Ali Journal of Inequalities and Applications 2014, 2014:269 http://www.journalofinequalitiesandapplications.com/content/2014/1/269

RESEARCH

Open Access

Lp Estimates for Marcinkiewicz integral operators and extrapolation Mohammed Ali* *

Correspondence: [email protected] Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Abstract In this article, we establish Lp estimates for parametric Marcinkiewicz integral operators with rough kernels. These estimates and extrapolation arguments improve and extend some known results on Marcinkiewicz integrals. MSC: Primary 40B20; secondary 40B15; 40B25 Keywords: Lp boundedness; Marcinkiewicz integrals; rough kernels; extrapolation

1 Introduction Throughout this article, let Sn– , n ≥  be the unit sphere in Rn which is equipped with the normalized Lebesgue surface measure dσ = dσ (·). Also, we let u = u/|u| for u ∈ Rn \ {} and p denote the exponent conjugate to p; that is /p + /p = . Let K,h = (u )h(|u|)|u|ρ–n , where ρ = a + ib (a, b ∈ R with a > ), h is a measurable function on R+ and  is a function on Sn– with  ∈ L (Sn– ) and  (u) dσ (u) = .

(.)

Sn–

For a suitable mapping φ : R+ → R, a measurable function h on Sn– and an  satisfying (.), we define the Marcinkiewicz integral operator Mρ,h,φ for f ∈ S (Rn ) by 

Mρ,φ,h f (x) =



  t –ρ 

∞

ρ

 /  dt   f x – φ |u| u K,h du . t |u|≤t 



ρ

ρ

If φ(t) = t, we denote M,φ,h by M,h . The operators M,φ,h have their roots in the classical Marcinkiewicz integral operators M, which were introduced by Stein in [] in which he studied the Lp Boundedness of M, when  ∈ Lipα (Sn– ) ( < α ≤ ). More precisely, he proved that M, is of type (p, p) for  < p ≤  and of weak type (, ). The Marcinkiewicz integral operators play an important role in many fields in mathematics such as Poisson integrals, singular integrals and singular Radon transforms. They have received much attention from many authors (we refer the readers to [–], as well as [], and the references therein). Before introducing our results, let us recall the definition of the space L(log L)α (Sn– ) and the definition of the block space Bq(,ν) (Sn– ), which are related to our work. For α > , ©2014 Ali; licensee Springer. 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 cited.

Ali Journal of Inequalities and Applications 2014, 2014:269 http://www.journalofinequalitiesandapplications.com/content/2014/1/269

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let L(log L)α (Sn– ) denote the class of all measurable functions  on Sn– that satisfy  L(log L)α (Sn– ) =

Sn–

     (x) logα  + (x) dσ (x) < ∞.

The special class of block spaces Bq(,ν) (Sn– ) (for ν > – and q > ) was introduced by Jiang and Lu in the study of the singular integral operators (see []), and it is defined as follows: A  q-block on Sn– is an Lq function b(x) that satisfies (i) supp(b) ⊆ I, (ii) bLq (Sn– ) ≤ |I|–/q , where |I| = σ (I) and I = B(x , δ) = {x ∈ Sn– : |x –x | < δ} is a cap on Sn– for some x ∈ Sn– and δ ∈ (, ]. The block space Bq(,ν) (Sn– ) is defined by Bq(,ν)

 ∞   n–      n– (,ν) Cμ bμ with Mq {Cμ } < ∞ , S = ∈L S := μ=

where each Cμ is a complex number; each bμ is a q-block supported on a cap Iμ on Sn– , and ∞      |Cμ |  + log(ν+) |Iμ |– . Mq(,ν) {Cμ } = μ=

Define B(,ν) (Sn– ) = inf{Mq(,ν) ({Cμ }) :  = q

∞

μ= Cμ bμ },

where the infimum is taken

over the whole q-block decomposition of , then  · B(,ν) (Sn– ) is a norm on the space q

Bq(,ν) (Sn– ), and the space (Bq(,ν) (Sn– ),  · B(,ν) (Sn– ) ) is a Banach space. q Employing the ideas of [], Wu [] pointed out that for q >  and for ν > ν > –,

       ) Sn– . Lr Sn– ⊂ Bq(,ν ) Sn– ⊂ B(,ν q

r> ρ

The study of parametric Marcinkiewicz integral operator M,h was initiated by Hörρ mander in [] in which he showed that M, is bounded on Lp (Rn ) for  < p < ∞ when ρ n– ρ >  and  ∈ Lipα (S ) with α > . However, the authors of [] proved that M, is bounded on Lp (Rn ) for  < p < ∞ when Re(ρ) >  and  ∈ Lipα (Sn– ) with  < α ≤ . This ρ result was improved in [] in which the authors established that M,h is bounded on L (Rn ) if  ∈ L(log L)(Sn– ) and h ∈ γ (R+ , dtt ), where γ (R+ , dtt ) is the collection of all

R measurable functions h : [, ∞) → C satisfying h γ (R+ , dt ) = supR∈Z (  |h(t)|γ dtt )/γ < ∞. t

On the other hand, Al-Qassem and Al-Salman in [] found that if  ∈ B(,–/) (Sn– ) q with q > , then M, is bounded on Lp (Rn ) for  < p < ∞. Furthermore, they proved that ν = –/ is sharp on L (Rn ). Walsh in [] found that M, is bounded on L (Rn ) if  ∈ L(log L)/ (Sn– ), and the exponent / is the best possible. However, under the same conditions, Al-Salman et al. in [] improved this result for any  < p < ∞. (Sn– ) for some q >  and h ∈ γ (R+ , dtt ) Recently, it was proved in [] that if  ∈ B(,–/) q for some  < γ ≤ , then M,φ,h is bounded on Lp (Rn ) for any p satisfying |/p – /| < min{/, /γ  }, where φ is C  ([, ∞)), a convex and increasing function with φ() = . Very recently, Al-Qassem and Pan established in [] that if  ∈ Lq (Sn– ) for some q ∈ (, ] and ρ h ∈ γ (R+ , dtt ) for some  < γ ≤ , then M,P ,h is bounded on Lp (Rn ) for any p satisfying  |/p – /| < min{/, /γ }, where P (x) = (P (x), P (x), . . . , Pm (x)) is a polynomial mapping and each Pi is a real valued polynomial on Rn .

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Our main concern in this work is in dealing with Marcinkiewicz operators under very weak conditions on the singular kernels. In fact, we establish certain estimates for Mρ,φ,h , and then we apply an extrapolation argument to obtain and improve some results on Marcinkiewicz integrals. Our approach in this work provides an alternative way in dealing with such kind of operators. Our main result is described in the following theorem. Theorem . Let  ∈ Lq (Sn– ) for some  < q ≤ , h ∈ Lγ (R+ , dtt ) for some γ > . Suppose that φ is C  ([, ∞)), a convex and increasing function with φ() = . Then for any f ∈ Lp (Rm ) with p satisfying |/p – /| < min{/, /γ  }, there exists a constant Cp (independent of , h, γ , and q) such that   ρ –/  M hLγ (R+ , ds ) Lq (Sn– ) f Lp (Rm ) , ,φ,h f Lp (Rm ) ≤ Cp A(γ )(q – ) s

where A(γ ) =

 γ /

if γ > , if  < γ ≤ .

(γ – )–/

Throughout this paper, the letter C denotes a bounded positive constant that may vary at each occurrence but independent of the essential variables.

2 Definitions and lemmas In this section, we present and establish some lemmas used in the sequel. Let us start this section by introducing the following. Definition . Let θ ≥ . For a suitable function φ defined on R+ , a measurable function h : R+ → C and  : Sn– → R, we define the family of measures {σ,φ,h,t : t ∈ R+ } and the ∗ and M,φ,h,θ on Rm by corresponding maximal operators σ,φ,h 



      (u ) f φ |u| u h |u| du, |u|n–ρ Rm /t≤|u|≤t   ∗ f (x) = sup |σ,φ,h,t | ∗ f (x), σ,φ,h f dσ,φ,h,t = t –ρ

t∈R+

M,φ,h,θ f (x) = sup k∈Z



  |σ,φ,h,t | ∗ f (x) dt , t

θ k+  θk

where |σ,φ,h,t | is defined in the same way as σ,φ,h,t , but with replacing , h by ||, |h|, respectively. We write σ  for the total variation of σ . In order to prove Theorem ., it suffices to prove the following lemmas. Lemma . Let θ ≥ ,  ∈ Lq (Sn– ) for some q >  and h ∈ Lγ (R+ , dss ) for some γ > . Suppose that φ is C  ([, ∞)), a convex and increasing function with φ() = . Then there are constants C and α with  < α < q  such that σ,φ,h,t  ≤ C;  θ k+       –α σˆ ,φ,h,t (ξ ) dt ≤ C(ln θ )ξ φ(θ) k–  q γ  h Lq (Sn– ) ; Lγ (R+ , ds t s ) θk  θ k+       α σˆ ,φ,h,t (ξ ) dt ≤ C(ln θ )ξ φ(θ) k+  q γ  h Lq (Sn– ) Lγ (R+ , ds t s ) θk hold for all k ∈ Z. The constant C is independent of k, ξ and φ.

(.) (.) (.)

Ali Journal of Inequalities and Applications 2014, 2014:269 http://www.journalofinequalitiesandapplications.com/content/2014/1/269

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Proof As Lq (Sn– ) ⊆ L (Sn– ) for q ≥ , it is enough to prove this lemma for  < q ≤ . By Hölder’s inequality, we get   σˆ ,φ,h,t (ξ ) ≤



  h(s) 

t  t

–iφ(s)x·ξ

e

Sn–

 t    ≤ hLγ (R+ , ds ) s   t

  ds (x) dσ (x) s –iφ(s)ξ ·x

e

Sn–

γ  /γ   ds (x) dσ (x) . s

Let us first consider the case  < γ ≤ . By a change of variable, we obtain   ) σˆ ,φ,h,t (ξ ) ≤ h γ + ds (–/γ L (R , ) L (Sn– ) s

(–/γ  )

 t      

≤ hLγ (R+ , ds ) L (Sn– ) s

where J(ξ , x, y) = 

s

Yt (s) =



–iφ(ts)ξ ·(x–y) ds . / e s

e–iφ(tw)ξ ·(x–y) dw,

t

 /γ   ds e–iφ(s)ξ ·x (x) dσ (x) s Sn– /γ 

Sn– ×Sn–

Write J(ξ , x, y) =

(x)(y)J(ξ , x, y) dσ (x) dσ (y)



ds  / Yt (s) s ,

,

where

/ ≤ w ≤ s ≤ .

/

By the conditions on φ and the mean value theorem we have  φ(tw) φ(t/) d  φ(tw) = tφ  (tw) ≥ ≥ dw w s

for / ≤ w ≤ s ≤ .

Hence, by Van der Corput’s lemma, |Yt (s)| ≤ s|φ(t/)ξ |– |ξ  · (x – y)|– , and then by integration by parts, we conclude       J(ξ , x, y) ≤ C φ(t/)ξ – ξ  · (x – y)– . Combining the last estimate with the trivial estimate |J(ξ , x, y)| ≤ C, and choosing  < αq < , we get       J(ξ , x, y) ≤ C φ(t/)ξ –α ξ  · (x – y)–α , which leads to  –α    ) (/γ  ) σˆ ,φ,h,t (ξ ) ≤ C φ(t/)ξ  γ  h γ + ds (–/γ Lq (Sn– ) L (R , ) L (Sn– )  ×

Sn– ×Sn–

s

/q γ 

    ξ · (x – y)–αq dσ (x) dσ (y)

.

By the assumption of φ, and since the last integral is finite, we obtain 

     –α σˆ ,φ,h,t (ξ ) dt ≤ C(ln θ )ξ φ(θ) k–  γ  h Lq (Sn– ) . Lγ (R+ , ds t s )

θ k+ 

θk

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For the case γ > , we use Hölder’s inequality to obtain   σˆ ,φ,h,t (ξ ) ≤ h γ + ds L (R , ) s

 ≤ hLγ (R+ , ds ) s

 ×

  

 /  ds e–iφ(s)ξ ·x (x) dσ (x) s Sn–

 t      t

(x)(y) Sn– ×Sn–

e–iφ(st)ξ ·x eiφ(st)ξ ·y

 / ds dσ (x) dσ (y) . s

By this, Van der Corput’s lemma, and the above procedure, we obtain    –α  σˆ ,φ,h,t (ξ ) ≤ CLq (Sn– ) h γ + ds ξ φ(t/) q γ  , L (R , ) s

and therefore 

     –α σˆ ,φ,h,t (ξ ) dt ≤ C(ln θ )ξ φ(θ) k–  q γ  h Lq (Sn– ) . Lγ (R+ , ds t s )

θ k+ 

θk

The estimate in (.) can be proved by using the cancellation property of . By a change of variable, we have       –iφ(ts)ξ ·x   ds e σˆ ,φ,h,t (ξ ) ≤ – (x)h(st) dσ (x)  n– s S       h(st)φ(ts) ds . ≤ |ξ |L (Sn– )  s  Since φ(t) is increasing and

 

< s < , we obtain

    σˆ ,φ,h,t (ξ ) ≤ CLq (Sn– ) h γ + ds ξ φ(t), L (R , ) s

which when combined with the trivial estimate |σˆ ,φ,h,t (ξ )| ≤ (ln ), we derive 

   α σˆ ,φ,h,t (ξ ) dt ≤ C(ln θ )ξ φ(θ)k+  q γ  h Lq (Sn– ) . Lγ (R+ , ds t s )

θ k+ 

θk

The proof is complete.



Following a similar argument to the one used in [, Lemma .], we achieve the following lemma. Lemma . Suppose that φ is given as in Lemma .. Let Mφ,y f be the maximal function of f in the direction y defined by       t  Mφ,y f (x) = sup  f x – φ(r)y dr. t∈R+ t t/ Then there exists a constant Cp such that   Mφ,y (f ) p m ≤ Cp f Lp (Rm ) L (R ) for any f ∈ Lp (Rm ) with  < p ≤ ∞.

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Proof By a change of variable, we get 

Mφ,y f (x) ≤ sup

t∈R+

 f (x – ry)

φ(t) 

φ(t/)

 dr . φ – (r)φ  (φ – (r))

Since the function φ – (r)φ (φ – (r)) is non-negative, decreasing and its integral over [φ(t/), φ(t)] is equal to ln(), then by [, Lemma .] we obtain

Mφ,y (f ) ≤ C My f (x),

L where My f (x) = supL∈R L  |f (x – ry)| dr is the Hardy-Littlewood maximal function of f in the direction of y. By this, and since My (f ) is bounded in Lp (Rm ) with bounded independent of y, we obtain our desired result.  Lemma . Let  ∈ Lq (Sn– ) for some  < q ≤  and h ∈ Lγ (R+ , dss ) for some γ > . Assume ∗ and φ are given as in Definition . and Lemma ., respectively. Then for any that σ,φ,h p m f ∈ L (R ) with γ  < p ≤ ∞, there exists a constant Cp (independent of , h and f ) such that   ∗  σ ,φ,h f (x) Lp (Rm ) ≤ Cp hLγ (R+ , ds ) Lq (Sn– ) f Lp (Rm ) .

(.)

s

Proof By Hölder’s inequality, we have   |σ,φ,h | ∗ f (x) ≤ h γ + ds /γ L (R , ) L (Sn– ) s

× sup

 t  t 

t∈R+

     (y)f x – φ(s)y γ dσ (y) ds n– s S

/γ  .

Using Minkowski’s inequality for integrals gives  

 ∗ σ

,φ,h f (x) Lp (Rm )

/γ

≤ ChLγ (R+ , ds ) L (Sn– ) s

 ×

Sn–

    (y) Mφ,y |f |γ  

p/γ 

L

 (Rm )

/γ  dσ (y)

By using Hölder’s inequality plus Lemma ., we finish the proof.

. 

Lemma . Let h ∈ Lγ (R+ , dtt ) for some  < γ ≤ ,  ∈ Lq (Sn– ) for some  < q ≤  and   θ = q γ . Assume that {σ,φ,h,t , t ∈ R+ } and φ are given as in Definition . and Lemma ., respectively. Then for any p satisfying |/p – /| < /γ  , there is a positive constant Cp such that      k∈Z

≤ Cp

θ k+

θk

     |σ,φ,h,t ∗ gk | p m t L (R )  dt

 /  hLγ (R+ , ds ) Lq (Sn– )    s    |g | k  p / [(q – )(γ – )] k∈Z

holds for arbitrary functions {gk (·), k ∈ Z} on Rm .

L (Rm )

Ali Journal of Inequalities and Applications 2014, 2014:269 http://www.journalofinequalitiesandapplications.com/content/2014/1/269

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Proof We employ some ideas from [, ], and []. By Schwarz’s inequality, we obtain |σ,φ,h,t ∗ gk | ≤ Ch ×

γ Lq (Sn– ) Lγ (R+ , ds s )

       gk x – φ(s)y  (y)h(s)–γ dσ (y) ds . s Sn–

 t   t

(.)

γ Let us first prove this lemma for the case  ≤ p < –γ . By duality, there is a non-negative (p/) m (R ) with ψL(p/) (Rm ) ≤  such that function ψ ∈ L

     k∈Z

θ k+

|σ,φ,h,t ∗ gk |

θk

dt t

/    p





=

L (Rm )

Rm k∈Z

 σ,φ,h,t ∗ gk (x) dt ψ(x) dx. t

θ k+ 

θk

By this, (.), and a change of variable we derive      k∈Z

/    |σ,φ,h,t ∗ gk | p m t L (R )         (x) g M,φ,|h|–γ ,θ ψ(–x) dx. q n– k L (S ) + dt

θ k+

 dt

θk

≤ Ch

γ Lγ (R , t )

Rm

k∈Z

γ  ) , then by Since h ∈ Lγ (R+ , dtt ), then |h(·)|–γ ∈ Lγ /(–γ ) (R+ , dtt ), and since ( p ) > ( –γ Lemma ., Hölder’s inequality, and the same arguments that Stein and Wainger used in [], we obtain

     k∈Z

/    |σ,φ,h,t ∗ gk | p m t L (R )

θ k+

 dt

θk



 γ ≤ Ch γ + dt Lq (Sn– )   L (R , t )

k∈Z

/    |gk | p 

L (Rm )



 γ ≤ C ln(θ)h γ + dt Lq (Sn– )   L (R , t ) ≤ Cp

k∈Z

  M,φ,|h|–γ ,θ ψ(–x)

/    |gk | p 

L (Rm )

 /  Lq (Sn– )       |gk |  p (q – )(γ – )



L(p/) (Rm )

  ∗ σ ,φ,|h|–γ ,θ ψ(–x)



L(p/) (Rm )

hγ

L (R+ , dt t )

.

L (Rm )

k∈Z

For the case γγ– < p < , by the duality, there are functions ζ = ζk (x, t) defined on Rm × R+ with ζk L ([θ k ,θ k+ ], dt ) l Lp (Rm ) ≤  such that t

     k∈Z

≤ Cp

θ k+

θk

|σ,φ,h,t ∗ gk |

dt t

     

Lp (Rm )

 /  (ϒ(ζ ))/ Lp (Rm )      |g | k  p / [(q – )(γ – )] k∈Z

where ϒ(ζ ) =

 k∈Z

 σ,φ,h,t ∗ ζk (x, t) dt . t

θ k+  θk

L (Rm )

,

(.)

Ali Journal of Inequalities and Applications 2014, 2014:269 http://www.journalofinequalitiesandapplications.com/content/2014/1/269

As

p 

Page 8 of 10

> , we obtain, by applying the above procedure,   ϒ(ζ )

 L(p /) (Rm )

≤

 

 γ Ch γ + dt Lq (Sn– )   L (R , t )

k∈Z

   ζk (·, t) dt  t L(p /) (Rm )

θ k+  θk

  × σ ∗ ,φ,|h|–γ ,θ (ς)L(p /) (Rm ) ≤ ChLγ (R+ , dt ) Lq (Sn– ) ,

(.)

t





where ς is a function in L(p /) (Rm ) with ςL(p /) (Rm ) ≤ . Thus, by (.) and (.), our estimate holds for

γ γ –

≤ p < ; and therefore the proof of Lemma . is complete.



In the same manner, we prove the following lemma. Lemma . Let h ∈ Lγ (R+ , dtt ) for some γ ≥ ,  ∈ Lq (Sn– ) for some  < q ≤  and θ =   q γ . Assume that {σ,φ,h,t , t ∈ R+ } and φ are given as in Definition . and Lemma ., respectively. Then for any p satisfying  < p < ∞, there exists a constant Cp such that      k∈Z

θ k+

θk

     |σ,φ,h,t ∗ gk | p m t L (R )  dt

 /  γ / hLγ (R+ , ds ) Lq (Sn– )    s    ≤ Cp |g | k  p (q – )/

L (Rm )

k∈Z

holds for arbitrary functions {gk (·), k ∈ Z} on Rm .

3 Proof of the main result We prove Theorem . by applying the same approaches that Al-Qassem and Al-Salman [] as well as Fan and Pan [] used. Let us first assume that h ∈ Lγ (R+ , dtt ) for some  < γ ≤ ; and φ is C  ([, ∞)), a convex and increasing function with φ() = . By Minkowski’s inequality, we get 

Mρ,φ,h f (x)

= 

≤

   

∞ ∞ 

 t –ρ

k=

∞  ∞   

k=

a = a  –

t –ρ 





 /  dt      f x – φ |u| u K,h du  t –k– –k  t<|u|≤ t

 /  dt    f x – φ |u| u K,h du t –k– t<|u|≤–k t 

 σ,φ,h,t ∗ f (x) dt t

∞ 

/ .

 

(.)

Take θ = q γ ; and for k ∈ Z, let {k,θ }∞ –∞ be a smooth partition of unity in (, ∞) adapted –k– –k+ ,θ ]. More precisely, we require the following: to the interval Ik,θ = [θ k,θ ∈ C ∞ ,

 ≤ k,θ ≤ ,

k

supp k,θ ⊆ Ik,θ

and

k,θ (t) = ,

 s   d k,θ (t)  Cs    dt s  ≤ t s ,

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 where Cs is independent of θ . Let  k,θ (ξ ) = k,θ (|ξ |). Decompose σ,φ,h,t ∗ f (x) =  Y (x, t), where j∈Z ,φ,h,j,θ Y,φ,h,j,θ (x, t) =



σ,φ,h,t ∗ k+j,θ f (x)χ[θ k ,θ k+ ) (t).

k∈Z

∞ Define S,φ,h,j,θ f (x) = (  |Y,φ,h,j,θ (x, t)| dtt )/ . Then for any f ∈ S (Rm ),

Mρ,φ,h f (x) ≤

a S,φ,h,j,θ (f ). –  j∈Z

(.)

a

Let us first compute the L -norm of S,φ,h,θ ,j (f ). By using Plancherel’s theorem and Lemma ., we obtain   S,φ,h,θ ,j (f ) L

   dt  ˆf (ξ ) dξ   ≤ σˆ ,φ,h,θ ,t (ξ ) (Rm ) t θk k∈Z k+j,θ    ˆf (ξ ) dξ ≤ Cp (ln θ )hLγ (R+ , dt ) Lq (Sn– ) –α|j| 



θ k+ 

t

k∈Z

k+j,θ

≤ Cp (ln θ )hLγ (R+ , dt ) Lq (Sn– ) –α|j| f L (Rm ) , t

where k,θ = {ξ ∈ Rm : |ξ | ∈ Ik,θ }. Thus, hLγ (R+ , dt ) Lq (Sn– ) –α|j|   t S,φ,h,θ ,j (f )  m ≤ Cp   f L (Rm ) . L (R ) [(q – )(γ – )]/

(.)

Applying the Littlewood-Paley theory and Theorem  along with the remark that follows its statement in [, p.], plus using Lemma ., we see that   S,φ,h,θ ,j (f )

Lp (Rm )

≤ Cp

hLγ (R+ , dt ) Lq (Sn– ) t

[(q – )(γ – )]/

f Lp (Rm )

(.)

holds for |/p – /| < /γ  . By interpolation between (.) and (.) we obtain   –α|j| hLγ (R+ , dt ) Lq (Sn– ) t S,φ,h,θ ,j (f ) p m ≤ Cp   f Lp (Rm ) . L (R ) [(q – )(γ – )]/

(.)

Consequently, by (.) and (.), we get our result for the case h ∈ Lγ (R+ , dtt ) for some  < γ ≤ . The proof of our theorem for the case h ∈ Lγ (R+ , dtt ) for some γ ≥  is obtained by following the above argument, except that we need to invoke Lemma . instead of Lemma .. Therefore, the proof of Theorem . is complete.

4 Further results The power of our theorem is in applying the extrapolation method on it (see []). In particular, Theorem . and extrapolation lead to the following theorem. Theorem . Suppose that h ∈ Lγ (R+ , dtt ) for some γ >  and  satisfies (.). Let φ be C  ([, ∞)), a convex and increasing function with φ() = .

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(i) If  ∈ B(,–/) (Sn– ) for some q > , then q  ρ M

,φ,h f

 

Lp (Rm )

  ≤ Cp A(γ )hLγ (R+ , dt ) f Lp (Rm )  + B(,–/) (Sn– ) q

t

for |/p – /| < min{/, /γ  }. (ii) If  ∈ L(log L)/ (Sn– ), then  ρ M

,φ,h f

 

Lp (Rm )

  ≤ Cp A(γ )hLγ (R+ , dt ) f Lp (Rm )  + L(log L)/ (Sn– ) t

for |/p – /| < min{/, /γ  }, where A(γ ) =

 γ / (γ – )–/

if γ > , if  < γ ≤ .

We point out that the Lp boundedness of M,φ,h was obtained in [] if  ∈ B(,–/) (Sn– ) q  p for some q > , and the L boundedness ( < p < ∞) of M, was investigated in [] if  ∈ L(log L)/ (Sn– ). Competing interests The author declares that they have no competing interests. Acknowledgements The author would like to thank Dr. Hussain Al-Qassem for his suggestions and comments on this note. Received: 2 October 2013 Accepted: 27 June 2014 Published: 23 Jul 2014 References 1. Stein, E: On the functions of Littlewood-Paley, Lusin and Marcinkiewicz. Trans. Am. Math. Soc. 88, 430-466 (1958) 2. Al-Qassem, H, Al-Salman, A: A note on Marcinkiewicz integral operators. J. Math. Anal. Appl. 282, 698-710 (2003) 3. Al-Qassem, H, Pan, Y: Lp Estimates for singular integrals with kernels belonging to certain block spaces. Rev. Mat. Iberoam. 18, 701-730 (2002) 4. Al-Salman, A, Al-Qassem, H, Cheng, L, Pan, Y: Lp Bounds for the function of Marcinkiewicz. Math. Res. Lett. 9, 697-700 (2002) 5. Ding, Y: On Marcinkiewicz integral. In: Proc. of the Conference Singular Integrals and Related Topics III, Osaka, Japan (2001) 6. Ding, Y, Fan, D, Pan, Y: On the Lp boundedness of Marcinkiewicz integrals. Mich. Math. J. 50, 17-26 (2002) 7. Walsh, T: On the function of Marcinkiewicz. Stud. Math. 44, 203-217 (1972) 8. Jiang, Y, Lu, S: A class of singular integral operators with rough kernel on product domains. Hokkaido Math. J. 24, 1-7 (1995) 9. Keitoku, M, Sato, E: Block spaces on the unit sphere in Rn . Proc. Am. Math. Soc. 119, 453-455 (1993) 10. Wu, H: A rough multiple Marcinkiewicz integral along continuous surfaces. Tohoku Math. J. 59, 145-166 (2007) 11. Hörmander, L: Estimates for translation invariant operators in Lp space. Acta Math. 104, 93-139 (1960) 12. Sakamota, M, Yabuta, K: Boundedness of Marcinkiewicz functions. Stud. Math. 135, 103-142 (1999) 13. Ding, Y, Lu, S, Yabuta, K: A problem on rough parametric Marcinkiewicz functions. J. Aust. Math. Soc. 72, 13-21 (2002) 14. Al-Qassem, H: Lp Estimates for rough parametric Marcinkiewicz integrals. SUT J. Math. 40, 117-131 (2004) 15. Al-Qassem, H, Pan, Y: On certain estimates for Marcinkiewicz integrals and extrapolation. Collect. Math. 60, 123-145 (2009) 16. Al-Qassem, H, Cheng, L, Pan, Y: On the boundedness of a class of rough maximal operators on product spaces. Hokkaido Math. J. 40, 1-13 (2011) 17. Fan, D, Pan, Y: Singular integral operators with rough kernels supported by subvarieties. Am. J. Math. 119, 799-839 (1997) 18. Stein, E: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993) 19. Stein, E: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

10.1186/1029-242X-2014-269 Cite this article as: Ali: Lp Estimates for Marcinkiewicz integral operators and extrapolation. Journal of Inequalities and Applications 2014, 2014:269