = f δ # if δ=1. We say that f belongs to BMO( R n ) if f # belongs to L ( R n ) and define f BMO = f # L . Let M be the Hardy-Littlewood maximal operator defined by

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

For kN, 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 k2.

Let Φ be a Young function and let Φ ˜ be the complementary associated to Φ. For a function f, we denote the Φ-average by

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 Q x f Φ , Q .

The Young functions to be used in this paper are Φ(t)=t ( 1 + log t ) r and Φ ˜ (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, 5], we know the generalized Hölder inequality

1 | Q | Q |f(y)g(y)|dy f Φ , Q g Φ ˜ , 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 , bBMO( 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 BMO , | f 2 k + 1 Q f 2 Q | C k f BMO .

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 the 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(λf,p)< for some λ=λ(f)>0, where

m(f,p)= R n |f(x) | p ( x ) dx.

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 hold:

1< p =ess inf x R n p(x),ess sup x R n p(x)= p + <.
(1)

In this paper, we study some integral operators as follows (see [14, 15]).

Definition Let F t (x,y) be defined on R n × R n ×[0,+), set

F t (f)(x)= R n F t (x,y)f(y)dy

for every bounded and compactly supported function f. F t satisfies: for fixed δ>0,

F t ( x y ) C|xy | n

and

F t ( y x ) F t ( z x ) C|yz | δ |xz | n δ

if 2|yz||xz|. We define that T(f)(x)= F t (f)(x).

Let H be the Banach space H={h:h<}. For each fixed x R n , we view F t (f)(x) and F t b (f)(x) as the mappings from [0,+) to H. Set

T(f)(x)= F t ( f ) ( x ) .

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

T b (f)= F t b ( f ) ,

where

F t b (f)= k = 1 m F t k , 1 M b F t k , 2 (f),

F t k , 1 (f) are F t (f) or ±I (the identity operator), T k , 2 (f)= F t k , 2 (f) are the operators for k=1,,m and M b (f)=bf.

Note that the commutator is a particular operator of the Toeplitz-type operator 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 they have been widely studied by many authors (see [4, 5]). In recent years, the boundedness of classical operators on spaces L p ( ) ( R n ) have attracted great attention (see [913] and their references). The main purpose of this paper is twofold. First, we establish a sharp estimate for the operator T b ; and second, we prove the boundedness for the operator on L p spaces with variable exponent by using the sharp estimate. In Section 4, we give some applications of the theorems in this paper.

We shall prove the following theorems.

Theorem 1 Let T be the integral operator as defined in Definition, 0<δ<1 and bBMO( R n ). If F 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 ,

( T b ( f ) ) δ # ( x ˜ )C b BMO k = 1 m M 2 ( T k , 2 ( f ) ) ( x ˜ ).

Theorem 2 Let T be the integral operator as defined in Definition, p()M( R n ) and bBMO( R n ). If F t 1 (g)=0 for any g L u ( R n ) (1<u<) and T k , 2 are the bounded 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 BMO f L p ( ) .

Corollary Let [b,T](f)=bT(f)T(bf) be the commutator generated by the integral operator 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 [5, p.485]

Let 0<p<q<. We define that for any function f0 and 1/r=1/p1/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)|dx f exp L r 1 , Q f exp L r l , Q g L ( log L ) 1 / r , Q .

Lemma 3 [14]

Let T be the integral operator as defined in Definition. Then T is bounded from L 1 ( R n ) to W L 1 ( R n ).

Lemma 4 [12]

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 [12]

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)|dx C n R n M λ n # (f)(x)M(g)(x)dx.

Lemma 6 [12, 16]

Let δ>0, 0<λ<1 and f L loc δ ( R n ). Then

M λ # (f)(x) ( 1 / λ ) 1 / δ f δ # (x).

Lemma 7 [17]

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)|dxC f L p ( ) g L p ( ) .

Lemma 8 [12]

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 C 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 BMO k = 1 m M 2 ( T k , 2 ( f ) ) ( x ˜ ).

Without loss of generality, we may assume that T k , 1 are T (k=1,,m). Fix a cube Q=Q( x 0 ,d) and x ˜ Q. We write, by F t 1 (g)=0,

F t b (f)(x)= F t b b 2 Q (f)(x)= F t ( b b 2 Q ) χ 2 Q (f)(x)+ F 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 ) f 2 ( x 0 ) | δ d x ) 1 / δ = ( 1 | Q | Q | F t b ( f ) ( x ) f 2 ( x 0 ) | δ d x ) 1 / δ ( 1 | Q | Q F t b ( f ) ( x ) f 2 ( x 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 ) f 2 ( x 0 ) δ d x ) 1 / δ = I 1 + I 2 .

For I 1 , by the weak ( L 1 , L 1 ) boundedness of T (see Lemma 3) and Kolmogorov’s inequality (see Lemma 1), we obtain

( 1 | Q | Q F t k , 1 M ( b b 2 Q ) χ 2 Q F t k , 2 ( f ) ( x ) δ d x ) 1 / δ = ( 1 | Q | Q | T k , 1 M ( b b 2 Q ) χ 2 Q T k , 2 ( f ) ( x ) | δ d x ) 1 / δ | Q | 1 / δ 1 | Q | 1 / δ T k , 1 M ( b b 2 Q ) χ 2 Q T k , 2 ( f ) χ Q L δ χ Q L δ / ( 1 δ ) C | Q | T k , 1 M ( b b 2 Q ) χ 2 Q T k , 2 ( f ) W L 1 C | Q | R n | M ( b b 2 Q ) χ 2 Q T k , 2 ( f ) ( x ) | d x C | Q | 1 M ( b b 2 Q ) χ 2 Q T k , 2 ( f ) L 1 C | Q | 1 2 Q | b ( x ) b 2 Q | | T k , 2 ( f ) ( x ) | d x C b b Q exp L , 2 Q T k , 2 ( f ) L ( log L ) , 2 Q C b BMO M 2 ( T k , 2 ( f ) ) ( x ˜ ) .

Thus

I 1 C k = 1 m ( 1 | Q | Q F t k , 1 M ( b b 2 Q ) χ 2 Q F t k , 2 ( f ) ( x ) δ d x ) 1 / δ C b BMO k = 1 m M 2 ( T k , 2 ( f ) ) ( x ˜ ) .

For I 2 , we get, for xQ,

F t k , 1 M ( b b 2 Q ) χ ( 2 Q ) c F t k , 2 ( f ) ( x ) F t k , 1 M ( b b 2 Q ) χ ( 2 Q ) c F t k , 2 ( f ) ( x 0 ) ( 2 Q ) c | b ( y ) b 2 Q | F t ( x , y ) F t ( x 0 , y ) | T k , 2 ( f ) ( y ) | d y C j = 1 2 j d | y x 0 | < 2 j + 1 d | b ( y ) b 2 Q | | x x 0 | δ | x 0 y | n + δ | T k , 2 ( f ) ( y ) | d y C j = 1 2 j δ 1 | 2 j + 1 Q | 2 j + 1 Q | b ( y ) b 2 Q | | T k , 2 ( f ) ( y ) | d y C j = 1 2 j δ b b 2 Q exp L , 2 j + 1 Q T k , 2 ( f ) L ( log L ) , 2 j + 1 Q C j = 1 j 2 j δ b BMO M 2 ( T k , 2 ( f ) ) ( x ˜ ) C b BMO M 2 ( T k , 2 ( f ) ) ( x ˜ ) .

Thus

I 2 C | Q | Q k = 1 m F t k , 1 M ( b b 2 Q ) χ ( 2 Q ) c F t k , 2 ( f ) ( x ) F t k , 1 M ( b b 2 Q ) χ ( 2 Q ) c F t k , 2 ( f ) ( x 0 ) d x C b BMO k = 1 m M 2 ( T k , 2 ( f ) ) ( x ˜ ) .

These complete 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 ( T b ( f ) ) δ # ( x ) M ( g ) ( x ) d x C b BMO k = 1 m R n M 2 ( T k , 2 ( f ) ) ( x ) M ( g ) ( x ) d x C b BMO k = 1 m M 2 ( T k , 2 ( f ) ) L p ( ) M ( g ) L p ( ) C b BMO k = 1 m T k , 2 ( f ) L p ( ) M ( g ) L p ( ) C b BMO f L p ( ) g L p ( ) .

Thus, by Lemma 8,

T b ( f ) L p ( ) f L p ( ) .

This completes the proof of Theorem 2. □

4 Applications

In this section we shall apply Theorems 1 and 2 of the paper to some particular operators such as Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Application 1 Littlewood-Paley operator.

Fixed ε>0. Let ψ be a fixed function which satisfies:

  1. (1)

    R n ψ(x)dx=0,

  2. (2)

    |ψ(x)|C ( 1 + | x | ) ( n + 1 ) ,

  3. (3)

    |ψ(x+y)ψ(x)|C|y | ε ( 1 + | x | ) ( n + 1 + ε ) when 2|y|<|x|.

Let ψ t (x)= t n ψ(x/t) for t>0 and F t (f)(x)= R n f(y) ψ t (xy)dy. The Littlewood-Paley operator is defined by (see [18])

g ψ (f)(x)= ( 0 | F t ( f ) ( x ) | 2 d t t ) 1 / 2 .

Set H to be the space

H= { h : h = ( 0 | h ( t ) | 2 d t / t ) 1 / 2 < } .

Let b be a locally integrable function on R n . The Toeplitz-type operator related to the Littlewood-Paley operator is defined by

g ψ b (f)(x)= ( 0 | F t b ( f ) ( x ) | 2 d t t ) 1 / 2 ,

where

F t b = k = 1 m F t k , 1 M b F t k , 2 ,

F t k , 1 are F t or ±I (the identity operator), T k , 2 = F t k , 2 are the bounded linear operators on L p ( R n ) for 1<p< and k=1,,m, M b (f)=bf. Then, for each fixed x R n , F t b (f)(x) may be viewed as the mapping from [0,+) to H, and it is clear that

g ψ b (f)(x)= F t b ( f ) ( x ) , g ψ (f)(x)= F t ( f ) ( x ) .

It is easy to see that g ψ b satisfies the conditions of Theorems 1 and 2 (see [14, 15, 19]), thus Theorems 1 and 2 hold for g ψ b .

Application 2 Marcinkiewicz operator.

Fixed 0<γ1. Let Ω be homogeneous of degree zero on R n with S n 1 Ω( x )dσ( x )=0. Assume that Ω Lip γ ( S n 1 ). Set F t (f)(x)= | x y | t Ω ( x y ) | x y | n 1 f(y)dy. The Marcinkiewicz operator is defined by (see [20])

μ Ω (f)(x)= ( 0 | F t ( f ) ( x ) | 2 d t t 3 ) 1 / 2 .

Set H to be the space

H= { h : h = ( 0 | h ( t ) | 2 d t / t 3 ) 1 / 2 < } .

Let b be a locally integrable function on R n . The Toeplitz-type operator related to the Marcinkiewicz operator is defined by

μ Ω b (f)(x)= ( 0 | F t b ( f ) ( x ) | 2 d t t 3 ) 1 / 2 ,

where

F t b = k = 1 m F t k , 1 M b F t k , 2 ,

F t k , 1 are F t or ±I (the identity operator), T k , 2 = F t k , 2 are the bounded linear operators on L p ( R n ) for 1<p< and k=1,,m, M b (f)=bf. Then it is clear that

μ Ω b (f)(x)= F t b ( f ) ( x ) , μ Ω (f)(x)= F t ( f ) ( x ) .

It is easy to see that μ Ω b satisfies the conditions of Theorems 1 and 2 (see [14, 15, 20]), thus Theorems 1 and 2 hold for μ Ω b .

Application 3 Bochner-Riesz operator.

Let δ>(n1)/2, F t δ (f ) ( ˆ ξ)= ( 1 t 2 | ξ | 2 ) + δ f ˆ (ξ) and B t δ (z)= t n B δ (z/t) for t>0. The maximal Bochner-Riesz operator is defined by (see [17])

B δ , (f)(x)= sup t > 0 | F t δ (f)(x)|.

Set H to be the space H={h:h= sup t > 0 |h(t)|<}. Let b be a locally integrable function on R n . The Toeplitz-type operator related to the maximal Bochner-Riesz operator is defined by

B δ , b (f)(x)= sup t > 0 | B δ , t b (f)(x)|,

where

B δ , t b = k = 1 m F t k , 1 M b F t k , 2 ,

F t k , 1 are F t or ±I (the identity operator), T k , 2 = F t k , 2 are the bounded linear operators on L p ( R n ) for 1<p< and k=1,,m, M b (f)=bf. Then

B δ , b (f)(x)= B δ , t b ( f ) ( x ) , B δ (f)(x)= B t δ ( f ) ( x ) .

It is easy to see that B δ , b satisfies the conditions of Theorems 1 and 2 (see [14, 15]), thus Theorems 1 and 2 hold for B δ , b .

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References

  1. Garcia-Cuerva J, Rubio de Francia JL North-Holland Math. 116. In Weighted Norm Inequalities and Related Topics. Elsevier, Amsterdam; 1985.

    Google Scholar 

  2. Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.

    Google Scholar 

  3. Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611–635. 10.2307/1970954

    Article  MathSciNet  Google Scholar 

  4. Pérez C: Endpoint estimate for commutators of singular integral operators. J. Funct. Anal. 1995, 128: 163–185. 10.1006/jfan.1995.1027

    Article  MathSciNet  Google Scholar 

  5. Pérez C, Trujillo-Gonzalez R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672–692. 10.1112/S0024610702003174

    Article  Google Scholar 

  6. 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.2000.7402

    Article  MathSciNet  Google Scholar 

  7. Lin Y, Lu SZ: Toeplitz type operators associated to strongly singular integral operator. Sci. China Ser. A 2006, 36: 615–630.

    Google Scholar 

  8. Lu SZ, Mo HX: Toeplitz type operators on Lebesgue spaces. Acta Math. Sci. Ser. B 2009, 29(1):140–150.

    Article  MathSciNet  Google Scholar 

  9. Cruz-Uribe D, Fiorenza A, Neugebauer CJ:The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 2003, 28: 223–238.

    MathSciNet  Google Scholar 

  10. Diening L:Maximal function on generalized Lebesgue spaces L p ( ) . Math. Inequal. Appl. 2004, 7: 245–253.

    MathSciNet  Google Scholar 

  11. 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.

    MathSciNet  Google Scholar 

  12. Karlovich AY, Lerner AK:Commutators of singular integral on generalized L p spaces with variable exponent. Publ. Mat. 2005, 49: 111–125.

    Article  MathSciNet  Google Scholar 

  13. Nekvinda A:Hardy-Littlewood maximal operator on L p ( x ) ( R n ). Math. Inequal. Appl. 2004, 7: 255–265.

    MathSciNet  Google Scholar 

  14. Liu LZ: The continuity of commutators on Triebel-Lizorkin spaces. Integral Equ. Oper. Theory 2004, 49: 65–76. 10.1007/s00020-002-1186-8

    Article  Google Scholar 

  15. Liu LZ: Triebel-Lizorkin space estimates for multilinear operators of sublinear operators. Proc. Indian Acad. Sci. Math. Sci. 2003, 113: 379–393. 10.1007/BF02829632

    Article  MathSciNet  Google Scholar 

  16. Lerner AK: Weighted norm inequalities for the local sharp maximal function. J. Fourier Anal. Appl. 2004, 10: 465–474.

    Article  MathSciNet  Google Scholar 

  17. Lu SZ: Four Lectures on Real Hp$H^{p}$ Spaces. World Scientific, River Edge; 1995.

    Book  Google Scholar 

  18. Torchinsky A Pure and Applied Math. 123. In Real Variable Methods in Harmonic Analysis. Academic Press, New York; 1986.

    Google Scholar 

  19. Liu LZ: Boundedness for multilinear Littlewood-Paley operators on Triebel-Lizorkin spaces. Methods Appl. Anal. 2004, 10(4):603–614.

    Google Scholar 

  20. Torchinsky A, Wang S: A note on the Marcinkiewicz integral. Colloq. Math. 1990, 60/61: 235–243.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The present investigation was supported by the National Natural Science Foundation under Grants 11226088, 11301008 and 11101053, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002 of the People’s Republic of China.

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Yuan, XS., Wang, ZG. Boundedness of Toeplitz-type operator associated to general integral operator on L p spaces with variable exponent. J Inequal Appl 2013, 344 (2013). https://doi.org/10.1186/1029-242X-2013-344

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