Guliyev et al. Journal of Inequalities and Applications (2015) 2015:71 DOI 10.1186/s13660-015-0584-9

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The Lpr × Lpr × · · · × Lpk rk boundedness of rough multilinear fractional integral operators in the Lorentz spaces Vagif S Guliyev1,2 , Ismail Ekincioglu3* and Sh A Nazirova4 *

Correspondence: [email protected] 3 Department of Mathematics, Dumlupınar University, Kütahya, Turkey Full list of author information is available at the end of the article

Abstract In this paper, we prove the O’Neil inequality for the k-linear convolution operator in the Lorentz spaces. As an application, we obtain the necessary and sufficient conditions on the parameters for the boundedness of the k-sublinear fractional maximal operator M,α (f) and the k-linear fractional integral operator I,α (f) with rough kernels from the spaces Lp1 r1 × Lp2 r2 × · · · × Lpk rk to Lqs , where n/(n + α ) ≤ p < q < ∞, 0 < r ≤ s < ∞, p is the harmonic mean of p1 , p2 , . . . , pk > 1 and r is the harmonic mean of r1 , r2 , . . . , rk > 0. MSC: Primary 42B20; 42B25; 42B35; secondary 47G10 Keywords: O’Neil inequality; k-linear convolution; rearrangement estimate; k-sublinear fractional maximal function; k-linear fractional integral; harmonic mean; Lorentz space

1 Introduction Fractional maximal and fractional integral operators are two important operators in harmonic analysis and partial differential equations. Multilinear maximal operator and multilinear fractional integral operator and related topics have been areas of research of many mathematicians such as Coifman and Grafakos [], Grafakos [, ], Grafakos and Kalton [], Kenig and Stein [], Ding and Lu [], Guliyev and Nazirova [, ], Ragusa [] and others. Let k ≥  be an integer and θj (j = , , . . . , k) be fixed, distinct and nonzero real numbers, and let f = (f , . . . , fk ). The k-linear convolution operator f ⊗ g is defined by  (f ⊗ g)(x) =

Rn

f (x – θ y) · · · fk (x – θk y)g(y) dy.

Let  ∈ Ls (Sn– ), s ≥  and  be homogeneous of degree zero on Rn , and let  < α < n, where Sn– is the unit sphere in Rn . The k-sublinear fractional maximal function with rough kernel is defined by

M,α (f)(x) = sup r>

 rn–α

 |y|
   (y)f (x – θ y) . . . fk (x – θk y) dy,

© 2015 Guliyev et al.; 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 credited.

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and the k-linear fractional integral with rough kernel is defined by  I,α (f)(x) =

Rn

(y) f (x – θ y) · · · fk (x – θk y) dy. |y|n–α

This paper consists of four sections. In Section , some lemmas needed to facilitate the proofs of our theorems and the O’Neil inequality for rearrangements of the k-linear convolution operator f ⊗ g proved in [] are given. In Section , we prove the O’Neil inequality for the k-linear convolution operator in the Lorentz spaces. Finally, in Section , we obtain rearrangement estimates for the multilinear fractional maximal function and multilinear fractional integral with rough kernels. We prove the boundedness of the multilinear fractional maximal operator M,α and the multilinear fractional integral operator I,α with rough kernels from the spaces Lp r × Lp r × · · · × Lpk rk to Lqs , n/(n + α) ≤ p < q < ∞,  < r ≤ s ≤ ∞, where p and r are the harmonic means of p , p , . . . , pk >  and r , r , . . . , rk > , respectively. We show that the conditions on the parameters ensuring the boundedness cannot be weakened.

2 Preliminaries We need the following two generalized Hardy inequalities (see []) which are to be used in the proof of Theorem .. We denote by M(Rn ) the set of all extended real-valued measurable functions on Rn . When v is a non-negative measurable function on (, ∞), we say that v is a weight. We t t r denote W (t) =  w(τ ) dτ , V (t) =  v(τ ) dτ and U(r, t) = t u(τ ) dτ . For simplicity we suppose that  < V (t) < ∞,  < W (t) < ∞ for all t >  and V (∞) = ∞, W (∞) = ∞. Lemma . [] Let  < r ≤ s < ∞ and let v, w be weights. Then the inequality 

∞

/s

s

 ≤C

g(t) w(t) dt



∞

/r

r

(.)

g(t) v(t) dt



holds for all non-negative and non-increasing g on (, ∞) if and only if A ≡ sup W /s (t)V –/r (t) < ∞, t>

and the best constant C in (.) equals A . Lemma . [, ] Let r, s ∈ (, ∞) and let v, w be weights. (i) Let  < r ≤ s < ∞. Then the inequality 

∞ 

 t



s

t

g(τ ) dτ

/s



∞

≤C

w(t) dt



r

/r

g(t) v(t) dt



holds for all non-negative and non-increasing g on (, ∞) if and only if A < ∞,  A ≡ sup t>

t

∞

w(τ ) dτ τs

/s 

t 



v(τ )τ r dτ V r (τ )

/r < ∞,

and the best constant C in (.) satisfies C ≈ A + A .

(.)

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(ii) Let  < r ≤ , r ≤ s. Then (.) holds if and only if A < ∞, 

∞

A ≡ sup t t>

w(τ ) dτ τs

t

/s V –/r (t) < ∞,

and the best constant C in (.) satisfies C ≈ A + A . Lemma . [] Let r, s ∈ (, ∞) and let u, v, w be weight functions. (i) Let  < r ≤ s < ∞. Then the inequality s

∞  ∞



g(τ )u(τ ) dτ 

/s

 ≤C

w(t) dt

∞

/r

r

g(t) v(t) dt

(.)



t

holds for all non-negative and non-increasing g on (, ∞) if and only if  t /s s A ≡ sup U (t, τ )w(τ ) dτ V –/r (t) < ∞, t>



also 

∞

A ≡ sup W /s (t) t>



/r



< ∞,

U r (τ , t)V –r (τ )v(τ ) dτ

t

and the best constant C in (.) satisfies C ≈ A + A . (ii) Let  < r ≤ , r ≤ s. Then (.) holds if and only if A < ∞ and the best constant C in (.) equals A . Lemma . [] Let r ∈ (, ∞) and let u, v, w be weight functions. (i) Let  < r < ∞. Then the inequality  sup t>

∞

  g(τ )u(τ ) dτ w(t) ≤ C

∞

r g(t) v(t) dt

/r (.)



t

holds for all non-negative and non-increasing g on (, ∞) if and only if  A ≡ sup w(t) t>

∞



/r



U r (τ , t)V –r (τ )v(τ ) dτ

< ∞,

t

and the best constant C in (.) equals A . (ii) Let  < r ≤  and r ≤ s. Then (.) holds if and only if A ≡ sup sup U(τ , t)w(τ )V –/r (t) < ∞, t> <τ
and the best constant C in (.) equals A . Lemma . [] Let r ∈ (, ∞) and let u, v, w be weight functions. (i) Let  < r < ∞. Then the inequality  t   sup k(t, τ )g(τ )u(τ ) dτ w(t) ≤ C t>





∞

r g(t) v(t) dt

/r (.)

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holds for all non-negative and non-increasing g on (, ∞) if and only if A ≡ sup w(t) t>

 t 

r

t –

k(t, τ )V (τ ) dτ 

/r < ∞,

v(s) ds

s

and the best constant C in (.) equals A . (ii) Let  < r ≤ , r ≤ s. Then (.) holds if and only if   A ≡ sup sup K t, min(τ , t) w(τ )V –/r (t) < ∞, t> τ >

and the best constant C in (.) equals A . Let g be a measurable function on Rn . The distribution function of g is defined by the equality     λg (t) =  x ∈ Rn : g(x) > t ,

t ≥ .

We shall denote by L (Rn ) the class of all measurable functions g on Rn , which are finite almost everywhere and such that λg (t) < ∞ for all t >  (see []). If a function g belongs to L (Rn ), then its non-increasing rearrangement is defined to be the function g ∗ which is non-increasing on (, ∞) equi-measurable with |g(x)|: 

  t >  : g ∗ (t) > τ  = λg (τ ) for all τ ≥ . Moreover, by the Hardy-Littlewood theorem (see [], p.) and for every f , f ∈ L (Rn ),  Rn

  f (x)f (x) dx ≤



∞



f∗ (t)f∗ (t) dt.

Equi-measurable rearrangements of functions play an important role in various fields of mathematics. We give some of the main important properties (see, for example, []): () if  < t < t + τ , then (g + h)∗ (t + τ ) ≤ g ∗ (t) + h∗ (τ ), () if  < p < ∞, then     g(x)p dx = Rn

∞

p g ∗ (t) dt,



() for any t >  and for any set E,  sup

|E|=t E

  g(x) dx =



t

g ∗ (τ ) dτ .



We denote by WLp (Rn ) the weak Lp space of all measurable functions g with finite norm

f WLp = sup t /p f ∗ (t) < ∞,

 ≤ p < ∞.

t>

The function g ∗∗ : (, ∞) → [, ∞] is defined as g ∗∗ (t) =

 t

t 

f ∗ (s) ds.

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Definition . If  < p, q < ∞, then the Lorentz space Lpq (Rn ) is the set of all classes of measurable functions f with the finite quasi-norm 

∞

f pq ≡ f Lpq =

t

/p ∗

f (t)

/q

q dt

.

t



If  < p ≤ ∞, q = ∞, then Lp∞ (Rn ) = WLp (Rn ). If  ≤ q ≤ p or p = q = ∞, then the functional f pq is a norm (see []). If p = q = ∞, then the space L∞∞ (Rn ) is denoted by L∞ (Rn ). In the case  < p, q < ∞ we define 

∞

f (pq) =

t /p f ∗∗ (t)

q dt

/q

t



(with the usual modification if  < p ≤ ∞, q = ∞) which is a norm on Lpq (Rn ) for  < p < ∞,  ≤ q ≤ ∞ or p = q = ∞. If  < p ≤ ∞ and  ≤ q ≤ ∞, then

f pq ≤ f (pq) ≤ p f pq that is, the quasi-norms f pq and f (pq) are equivalent. Lemma . [] Let f , f , . . . , fk ∈ L (Rn ), k ≥ . Then, for all x ∈ Rn and nonzero real numbers θ , . . . , θk ,  Rn

  f (x – θ y)f (x – θ y) · · · fk (x – θk y) dy ≤ Cθ



∞



f∗ (t)f∗ (t) · · · fk∗ (t) dt,

(.)

where Cθ = |θ . . . θk |–n . Let f = (f , f , . . . , fk ) and define f

∗

(t) = f∗ (t) · · · fk∗ (t),



 f (t) = t ∗∗

t



f∗ (τ ) · · · fk∗ (τ ) dτ ,

t > .

In the following, we give the O’Neil inequality for rearrangements of the multilinear convolution operator f ⊗ g proved in []. Lemma . [] Let f , f , . . . , fk , g ∈ L (Rn ). Then, for all  < t < ∞, the following inequality holds:   (f ⊗ g)∗∗ (t) ≤ Cθ tf ∗∗ (t)g ∗∗ (t) +

∞

 f ∗ (s)g ∗ (s) ds .

(.)

t

Corollary . [] Let f , f , . . . , fk ∈ L (Rn ) and g ∈ WLm (Rn ),  < m < ∞. Then (f ⊗ g)∗ (t) ≤ (f ⊗ g)∗∗ (t)    t  –/m ∗ f (τ ) dτ + ≤ Cθ g WLm m t 

∞

τ t

 f (τ ) dτ .

–/m ∗

(.)

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Lemma . [] Let f , f , . . . , fk , g ∈ L (Rn ). Then for any t >  (f ⊗ g)∗∗ (t) ≤ Cθ



∞

f ∗∗ (t)g ∗∗ (t) dt.

(.)

t

Corollary . Let f , f , . . . , fk ∈ L (Rn ) and g ∈ WLm (Rn ),  < m < ∞. Then ∗

∗∗





(f ⊗ g) (t) ≤ (f ⊗ g) (t) ≤ m Cθ g WLm

∞

τ –/m f ∗∗ (τ ) dτ .

(.)

t

3 O’Neil inequality for the multilinear convolutions in the Lorentz spaces In this section, we prove the O’Neil inequality for the multilinear convolutions in the Lorentz spaces. It is said that p is the harmonic mean of p , p , . . . , pk >  if /p = /p +/p + · · · + /pk . If fj ∈ Lpj rj (Rn ), j = , , . . . , k, then we say that f ∈ Lp r × Lp r × · · · × Lpk rk (Rn ). Theorem . (O’Neil inequality for k-linear convolution in the Lorentz spaces) Suppose that  < m < ∞, g ∈ WLm (Rn ), p and r are the harmonic means of p , p , . . . , pk >  and r , r , . . . , rk > , respectively. If  < p < m ,  < r ≤ s < ∞ or m /( + m ) ≤ p ≤ ,  < r ≤ , r ≤ s < ∞ or p = m ,  < r < ∞, s = ∞ or p = m ,  < r ≤ , s = ∞ f ∈ Lp r ×Lp r ×· · ·×Lpk rk (Rn ) and /p – /q = /m , then f ⊗ g ∈ Lqs (Rn ) and

f ⊗ g qs  Cθ K(p, q, r, s, m)

k 

fj pj rj g WLm ,

j=

where K(p, q, r, s, m) = κ and ⎧   ⎪ ⎪ m A + m A + A + A , ⎪ ⎨ m A + m A + A ,    κ≈   ⎪ m A + m A , ⎪ ⎪ ⎩  m A  + m A  ,

if  < p < m ,  < r ≤ s < ∞, m if +m  ≤ p ≤ ,  < r ≤ , r ≤ s < ∞ if p = m ,  < r < ∞, s = ∞, if p = m ,  < r ≤ , s = ∞

and /s  /r  /s   /r r p mq m q r , A = ,  s(m + q) p p s(q – m) r /s  /r      +/s r /r   /s r mq B s + , sm /q A = , A = m , p s(q – m) p  /r     +/r r q /s         /r  r A = m , A = m , B r + , r m /p – r p s p /r  +/r    B r + , r m /p – r A  = m , 

A =

   /r  /r  r r p +/r  A = , B r + , p p–r p–r

Here B(s, r) =



 ( – τ )

s– r–

τ

 /r r A = . p

dτ is the beta function.

Proof Let  < m < ∞, m /( + m ) ≤ p < m , /p – /q = /m , p be the harmonic mean of p , p , . . . , pk > , r be the harmonic mean of r , r , . . . , rk > ,  < r ≤ s ≤ ∞ and f ∈

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Lp r × Lp r × · · · × Lpk rk (Rn ). By using inequality (.), we have  

f ⊗ g qs = (f ⊗ g)∗ (t)t /q–/s Ls (,∞)  ∞    t  –/m ∗ ≤ Cθ f (τ ) dτ + mt 

≤ Cθ m







∞  t

f (τ ) dτ





+ Cθ

τ 

f (τ ) dτ

t

f (τ ) dτ

t

dt

dt

s

–/m ∗

/s s/q–

/s

–s/m+s/q–



∞  ∞

τ

s

–/m ∗

t

s

∗

∞

/s t

s/q–

.

dt

t

Case I. Suppose that  < p < m (equivalently m < q < ∞),  < r ≤ s < ∞. From Lemma ., for the validity of the inequality for  < r ≤ s < ∞ ∞

 

 t



t

f ∗ (τ ) dτ

s

/s t s–s/m+s/q– dt



∞

≤ C



t /p f ∗ (t)

r dt



/r ,

t

(.)

the necessary and sufficient condition is /s  /r r m q  sup t /m +/q–/p < ∞  + q) s(m p t> t> /s  /r   r mq ⇔ /p – /q = /m and A = s(m + q) p 

A = sup W /s (t)V –/r (t) =

and  A = sup

/s 

/r  v(τ )τ r dτ p t> t  V (τ ) /s  t /r  ∞ r –s/m+s/q– r/p–+r –rr /p = sup τ dτ τ dτ p t> t   /s   /r p mq r  = sup t –/m+/q–/p < ∞  p s(q – m) r t>  /s   /r p mq r  ⇔ /p – /q = /m and A = . p s(q – m) r ∞

w(τ ) dτ τs

t

Note that the best constant C in (.) satisfies C ≈ A + A . Furthermore, from Lemma . for the validity of the inequality for  < r ≤ s < ∞ 

∞  ∞ 

τ –/m f ∗ (τ ) dτ

s

/s t s/q– dt



∞

t /p f ∗ (t)

≤ C

r dt



t

the necessary and sufficient condition is

A = m sup t>

 

t

  s t /m – τ /m τ s/q– dτ

/s 

–/r

t

τ r/p– dτ 

 /r  t /s  /m r  s t = m sup – τ /m τ s/q– dτ t –/p p t> 

t

/r ,

(.)

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   +/s r /r   /s  = m sup t –/m +/q–/p < ∞ B s + , sm /q p t>    +/s r /r   /s B s + , sm /q ⇔ /p – /q = /m and A = m p and 

A = sup W /s (t) t>

∞



/r



U r (τ , t)V –r (τ )v(τ ) dτ

t

   ∞ /r  /m /m r –rr /p+r/p– m r q /s /q τ = sup t –t τ dτ p s t> t    ∞ /r  /m r –r /p– m r q /s  λ = – λ dλ sup t /q+/m –/p p s t>  /r  /s     +/r r q  r       – λ/m λ–r /m +r /p– dλ = m sup t /q+/m –/p p s t>   /s   /r   +/r r q  r –r +r m /p– = m ( – τ ) τ dτ sup t /q+/m –/p p s t>   /s  +/r r q    /r  = m sup t /q+/m –/p < ∞ B r + , r m /p – r p s t>  /s    /r  +/r r q ⇔ /p – /q = /m and A = m B r + , r m /p – r . p s Note that the best constant C in (.) satisfies C ≈ A + A . Case II. Let m /( + m ) ≤ p ≤ ,  < r ≤  and r ≤ s < ∞. From Lemma ., for the validity of inequality (.), the necessary and sufficient condition is A < ∞ and 

/s w(τ ) dτ V –/r (t) s τ t> t  /r  ∞ /s r –s/m+s/q– = sup t τ dτ t –/p p t> t  /r  /s r mq = sup t –/m+/q–/p p s(q – m) t> /s  /r  mq r  ⇔ /p – /q = /m and A = . p s(q – m)

A = sup t

∞

Note that the best constant C in (.) satisfies C ≈ A + A . From Lemma ., for the validity of inequality (.), the necessary and sufficient condition is A < ∞. Consequently, using inequalities (.), (.) and applying the Hölder inequality, we obtain  

f ⊗ g qs ≤ Cθ m C + C



∞

t /p f ∗ (t)

r dt





k ∞ 

= Cθ K(p, q, r, s, m) 

j=

∗

fj (t)t

t

/r

g WLm

 /pj r dt t

/r

g WLm

Guliyev et al. Journal of Inequalities and Applications (2015) 2015:71

≤ Cθ K(p, q, r, s, m)

k  

= Cθ K(p, q, r, s, m)

∞



j= k 

Page 9 of 15

∗

fj (t)t

 /pj rj

dt t

/rj

g WLm

fj pj rj g WLm .

j=

Case III. Let p = m , q = s = ∞,  < r < ∞ or p = m , q = s = ∞,  < r ≤  and f ∈ Lp r × Lp r × · · · × Lpk rk (Rn ). By using inequality (.), we have

f ⊗ g ∞ = sup(f ⊗ g)∗ (t) t>

   t  –/m ∗ f (τ ) dτ + ≤ Cθ sup m t t>



∞

τ

 f (τ ) dτ g WLm

–/m ∗

t

    t ≤ Cθ m sup t –/m f ∗ (τ ) dτ + sup t>

≤ C θ m



 ∞

t /p f ∗ (t)

r dt t



t>



∞

 τ –/m f ∗ (τ ) dτ g WLm

t

g WLm .

From Lemma ., for the validity of the inequality for  < r < ∞     t –/m ∗ f (τ ) dτ ≤ C sup t t>



∞

t

/p ∗

f (t)

r dt



/r

t

,

(.)

the necessary and sufficient condition is  t  t  /r r /r r –/m –r/p r/p– A = sup t τ dτ s ds p t>  s  t  /r /r  –r/p r r r t sup t –/m = – τ –r/p τ r/p– dτ p p – r t>  +/r    /r  /r   p r –r/p s r/p– –τ = τ dτ sup t –/m–/p+ p p–r t>  +/r   /r  /r  r p r B r + , = sup t –/m–/p+ < ∞ p p–r p–r t>  /r  +/r   /r r p r B r + , . ⇔ p = m and A = p p–r p–r From Lemma ., for the validity of the inequality for  < r ≤      t sup t –/m f ∗ (τ ) dτ ≤ C t>



∞

t /p f ∗ (t)



r dt t

/r ,

the necessary and sufficient condition is   A = sup sup K t, min(τ , t) w(τ )V –/r (t) = t> τ >

⇔

p = m and A =

 /r r . p

 /r r  sup t /m –/p < ∞ p t>

(.)

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From Lemma ., for the validity of the inequality for  < r < ∞  sup t>

∞

τ





–/m ∗

∞

≤ C

f (τ ) dτ

t

/p ∗

f (t)

r dt

,

t



t

/r (.)

the necessary and sufficient condition is A  A = sup t>

∞

r

U (τ , t)V

–r

/r (τ )v(τ ) dτ

t



∞

=



 r



τ /m – t /m

/r



τ –rr /p+r/p– dτ

t



∞

=

λ

/m

r 

– λ

–r /p–

/r t>

 

mr = p =

m r p



sup t /m –/p

dλ





–λ

 /m r –r /m +r /p– λ

/r t>







sup t /m –/p

dλ









/r



( – τ )r τ –r +r m /p– dτ



sup t /m –/p t>



/r m r     B r + , r m /p – r = sup t /m –/p < ∞ p t> p = m and A =

⇔

/r m r    B r + , r m /p – r . p

Furthermore, from Lemma ., for the validity of the inequality for  < r ≤   sup t>

∞

τ





–/m ∗

f (τ ) dτ

∞

≤ C

t

/p ∗

f (t)



t

r dt t

/r ,

(.)

the necessary and sufficient condition is

A = sup sup U(τ , t)w(τ )V –/r (t) t> <τ




= m sup sup t t> <τ
/m

–τ

/m

 

–/r

t

τ

r/p–

dτ



 /r   r  =m sup sup t /m – τ /m t –/p p t> <τ   /r r . ⇔ p = m and A = m p 

Thus the proof of Theorem . is completed.



Corollary . [] Suppose that  < m < ∞, g ∈ WLm (Rn ) and p is the harmonic mean of p , p , . . . , pk > . If m /( + m ) ≤ p < m , f ∈ Lp × Lp × · · · × Lpk (Rn ) and q satisfy

Guliyev et al. Journal of Inequalities and Applications (2015) 2015:71

Page 11 of 15

/p – /q = /m , then f ⊗ g ∈ Lq (Rn ) and

f ⊗ g q ≤ Cθ K(p, q, m)

k 

fj pj g WLm ,

j=

where in the case  < p = r < m , q = s /q /q  m m  + m m + q q–m /q   +/p    /p   +/q   B q + , m B p + , p m /p – p + m + m ,

K(p, q, m) = m



and in the case m /( + m ) ≤ p = r ≤ , m < q = s K(p, q, m) = m





m m + q

/q

  + m + 



m q–m

/q

 +/q   /q + m . B q + , m

4 The Lp1 r1 × Lp2 r2 × ··· × Lpk rk boundedness of rough multilinear fractional integral operators In this section, we prove the Sobolev type theorem for the rough multilinear fractional integral I,α f. Lemma . Let  < α < n,  be homogeneous of degree zero on Rn ,  ∈ Ln/(n–α) (Sn– ) and g(x) =

(x) . |x|n–α

Then g ∈ WLn/(n–α) (Rn ) and

g WLn/(n–α) = nα/n–  Ln/(n–α) ,

(.)

where 

 Ln/(n–α) =

   n/(n–α)     x  dσ x

(n–α)/n .

Sn–

Proof Note that g ∗ (t) = (nt)α/n–  Ln/(n–α) ,

g ∗∗ (t) =

n ∗ g (t), α

therefore g ∈ WLn/(n–α) (Rn ) and equality (.) is valid.



Lemma . Suppose that  < α < n,  ∈ Ls (Sn– ) and s ≥ . Then   M,α f(x) ≤ I||,α |f| (x), where |f| = (|f |, . . . , |fk |).

(.)

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Proof Indeed, for all r > , we have 

 |(y)|  f (x – θ y) . . . fk (x – θk y) dy n–α  |y| E(,r)      (y)f (x – θ y) . . . fk (x – θk y) dy, ≥ n–α r E(,r)

  I||,α |f | (x) ≥

where E(, r) is the open ball centered at the origin of radius r. Taking supremum over all r > , we get (.).  By Lemmas . and ., we obtain a pointwise rearrangement estimate of the rough k-sublinear fractional maximal integral M,α f and k-linear fractional integral I,α f. Lemma . [] Suppose that  is homogeneous of degree zero on Rn and  ∈ Ln/(n–α) (Sn– ),  < α < n. Then the following inequalities hold: (I,α f)∗ (t) ≤ (I,α f)∗∗ (t)



≤ Cθ nα/n–  Ln/(n–α) (M,α f)∗ (t) ≤ (M,α f)∗∗ (t) ≤ Cθ nα/n–  Ln/(n–α)

n α/n– t α





n α/n– t α

t

f ∗ (τ ) dτ +





∞

 τ α/n– f ∗ (τ ) dτ ,

t



t

f ∗ (τ ) dτ +





∞

 τ α/n– f ∗ (τ ) dτ .

t

From Theorem . and Lemma ., we get the following. Theorem . Let  be homogeneous of degree zero on Rn ,  ∈ Ln/(n–α) (Sn– ),  < α < n, p and r be the harmonic means of p , p , . . . , pk >  and r , r , . . . , rk > , respectively, and  < r ≤ s ≤ ∞, q satisfy /q = /p – α/n. If  < p < n/α,  < r ≤ s < ∞ or n/(n + α) ≤ p ≤ ,  < r ≤ s < ∞ or p = n/α, r = , then I,α is a bounded operator from Lp r × Lp r × · · · × Lpk rk (Rn ) to Lqs (Rn ) and k   

fj pj rj .

I,α f qs ≤ Cθ nα/n– K p, q, r, s, n/(n – α)  Ln/(n–α) j=

Corollary . [] Let  be homogeneous of degree zero on Rn ,  ∈ Ln/(n–α) (Sn– ),  < α < n, p be the harmonic mean of p , p , . . . , pk > , and q satisfy /q = /p – α/n. Then I,α is a bounded operator from Lp × Lp × · · · × Lpk (Rn ) to Lq (Rn ) for n/(n + α) ≤ p < n/α (equivalently  ≤ q < ∞) and k   

I,α f q ≤ Cθ nα/n– K p, q, n/(n – α)  Ln/(n–α)

fj pj . j=

Corollary . [] Let  be homogeneous of degree zero on Rn ,  ∈ Ln/(n–α) (Sn– ),  < α < n, p be the harmonic mean of p , p , . . . , pk > , and q satisfy /q = /p – α/n. Then M,α is a bounded operator from Lp × Lp × · · · × Lpk (Rn ) to Lq (Rn ) for n/(n + α) ≤ p ≤ n/α (equiv-

Guliyev et al. Journal of Inequalities and Applications (2015) 2015:71

Page 13 of 15

alently  ≤ q ≤ ∞) and k   

M,α f q ≤ Cθ nα/n– K p, q, n/(n – α)  Ln/(n–α)

fj pj , j=

when n/(n + α) ≤ p < n/α, and

M,α f ∞ ≤ Cθ  Ln/(n–α)

k 

fj pj ,

p = n/α.

j=

Finally, in the following theorem we obtain the necessary and sufficient conditions for the rough k-linear fractional integral operator I,α to be bounded from the Lorentz spaces Lp r × Lp r × · · · × Lpk rk (Rn ) to Lqs (Rn ), n/(n + α) ≤ p < q < ∞,  < r ≤ s < ∞. Theorem . Let  < α < n,  be homogeneous of degree zero on Rn ,  ∈ Ln/(n–α) (Sn– ), p and r be the harmonic means of p , p , . . . , pk >  and r , r , . . . , rk > , respectively. If  < p < n/α,  < r ≤ s < ∞ or n/(n + α) ≤ p ≤ ,  < r ≤ s < ∞, then the condition /p – /q = α/n is necessary and sufficient for the boundedness of I,α from Lp r × Lp r × · · · × Lpk rk (Rn ) to Lqs (Rn ). Proof Sufficiency of the theorem follows from Theorem .. Necessity. Suppose that the operator I,α is bounded from Lp r × Lp r × · · · × Lpk rk (Rn ) to Lqs (Rn ), and n/(n + α) ≤ p < n/α (equivalently  ≤ q < ∞). Define ft (x) =: f(tx) for t >   and f pr = kj= fj pj rj . Then it can be easily shown that

ft pr =

k    (fj )t 

pj rj

=

j=

k 

t –n/pj fj pj rj = t –n/p f pr

j=

and I,α ft (x) = t –α I,α f(tx),

I,α ft qs = t –α–n/q I,α f qs .

Since the operator I,α is bounded from Lp r × Lp r × · · · × Lpk rk (Rn ) to Lqs (Rn ), we have

I,α f qs ≤ C f pr , where C is independent of f. Then we get

I,α f qs = t α+n/q I,α ft qs ≤ Ct α+n/q ft pr = Ct α+n/q–n/p f pr . If /p < /q + α/n, then for all f ∈ Lp r × Lp r × · · · × Lpk rk (Rn ) we have I,α f Lq,s =  as t → . If /p > /q + α/n, then for all f ∈ Lp r × Lp r × · · · × Lpk rk (Rn ) we have I,α f qs =  as t → ∞. Therefore we get /p = /q + α/n.  Corollary . [] Let  < α < n, p be the harmonic mean of p , p , . . . , pk > ,  be homogeneous of degree zero on Rn and  ∈ Ln/(n–α) (Sn– ). If n/(n + α) ≤ p < n/α, then the condition /p – /q = α/n is necessary and sufficient for the boundedness of I,α from Lp × Lp × · · · × Lpk (Rn ) to Lq (Rn ).

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Remark . Note that the sufficiency part of Corollary . was proved in [] and in the case  ≡  in [], and in the case  ∈ Ls (Sn– ), s > n/(n – α) in []. Theorem . Let  < α < n,  be homogeneous of degree zero on Rn ,  ∈ Ln/(n–α) (Sn– ), p and r be the harmonic means of p , p , . . . , pk >  and r , r , . . . , rk > , respectively. If  < p < n/α,  < r ≤ s < ∞ or n/(n + α) ≤ p ≤ ,  < r ≤ s < ∞, then the condition /p – /q = α/n is necessary and sufficient for the boundedness of M,α from Lp r × Lp r × · · · × Lpk rk (Rn ) to Lqs (Rn ). Proof Sufficiency part of the theorem follows from Theorem . and Lemma .. Necessity. Suppose that the operator M,α is bounded from Lp r × Lp r × · · · × Lpk rk (Rn ) to Lqs (Rn ), and n/(n + α) ≤ p < n/α,  < r ≤ s < ∞. Then we have M,α ft (x) = t –α M,α f (tx) and n

M,α ft qs = t –α– q M,α f qs . By the same argument in Theorem ., we obtain

 p

–

 q

= αn .



Corollary . [] Let  < α < n, p be the harmonic mean of p , p , . . . , pk > ,  be homogeneous of degree zero on Rn and  ∈ Ln/(n–α) (Sn– ). If n/(n + α) ≤ p ≤ n/α, then the condition /p – /q = α/n is necessary and sufficient for the boundedness of M,α from Lp × Lp × · · · × Lpk (Rn ) to Lq (Rn ).

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Institute of Mathematics and Mechanics, Baku, Azerbaijan. 2 Department of Mathematics, Ahi Evran University, Kirsehir, Turkey. 3 Department of Mathematics, Dumlupınar University, Kütahya, Turkey. 4 Khazar University, Baku, Azerbaijan. Acknowledgements The research of V Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2014-9(15)-46/10/1 and by the grant of Presidium of Azerbaijan National Academy of Science 2015. Received: 5 November 2014 Accepted: 30 January 2015 References 1. Coifman, R, Grafakos, L: Hardy spaces estimates for multilinear operators. I. Rev. Mat. Iberoam. 8, 45-68 (1992) 2. Grafakos, L: On multilinear fractional integrals. Stud. Math. 102, 49-56 (1992) 3. Grafakos, L: Hardy spaces estimates for multilinear operators. II. Rev. Mat. Iberoam. 8, 69-92 (1992) 4. Grafakos, L, Kalton, N: Some remarks on multilinear maps and interpolation. Math. Ann. 319, 49-56 (2001) 5. Kenig, CE, Stein, EM: Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1-15 (1999) 6. Ding, Y, Lu, S: The f ∈ Lp1 × Lp2 × · · · × Lpk boundedness for some rough operators. J. Math. Anal. Appl. 203, 151-180 (1996) 7. Guliyev, VS, Nazirova, SA: A rearrangement estimate for the rough multilinear fractional integrals. Sib. Math. J. 48(3), 1-12 (2007) 8. Guliyev, VS, Nazirova, SA: O’Neil inequality for multilinear convolutions and some applications. Integral Equ. Oper. Theory 60(4), 485-497 (2008) 9. Ragusa, MA: Necessary and sufficient condition for a VMO function. Appl. Math. Comput. 128, 11952-11958 (2012)

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