Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

RESEARCH

Open Access

Multilinear fractional integral operators on generalized weighted Morrey spaces Yue Hu1* and Yueshan Wang2 * Correspondence: [email protected] 1 College of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454003, P.R. China Full list of author information is available at the end of the article

Abstract Let Iα ,m be multilinear fractional integral operator and let (b1 , . . . , bm ) ∈ (BMO)m . In this b and the iterated paper, the estimates of Iα ,m , the m-linear commutators Iα,m  b commutators Iα ,m on the generalized weighted Morrey spaces are established. MSC: 42B35; 42B20 Keywords: multilinear fractional integral; generalized weighted Morrey space; commutator; Muckenhoupt weight

1 Introduction and results The classical Morrey spaces were introduced by Morrey [] in , have been studied intensively by various authors, and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces. See [–] for details. Moreover, various Morrey spaces have been defined in the process of this study. Mizuhara [] introduced p the generalized Morrey space Mϕ ; Komori and Shirai [] defined the weighted Morrey p spaces Lp,κ (ω); Guliyev [] gave the concept of generalized weighted Morrey space Mϕ (ω), p which could be viewed as an extension of both Mϕ and Lp,κ (ω). The boundedness of some operators on these Morrey spaces can be seen in [–]. Let Rn be the n-dimensional Euclidean space, (Rn )m = Rn × · · · × Rn be the m-fold product space (m ∈ N), and let f = (f , . . . , fm ) be a collection of m functions on Rn . Given α ∈ (, mn) and (b , . . . , bm ) ∈ (BMO)m . We consider the multilinear fractional integral operators Iα,m defined by Iα,m (f)(x) =

 (Rn )m

f (y ) · · · fm (ym ) dy · · · dym . (|x – y | + · · · + |x – ym |)mn–α

(.)

b b The corresponding m-linear commutators Iα,m and the iterated commutators Iα,m defined by, respectively,

b  Iα,m (f )(x) =

m   i=

(Rn )m

(bi (x) – bi (yi ))

m

j= fj (yj )

(|x – y | + · · · + |x – ym |)mn–α

dy · · · dym

(.)

©2014 Hu and Wang; 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.

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 2 of 18

and 

b  Iα,m (f )(x) =

m

i= (bi (x) – bi (yi ))fi (yi ) mn–α (Rn )m (|x – y | + · · · + |x – ym |)

dy · · · dym .

(.)

As is well known, multilinear fractional integral operator was first studied by Grafakos [], subsequently, by Kenig and Stein [], Grafakos and Kalton []. In , Moen [] introduced weight function AP,q  and gave weighted inequalities for multilinear fractional integral operators; In , Chen and Wu [] obtained the weighted norm inequalities b b for the multilinear commutators Iα,m and Iα,m . More results of the weighted inequalities for multilinear fractional integral and its commutators can be found in [–]. The aim of the present paper is to investigate the boundedness of multilinear fractional integral operator and its commutator on the generalized weighted Morrey spaces. Our results can be formulated as follows.  Theorem . Let m ≥  and let  < α < mn. Suppose /p = m i= /pi , /qi = /pi – m α/mn, and /q = i= /qi = /p – α/n, ω  = (ω , . . . , ωm ) satisfy the Ap,q condition with q q ω  , . . . , ωmm ∈ A∞ , and ϕk = (ϕk , . . . , ϕkm ), k = , , satisfy the condition  s

∞

  pi pi ess infr
(.)

 m where ϕ = m  = i= ϕi , νω i= ωi . If p , . . . , pm ∈ (, ∞), then there exists a constant C inde pendent of f such that Iα,m fMϕq

q )  (νω 

≤C

m 

fi Mϕpi

i=

i

p

(ωi i )

;

(.)

If p , . . . , pm ∈ [, ∞), and min{p , . . . , pm } = , then there exists a constant C independent of f such that Iα,m fWMϕq

q )  (νω 

≤C

m  i=

fi Mϕpi

i

p

(ωi i )

.

(.)

Theorem . Let m ≥  and let  < α < mn. Suppose p , . . . , pm ∈ (, ∞) with /p = m m  = (ω , . . . , ωm ) satisfy the i= /pi , /qi = /pi – α/mn and /q = i= /qi = /p – α/n, ω m p pm Ap,q ω , and ϕ  = (ϕ  condition with ω , . . . , ωm ∈ A∞ , νω  = i k k , . . . , ϕkm ), k = , , satisfy i= the condition 

∞

s

r  + ln s

m

  pi pi ess infr
(.)

 m m where ϕ = m  = i= ϕi , νω i= ωi . If (b , . . . , bm ) ∈ (BMO) , then there exists a constant C >  independent of f such that b I (f) α,m

q q Mϕ (νω )

≤C

m  i=

bi ∗ fi Mϕpi

i

p

(ωi i )

;

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 3 of 18

and b I (f) α,m

q q Mϕ (νω )

≤C

m 

bi ∗ fi Mϕpi

p

i

i=

(ωi i )

.

(.)

2 Definitions and preliminaries A weight ω is a nonnegative, locally integrable function on Rn . Let B = B(x , rB ) denote the ball with the center x and radius rB . For any ball B and λ > , λB denotes the ball concentric with B whose radius is λ times as long. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by |E| and set weighted

measure ω(E) = E ω(x) dx. The classical Ap weight theory was first introduced by Muckenhoupt in the study of weighted Lp boundedness of Hardy-Littlewood maximal functions in []. A weight ω is said to belong to Ap for  < p < ∞, if there exists a constant C such that for every ball B ⊂ Rn , 

 |B|



 ω(x) dx B

 |B|



p–



≤ C,

ω(x)–p dx

(.)

B

where p is the dual of p such that /p + /p = . The class A is defined by replacing the above inequality with  |B|

 w(y) dy ≤ C · ess inf w(x) for every ball B ⊂ Rn . x∈B

B

(.)

A weight ω is said to belong to A∞ if there are positive numbers C and δ so that  δ |E| ω(E) ≤C ω(B) |B|

(.)

for all balls B and all measurable E ⊂ B. It is well known that 

A∞ =

Ap .

(.)

≤p<∞

We need another weight class Ap,q introduced by Muckenhoupt and Wheeden in []. A weight function ω belongs to Ap,q for  < p < q < ∞ if there is a constant C >  such that, for every ball B ⊂ Rn , 

 |B|

/q 

 q

ω(x) dx B

 |B|

 ω(x)

–p

p

dx

≤ C.

(.)

B

When p = , ω is in the class A,q with  < q < ∞ if there is a constant C >  such that, for every ball B ⊂ Rn , 

 |B|

/q 

 ω(x)q dx B

ess sup x∈B

 ω(x)

 ≤ C.

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 4 of 18

Let us recall the definition of multiple weights. For m exponents p , . . . , pm , we write p =   = (ω , . . . , ωm ), (p , . . . , pm ). Let p , . . . , pm ∈ [, ∞), /p = m i= /pi , and let q > . Given ω m  satisfies the Ap,q condition if it satisfies set νω = i= ωi . We say that ω /q  /p i    m   

νω (x)q dx ωi (x)–pi dx ≤ C. sup |B| B |B| B B i=  When pi = , ( |B|



B ωi (x)

–p i



(x) dx)/pi is understood as (infx∈B ωi (x))– .

Lemma . [, ] Let  < α < mn, and p , . . . , pm ∈ [, ∞), let /p = /q = /p – α/n. If ω  ∈ Ap,q , then q

νω ∈ Amq where νω =

–p i

and ωi

(.)

∈ Amp i

for i = , . . . , m,

m

k= /pk ,

and let

(.)

m

i= ωi .

 Lemma . [] Let m ≥ , q , . . . , qm ∈ [, ∞) and q ∈ (, ∞) with /q = m i= /qi . Assume  q qm m that ω , . . . , ωm ∈ A∞ and νω = i= ωi . Then for any ball B, there exists a constant C >  such that m   i=

ωi (x)qi dx

q/qi

 ≤C

B

νω (x)q dx.

(.)

B

Let  ≤ p < ∞, let ϕ be a positive measurable function on Rn × (, ∞), and let ω be a nonp negative measurable function on Rn . Following [], we denote by Mϕ (ω) the generalized p weighted Morrey space and the space of all functions f ∈ Lloc (ω) with finite norm  /p   p f Lp (ω,B(x,r)) , f Mϕp (w) = sup x∈Rn ,r> ϕ(x, r) w(B(x, r))

(.)

where 

f (y) p w(y) dy.

f Lp (ω,B(x,r)) = B(x,r) p

Furthermore, by WMϕ (ω) we denote the weak generalized weighted Morrey space of all p function f ∈ WMϕ (ω) for which  /p   p f WLp (ω,B(x,r)) , x∈Rn ,r> ϕ(x, r) w(B(x, r))

f WMϕp (w) = sup

(.)

where  

 f WLp (ω,B(x,r)) = sup t ω y ∈ B(x, r) : f (y) > t p . t>

λ–n

p

() If ω =  and ϕ(x, r) = r p with  < λ < n, then Mϕ (ω) = Lp,λ is the classical Morrey space. κ– p () If ϕ(x, r) = ω(B(x, r)) p , then Mϕ (ω) = Lp,κ (ω) is the weighted Morrey space.

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323



κ

Page 5 of 18

p

() If ϕ(x, r) = ν(B(x, r)) p ω(B(x, r))– p , then Mϕ (ω) = Lp,κ (ν, ω) is the two weighted Morrey space. p p () If ω = , then Mϕ (ω) = Mϕ is the generalized Morrey space.  p () If ϕ(x, r) = ω(B(x, r))– p , then Mϕ (ω) = Lp (ω). Let us recall the definition and some properties of BMO. A locally integrable function b is said to be in BMO if   b(x) – bB dx = b∗ < ∞, sup B⊂Rn |B| B where bB = |B|–



B b(y) dy.

Lemma . (John-Nirenberg inequality; see []) Let b ∈ BMO. Then for any ball B ⊂ Rn , there exist positive constants C and C such that for all λ > , 

  x ∈ B : b(x) – bB > λ ≤ C |B| exp –C λ/b∗ .

(.)

By Lemma ., it is easy to get the following. Lemma . Suppose ω ∈ A∞ and b ∈ BMO. Then for any p ≥  we have 

 ω(B)



b(x) – bB p ω(x) dx

/p ≤ Cb∗ .

(.)

B

Lemma . [] Let b ∈ BMO,  ≤ p < ∞, and r , r > . Then 

  p  b(y) – bB(x ,r ) p dy ≤ Cb∗  + ln r ,   |B(x , r )| B(x ,r ) r 



(.)

where C >  is independent of f , x , r , and r . By Lemma . and Lemma ., it is easily to prove the following results. Lemma . Suppose ω ∈ A∞ and b ∈ BMO. Then for any  ≤ p < ∞ and r , r > , we have 

 ω(B(x , r ))



b(x) – bB(x

B(x ,r )

 ,r

p ) ω(x) dx

/p

 ≤ Cb∗

 r  + ln . r

(.)

We also need the following result. Lemma . [] Let f be a real-valued nonnegative function and measurable on E. Then 

–  . = ess sup ess inf f (x) x∈E f (x) x∈E

(.)

At the end of this section, we list some known results about weighted norm inequalities for the multilinear fractional integrals and their commutators. Lemma . [] Let m ≥  and let  < α < mn. Suppose /p = /p + · · · + /pm , /q = /p – α/n, ω  = (ω , . . . , ωm ) satisfies the Ap,q condition. If p , . . . , pm ∈ (, ∞), then there exists

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 6 of 18

a constant C independent of f = (f , . . . , fm ) such that Iα,m fLq (ν q ) ≤ C ω 

m 

fi Lpi (ωpi ) .

(.)

i

i=

If p , . . . , pm ∈ [, ∞), and min{p , . . . , pm } = , then there exists a constant C independent of f such that Iα,m fWLq (ν q ) ≤ C ω 

where νω =

m 

fi Lpi (ωpi ) ,

(.)

i

i=

m

i= ωi .

Lemma . [] Let m ≥ , let  < α < mn and let (b , . . . , bm ) ∈ (BMO)m . For  < p , . . . , pm < ∞, /p = /p + · · · + /pm , and /q = /p – α/n, if ω  ∈ Ap,q , then there exists a constant C >  such that b I (f) α,m

q Lq (νω )

≤C

m 

bi ∗ fi Lpi (ωpi ) ;

(.)

bi ∗ fi Lpi (ωpi ) ,

(.)

i

i=

and b I (f) α,m

where νω =

q Lq (νω )

≤C

m 

i

i=

m

i= ωi .

3 Proof of Theorem 1.1 We first prove the following conclusions.  Theorem . Let m ≥  and let  < α < mn. Suppose /p = m i= /pi , /qi = /pi – m  = (ω , . . . , ωm ) satisfy the AP,q α/mn, and /q = i= /qi = /p – α/n, ω  condition with q q ω  , . . . , ωmm ∈ A∞ . If p , . . . , pm ∈ (, ∞), then there exists a constant C independent of f such that Iα,m fLq (ν q ,B(x ,s)) ≤ C

m 

ω 

  q ωi i B(x , s) qi

i=

 ×

∞



m 

s

i=

–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α

(.)

If p , . . . , pm ∈ [, ∞), and min{p , . . . , pm } = , then there exists a constant C independent of f such that Iα,m fWLq (ν q ,B(x ,s)) ≤ C

m 

ω 

i=

 ×

∞ s

where νω =

m

i= ωi .

  q ωi i B(x , s) qi m  i=

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr , r–α

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 7 of 18

Proof We represent fi as fi = fi + fi∞ , where fi = fi χB(x ,s) , i = , . . . , m, and χB(x ,s) denotes the characteristic function of B(x , s). Then m 

fi (yi ) =

m 

i=

i=

=



fi (yi ) + fi∞ (yi )



fα (y ) · · · fmαm (ym )

α ,...,αm ∈{,∞}

=

m 

fi (yi ) +  fα (y ) · · · fmαm (ym ),

i=

where each term of  contains at least one αi = . Since Iα,m is an m-linear operator,

 Iα,m fLq (ν q ,B(x ,s)) ≤ C Iα,m f , . . . , fm Lq (ν q ,B(x ω 

ω 



 + C Iα,m f α , . . . , f αm 

 ,s)) q

Lq (νω ,B(x ,s))

m

= J ,..., +  J α ,...,αm

(.)

and

 Iα,m fWLq (ν q ,B(x ,s)) ≤ C Iα,m f , . . . , fm WLq (ν q ,B(x ω 

ω 



 + C Iα,m f α , . . . , f αm 

m

 ,s)) q

WLq (νω ,B(x ,s))

= K ,..., +  K α ,...,αm .

(.)

Then by (.), if  < pi < ∞, i = , . . . , m, we get J ,..., ≤C

m 

fi Lpi (ωpi ,B(x ,s)) .

(.)

i

i=

By (.), if min{p , . . . , pm } = , then K ,..., ≤C

m 

fi Lpi (ωpi ,B(x ,s)) .

(.)

i

i=

Applying Hölder’s inequality, for  ≤ pi ≤ qi < ∞, i = , . . . , m, we have  ≤  ≤

 |B|  |B|

 ωi (yi )pi dyi

 p  i

B

 qi

ωi (yi ) dyi B

 q  i

 |B|  |B|





ωi (yi )–pi dyi

 

pi

B

 ωi (yi )

–p i

 

dyi

pi

B

for any ball B ⊂ Rn . Then m α    –p  

qi B(x , s) m– n ≤ ωi B(x , s) qi ωi i B(x , s) pi . i=

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 8 of 18

Thus, for  ≤ pi < ∞, m 

fi Lpi (ωpi ,B(x ,s)) i

i=

≤C

m 

i

i=

≤C

m 

i=

·

∞

s

dr rmn–α+ 

i

m 



∞

   

–p q ωi i B(x , s) qi fi Lpi (ωpi ,B(x ,s)) ωi i B(x , s) pi

i=

≤C



m– α fi Lpi (ωpi ,B(x ,s)) · B(x , s) n

∞

dr rmn–α+

s

  q ωi i B(x , s) qi



s

m  i=



–p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i





dr . rmn–α+

From (.) and Lemma . we get m 



–p ωi i B(x , r) pi 

q +mi= p

i ≤ C B(x , r)

 q

– q

νω (x) dx B(x ,r)

i=

m m– α  – 

qi ≤ C B(x , r) n ωi B(x , r) qi .

(.)

i=

Using Hölder’s inequality, 

 |B|



 p   q  i i  qi ωi (y) dy ≤ ωi (y) dy . |B| B B pi

Note that /qi = /pi – α/mn, then

–  – 

p q ωi i B(x , r) qi ≤ Crα/m ωi i B(x , r) pi .

(.)

Then for  ≤ pi < ∞, i = , . . . , m, m 

fi Lpi (ωpi ,B(x ,s)) ≤

i=

i

m 

  q ωi i B(x , s) qi

i=

 ×

∞



m 

s

i=

–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α

(.)

This gives J ,..., and K ,..., are majored by m 

i=

  q ωi i B(x , s) qi

 ·

∞

s



m  i=

–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 9 of 18

For the other term, let us first consider the case when α = · · · = αm = ∞. For any x ∈ B(x , s), y ∈ B(x , j+ s) \ B(x , j s), we have |x – yi | ≈ |x – yj | for i = j. Then

 Iα,m f ∞ , . . . , f ∞ (x)  m  ≤C (Rn \B(x ,s))m

≤C

∞  

(B(x ,j+ s)\B(x ,j s))m

j=

≤C

|f (y ) · · · fm (ym )| dy · · · dym (|x – y | + · · · + |x – ym |)mn–α

m  ∞  

|fi (yi )|

B(x ,j+ s)\B(x ,j s)

j= i=

≤C

|f (y ) · · · fm (ym )| dy · · · dym (|x – y | + · · · + |x – ym |)mn–α

m  ∞  

j+ s

–n+ mα

α

|x – yi |n– m

 B(x ,j+ s)

j= i=

dyi

 fi (yi ) dyi .

Applying Hölder’s inequality, it can be found that supx∈B(x ,s) |Iα,m (f∞ , . . . , fm∞ )(x)| is less than C

m ∞  

j+ s

–n+ mα

  

–p fi Lpi (ωpi ,B(x ,j+ s)) ωi i B x , j+ s pi . i

j= i=

Hence,

 sup Iα,m f∞ , . . . , fm∞ (x)

x∈B(x ,s)

≤C

∞   j=

≤C

j=

≤C

∞



j+

s

m –nm+α– 

j+ s

∞  



j+ s

j+ s



j+ s



s

m 

 

–p fi Lpi (ωpi ,B(x ,j+ s)) ωi i B x , j+ s pi i

i= m  i=

  –p fi Lpi (ωpi ,B(x ,j+ s)) ωi i B x , j+ s pi i

 

–p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi



i

i=



 dr

dr rmn–α+

dr . rmn–α+

Substituting (.) and (.) into the above, we obtain  sup T f∞ , . . . , fm∞ (x)

x∈B(x ,s)

 ≤C

∞

s



m 

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i

i=



dr . r–α

(.)

Using Hölder’s inequality,  q

νω (x) dx B(x ,s)

 q

≤C

m 

  q ωi i B(x , s) qi .

(.)

i=

From (.) and (.) we know J ∞,...,∞ and K ∞,...,∞ are not greater than (.) for  ≤ pi < ∞, i = , . . . , m.

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 10 of 18

Now we consider the case where exactly τ of the αi are ∞ for some  ≤ τ < m. We only give the arguments for one of the cases. The rest is similar and can easily be obtained from the arguments below by permuting the indices. Then for any x ∈ B(x , s),

 Iα,m f ∞ , . . . , f ∞ , f  , . . . , f  (x)  τ τ + m   ≤C (Rn \B(x ,s))τ

≤C

m  

(B(x ,s))m–τ

|f (y ) · · · fm (ym )| dy · · · dym (|x – y | + · · · + |x – ym |)mn–α

fi (yi ) dyi

i=τ + B(x ,s)

×

∞  j=

≤C



 |B(x

, j+ s)|m–α/n

m  

(B(x ,j+ s)\B(x ,j s))τ

∞  fi (yi ) dyi ·

i=τ + B(x ,s)

≤C

m ∞  

j+ s

j=

–n+α/m

τ  

 |B(x , j+ s)|m–α/n

 B(x ,j+ s)

j= i=

f (y ) · · · fτ (yτ ) dy · · · dyτ

i=

B(x ,j+ s)\B(x ,j s)

fi (yi ) dyi

fi (yi ) dyi .

Similar to the estimates for J ∞,...,∞ , we get

 sup Iα,m f∞ , . . . , fτ∞ , fτ+ , . . . , fm (x)

x∈B(x ,s)

 ≤C

∞

s



m 

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i

i=



dr . r–α

(.)

Then J ∞,...,∞,,..., and K ∞,...,∞,,..., are all less than m 

  q ωi i B(x , s) qi

i=

 ·

∞



s

m  i=

–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α

Combining the above estimates, the proof of Theorem . is completed.

(.)



Now, we can give the proof of Theorem .. From the definition of generalized weighted q q Morrey space, the norm of Iα,m (f) on Mϕ (ν ) equals ω 

 sup ϕ (x, s)–

x∈Rn ,r>

 q νω (B(x, s))

 B(x,s)

/q Iα,m (f)(y) q ν q (y) dy . ω 

(.)

By Lemma . we have  B(x,s)

q νω (x) dx

– q

≤C

m   i=

B(x,s)

q ωi i (x) dx

– q

i

.

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 11 of 18

Combining (.) and (.), 

/q  q q   ( f )(y) ν (y) dy I α,m q ω  νω (B(x, s)) B(x,s)   ∞  m – p dr

pi i p ≤ fi Lpi (ω i ,B(x ,r)) ωi B(x , r) . i r–α s i= p

(.)

p

Since fi ∈ Mϕii (ωi i ), from Lemma . and the fact fi Lpi (ωpi ,B(x,r)) are all non-decreasing i functions of r, we get m

m

i= fi Lpi (ωi i ,B(x,r))

ess infr
i= fi Lpi (ωi i ,B(x,r))

p

m

 pi pi i= ϕi (x, t)(ωi (B(x, t)))

≤ ess sup  
p

 pi m pi i= ϕi (x, t)(ωi (B(x, t)))

m

i= fi Lpi (ωi i ,B(x,t))

≤ ess sup  m t>,x∈Rn

p

 pi pi i= ϕi (x, t)(ωi (B(x, t)))

≤C

m 

fi Mϕpi

i

i=

p

(ωi i )

.

(.)

Then 

∞

m 

s

p –  fi Lpi (ωpi ,B(x,r)) ωi i B(x, r) pi

i=



m



dr r–α

i= fi Lpi (ωi i ,B(x,r))

∞

=

p

  pi pi ess infr
s

(.)

By (.) we get 

∞

s



m 

p –  fi Lpi (ωpi ,B(x,r)) ωi i B(x, r) pi i

i=



 dr ≤ Cϕ (x, s) fi Mϕpi (ωpi ) . –α i i r i= m

(.)

Combining (.), (.), and (.), then Iα,m fMϕq

q (ν  )  ω

≤C

m  i=

fi Mϕpi

i

p

(ωi i )

.

This completes the proof of first part of Theorem .. p q Similarly, the norm of Iα,m (f) on WMϕ (νω ) equals  sup ϕ (x, s)–

x∈Rn ,r>

 Iα,m (f) q q q q WL (νω ,B(x,s)) νω (B(x, s))

/q .

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 12 of 18

Combining (.) and (.), 

/q q   ( f ) I q α,m q WLq (νω ,B(x,s)) νω (B(x, s))   ∞  m

pi – p dr i ≤C fi Lpi (ωpi ,B(x,r)) ωi B(x, r) . i r–α s i=

(.)

Substituting (.) into (.), 

 Iα,m (f) q q q q WL (νω ,B(x,s)) νω (B(x, s))

/q ≤ Cϕ (x, s)

m 

fi Mϕpi

i=

i

p

(ωi i )

.

(.)

Then Iα,m fWMϕq

q

(ν  )  ω

≤C

m 

fi Mϕpi

i

i=

p

(ωi i )

.

This completes the proof of second part of Theorem ..

4 Proof of Theorem 1.2  Theorem . Let m ≥  and let  < α < mn. Suppose /p = m i= /pi , /qi = /pi – m  = (ω , . . . , ωm ) satisfy the Ap,q condition with α/mn, and /q = i= /qi = /p – α/n, ω  q q m ω  , . . . , ωmm ∈ A∞ , νω = m i= ωi . If p , . . . , pm ∈ (, ∞), (b , . . . , bm ) ∈ (BMO) , then there exists a constant C independent of f such that b I f α,m

≤C

q

Lq (νω ,B(x ,s))

m 

 

q bi ∗ ωi i B(x , s) qi

i=

 ×

∞

s

r  + ln s

m  m

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi



i

i=

dr r–α

(.)

dr , r–α

(.)

and b I f α,m

≤C

q

Lq (νω ,B(x ,s))

m 

 

q bi ∗ ωi i B(x , s) qi

i=

 ×

∞

s

where νω =

r  + ln s

m  m i=

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



m

i= ωi .

b b because the proof for Iα,m is very similar but easier. Proof We will give the proof for Iα,m Moreover, for simplicity of the expansion, we only present the case m = .

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 13 of 18

We represent fi as fi = fi + fi∞ , where fi = fi χB(x ,s) , i = , , and χB(x ,s) denotes the characteristic function of B(x , s). Then b I (f) α,

 q

Lq (νω ,B(x ,s))

≤C B(x ,s)

b    q q I f , f (x) ν (x) dx α,   ω 

 +C B(x ,s)

 +C B(x ,s)

 +C B(x ,s)

 q

b  ∞  q q I f , f (x) ν (x) dx α,   ω  b ∞   q q I f , f (x) ν (x) dx α,   ω 

 q  q

b ∞ ∞  q q I f , f (x) ν (x) dx α,   ω 

 q

= I + II + III + IV . p

(.)

p

q

b bounded from Lp (ω  ) × Lp (ω  ) to Lq (νω ), we get Since Iα,

 B(x ,s)

b    q q I f , f (x) ν (x) dx α,   ω 

 q

≤C

 

bi ∗ fi Lpi (ωpi ,B(x ,s)) .

i=

Then by (.) we get

I ≤C

 

 

q bi ∗ ωi i B(x , s) qi

i=

 ·

∞



 

s

–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi



i=

dr . r–α

(.)

Owing to the symmetry of II and III, we only estimate II. Taking λi = (bi )B(x ,s) , then



   b  ∞ f , f (x) = b (x) – λ b (x) – λ Iα, f , f∞ (x) Iα,  

– b (x) – λ Iα, f , (b – λ )f∞ (x)  

– b (x) – λ Iα, (b – λ )f , f∞ (x)

 + Iα, (b – λ )f , (b – λ )f∞ (x) = II + II + II + II .

(.)

Similar to the estimate of (.), for any x ∈ B(x , s) we can deduce  sup Iα, f , f∞ (x)

x∈B(x ,s)

 ≤C

∞

s



  i=

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 14 of 18

q

By Lemma . we know νω ∈ A∞ . Applying Hölder’s inequality and (.), we have 

  b (x) – λ b (x) – λ q ν q (x) dx ω 

B(x ,s)

≤C

 q

   B(x ,s)

i=

bi (x) – λi q ν q (x) dx ω 

 q

≤C

 

q   bi ∗ · νω B(x , s) q .

(.)

i=

Then by (.), (.), and (.), we have  q |II |q νω (x) dx

B(x ,s)



 q

  b (x) – λ b (x) – λ q ν q (x) dx

≤

ω 

B(x ,s)

≤C

  i=

 ·

∞

s

 q

 sup Iα, f , f∞ (x)

x∈B(x ,s)

 

q bi ∗ ωi i B(x , s) qi 

 

–  p fi Lpi (ωi ,B(x ,r)) ωi i B(x , r) pi



i=

dr . r–α

(.)

For any x ∈ B(x , s), we have   Iα, f , (b – λ )f ∞ (x)     |f (y )(b (y ) – λ )f (y )| dy dy ≤C n–α B(x ,s) Rn \B(x ,s) (|x – y | + |x – y |)   ∞ 

j+ –n+α  b (y ) – λ f (y ) dy . ≤C f (y ) dy  s

(.)

B(x ,j+ s)

B(x ,s)

j=

Note that  B(x ,s)

 

–p  f (y ) dy ≤ Cf  p p B(x , s) p L  (ω ,B(x ,s)) ω 

(.)

and  B(x ,j+ s)

 b (y ) – λ f (y ) dy

≤ Cf Lp (ωp ,B(x ,j+ s)) b (·) – λ

Then  sup Iα, f , (b – λ )f∞ (x)

x∈B(x ,s)

≤C

 ∞ 

j+ –n+α  fi Lpi (ωpi ,B(x ,j+ s))  s j=



–p p

L  (ω  ,B(x ,j+ s))



i=

i

.

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323



–p   × ω  B x , j+ s p b (·) – λ

Page 15 of 18



–p p

L  (ω  ,B(x ,j+ s))



 ∞

≤C

s

 

–p fi Lpi (ωpi ,B(x ,r)) ω  B(x , r) p i

i=

× b (·) – λ

dr



–p p

L  (ω  ,B(x ,r)) r n+–α

–p 

From Lemma . we know ω b (·) – λ

.

(.)

∈ Ap  , then by Lemma . we get



–p p

L  (ω  ,B(x ,r))

 ≤C B(x ,r)



b (z) – λ p ω–p (z) dz 



 p 

 r  

–p ≤ C  + ln b ∗ (ω  B(x , r) p . s 

(.)

By (.) and (.) we have  

–p i

ωi

     – 

pi B(x , r) pi ≤C B(x , r) ωi B(x , r) pi .

i=

(.)

i=

From (.), (.), and (.) we can deduce  sup Iα, f , (b – λ )f∞ (x)

x∈B(x ,s)

∞

 ≤ Cb ∗

s

r  + ln s

  i=

–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α

(.)

Applying (.) and (.) we have  B(x ,s)

b (x) – λ q ν q (x) dx ω 

 q

 

q ≤Cb ∗ νω B(x , s) q ≤Cb ∗

 

  q ωi i B(x , r) qi .

(.)

i=

Then by (.) and (.),  q

|II |q νω (x) dx

B(x ,s)

 ≤ B(x ,s)

≤C

 

b (x) – λ q ν q (x) dx ω 

×

∞

s

 q

 sup Iα, f , (b – λ )f∞ (x)

x∈B(x ,s)



bi ∗ ωi B(x , s) pi

i=



 q

r  + ln s

   i=

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 16 of 18

Similarly, we also have  |II | νω (x) dx p

 p

B(x ,s)

≤C

 



bi ∗ ωi B(x , s) pi

i= ∞

 ×

s

r  + ln s

  

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi



i

i=

dr . r–α

(.)

For any x ∈ B(x , s), with the same method of estimate for (.) we have  Iα, (b – λ )f  , (b – λ )f ∞ (x)   ≤C

  ∞ 

j+ –n+α   s j=



i=

 ∞

≤C

s

≤C

×

∞

 + ln s

  i=

dr

–p

L (ωi i ,B(x ,r)) n–α+

i

 bi ∗

i=



fi Lpi (ωpi ,B(x ,r) · bi (·) – λi

i=

 

B(x ,j+ s)

 bi (yi ) – λi fi (yi ) dyi

r s

p i

r



–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α

(.)

Then  B(x ,s)

q |II |q νω (x) dx

 q

q   ≤ C νω B(x , s) q ≤C

 

×

x∈B(x .s)



bi ∗ ωi B(x , s) pi

i=



 sup Iα, (b – λ )f , (b – λ )f∞ (x)

∞

s

r  + ln s

  

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi



i

i=

dr . r–α

(.)

dr . r–α

(.)

Then combining (.), (.), (.), and (.) we get  q

B(x ,s)

≤C

|II|q νω (x) dx

 



bi ∗ ωi B(x , s) pi

i=

 ×

∞

s

 q

r  + ln s

   i=

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Page 17 of 18

b ∞ ∞ Finally, we still decompose Iα, (f , f )(x) as follows:



   b ∞ ∞ f , f (x) = b (x) – λ b (x) – λ Iα, f∞ , f∞ (x) Iα,  

– b (x) – λ Iα, f∞ , (b – λ )f∞ (x)  

– b (x) – λ Iα, (b – λ )f∞ , f∞ (x)

 + Iα, (b – λ )f∞ , (b – λ )f∞ (x) = IV + IV + IV + IV .

(.)

Because each term IVj is completely analogous to IIj , j = , , , , being slightly different, we get the following estimate without details:  q

B(x ,s)

≤C

|IV |q νω (x) dx

 



bi ∗ ωi B(x , s) pi

i= ∞

 ×

 q

s

r  + ln s

  

– 

p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i

i=



dr . r–α

(.) 

Summing up the above estimates, (.) is proved for m = . In the following we give the proof of Theorem .. From (.) and (.), 

 q νω (B(x, s)) ≤C

m 

 B(x,s)



bi ∗

×

m  i=

∞ s

i=



/q b I (f)(y) q ν q (y) dy α,m ω  r  + ln s

m

–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i



dr . r–α p

(.) p

Since ϕk , k = , , satisfy the condition (.), and fi ∈ Mϕii (ωi i ), by (.) we get 

∞ s

r  + ln s



m  m i=

m



dr r–α

i= fi Lpi (ωi i ,B(x ,r))

∞

= s

–  p fi Lpi (ωpi ,B(x ,r)) ωi i B(x , r) pi i

ess infr
p

m

 pi pi i= ϕi (x, t)(ωi (B(x, t)))

  pi pi ess infr


r ×  + ln s

≤ Cϕ (x, s)

m

m  i=

fi Mϕpi

i

p

(ωi i )

.

(.)

Hu and Wang Journal of Inequalities and Applications 2014, 2014:323 http://www.journalofinequalitiesandapplications.com/content/2014/1/323

Combining (.) and (.), we have b I (f) α,m

q q Mϕ (νω )

≤C

m  i=

bi ∗ fi Mϕpi

i

p

(ωi i )

.

Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors completed the paper together. They also read and approved the final manuscript. Author details 1 College of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454003, P.R. China. 2 Department of Mathematics, Jiaozuo University, Jiaozuo, 454003, P.R. China. Acknowledgements The authors would like to thank the referees and the Editors for carefully reading the manuscript and making several useful suggestions. Received: 30 April 2014 Accepted: 28 July 2014 Published: 22 Aug 2014 References 1. Morrey, CB: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126-166 (1938) 2. Di Fazio, G, Ragusa, MA: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. J. Funct. Anal. 112, 241-256 (1993) 3. Palagachev, DK, Softova, LG: Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s. Potential Anal. 20, 237-263 (2004) 4. Ragusa, MA: Regularity of solutions of divergence form elliptic equation. Proc. Am. Math. Soc. 128, 533-540 (2000) 5. Mizuhara, T: Boundedness of some classical operators on generalized Morrey spaces. In: Harmonic Analysis (Sendai, 1990). ICM-90 Satell. Conf. Proc., pp. 183-189 (1991) 6. Komori, Y, Shirai, S: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282, 219-231 (2009) 7. Guliyev, VS: Generalized weighted Morrey spaces and higher order commutators of sublinear operators. Eurasian Math. J. 3, 33-61 (2012) 8. Guliyev, VS, Aliyev, SS, Karaman, T, Shukurov, P: Boundedness of sublinear operators and commutators on generalized Morrey spaces. Integral Equ. Oper. Theory 71, 327-355 (2011) 9. Eroglu, A: Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces. Bound. Value Probl. 2013, Article ID 70 (2013) 10. Grafakos, L: On multilinear fractional integrals. Stud. Math. 102, 49-56 (1992) 11. Kenig, CE, Stein, EM: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1-15 (1996) 12. Grafakos, L, Kalton, N: Some remarks on multilinear maps and interpolation. Math. Ann. 319, 151-180 (2001) 13. Moen, K: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60, 213-238 (2009) 14. Chen, S, Wu, H: Multiple weighted estimates for commutators of multilinear fractional integral operators. Sci. China Math. 56, 1879-1894 (2013) 15. Chen, X, Xue, Q: Weighted estimates for a class of multilinear fractional type operators. J. Math. Anal. Appl. 362, 355-373 (2010) 16. Si, Z, Lu, S: Weighted estimates for iterated commutators of multilinear fractional operators. Acta Math. Sin. Engl. Ser. 28, 1769-1778 (2012) 17. Pradolini, G: Weighted inequalities and point-wise estimates for the multilinear fractional integral and maximal operators. J. Math. Anal. Appl. 367, 640-656 (2010) 18. Muckenhoupt, B: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207-226 (1972) 19. Muckenhoupt, B, Wheeden, R: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261-274 (1974) 20. Wang, H, Yi, W: Multilinear singular and fractional integral operators on weighted Morrey spaces. J. Funct. Spaces Appl. 2013, Article ID 735795 (2013) 21. John, F, Nirenberg, L: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415-426 (1961) 22. Lin, Y, Lu, S: Strongly singular Calderón-Zygmund operators and their commutators. Jordan J. Math. Stat. 1, 31-49 (2008) 23. Wheeden, RL, Zygmund, A: Measure and Integral. An Introduction to Real Analysis. Pure and Applied Mathematics, vol. 43. Dekker, New York (1977) 10.1186/1029-242X-2014-323 Cite this article as: Hu and Wang: Multilinear fractional integral operators on generalized weighted Morrey spaces. Journal of Inequalities and Applications 2014, 2014:323

Page 18 of 18