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Weighted Trudinger inequality associated with rough multilinear fractional type operators

Journal of Inequalities and Applications20122012:179

https://doi.org/10.1186/1029-242X-2012-179

  • Received: 7 June 2012
  • Accepted: 3 August 2012
  • Published:

Abstract

Let I Ω , α Θ be the multilinear fractional type operator defined by I Ω , α Θ ( f ) ( x ) = R n Ω ( y ) j = 1 m f j ( x θ j y ) | y | ( α n ) d y . In this paper, we study the weighted estimates for the Trudinger inequality associated to I Ω , α Θ with rough homogeneous kernels, which improve some known results significantly. A similar Trudinger inequality holds for another type of fractional integral defined by I ¯ Ω , α ( f ) ( x ) = ( R n ) m j = 1 m | f j ( y j ) | | Ω j ( x y j ) | | ( x y 1 , x y 2 , , x y m ) | m n α d y , where d y = d y 1 d y m .

Keywords

  • Riesz potential
  • multilinear fractional integral
  • A p weights
  • A p , q weights
  • Trudinger inequality

1 Introduction

The Trudinger inequality (also sometimes called the Moser-Trudinger inequality) is named after N. Trudinger who first put forward this inequality in [22]. Later, J. Moser [14] gave a sharp form of this Trudinger inequality. It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a function. In [14], J. Moser gave the largest positive number β 0 , such that if u C 1 ( R n ) , normalized and supported in a domain D with finite measure in R n , such that D | u ( x ) | n d x 1 , then there is a constant c 0 depending only on n such that for all β β 0 = n w n 1 1 / ( n 1 ) , where w n 1 is the area of the surface of the unit n-ball. The following inequality holds:
D exp ( β | u ( x ) | n / ( n 1 ) ) d x c 0 | D | .
(1.1)
In 1971, D. Adams [1] considered the similar inequality of J. Moser for higher order derivatives. The key, for him, was to write the function u as a potential I α (see the definition below) and prove the analogue of (1.1) as follows:
D exp ( n w n 1 | I α f ( x ) f p | n / ( n α ) ) d x c 0 | D | , for  α = n / p , f L p ( 1 < p < ) .
(1.2)

Variant forms of the Trudinger inequality as a generalization of the classical results, especially in the literature associated with multilinear Riesz potential or multilinear fractional integral, have been studied in recently years (see, for example, [2, 3, 6, 7, 10, 14, 1618, 20, 21]). This kind of inequality plays an important role in Harmonic analysis and other fields, such as PDE.

We begin by introducing a class of multilinear maximal function and multilinear fractional integral operators. Suppose that n 2 , 0 < α < n , Ω is homogeneous of degree zero, and Ω L s ( S n 1 ) ( s > 1 ), where S n 1 denotes the unit sphere of R n . The multilinear maximal function and multilinear fractional integral is defined by
I Ω , α Θ ( f ) ( x ) = R n Ω ( y ) j = 1 m f j ( x θ j y ) | y | ( α n ) d y
(1.3)
and the fractional maximal operator M Ω , α defined by
M Ω , α Θ ( f ) ( x ) = sup r > 0 1 r n α | y | < r | Ω ( y ) | j = 1 m | f j ( x θ j y ) | d y .
(1.4)

Multilinear fractional integral I Ω , α Θ can be looked at as a natural generalization of the classical fractional integral, which has a very profound background of partial differential equations and is a very important operator in Harmonic analysis. In fact, if we take K = 1 , θ j = 1 , and Ω = 1 , then I Ω , α Θ is just the well-known classical fractional integral operator studied by Muckenhoupt and Wheeden in [15]. We denote it by I α . If Ω 1 , we simply denote I Ω , α Θ = I α Θ . In recent years, the study of the Trudinger inequality associated to multilinear type operators has received increasing attention. Among them, it is well known that Grafakos considered the boundedness of a family of related fractional integrals in [7]. After that, in [6], Y. Ding and S. Lu gave the following Trudinger inequality with rough kernels.

Theorem A ([6])

Let 0 < α < n , s = n α , 1 s = 1 p 1 + 1 p 2 + + 1 p m , p j > 1 , j = 1 , 2 , , m , m 2 . Denote B as a ball with a radius R in R n . If f j L p j ( B ) , supp ( f j ) B , and Ω L n / ( n α ) ( S n 1 ) , then for any γ < 1 , there is a constant C, independent of n, α, θ j , γ, such that
B exp ( n γ ( L I Ω , α Θ ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L p j ) n / ( n α ) ) d x C R n ,
where L = j = 1 m | θ j | n / p j , Θ = ( θ 1 , θ 2 , , θ m ) , f = ( f 1 , f 2 , , f m ) and
Ω L n / ( n α ) = ( S n 1 | Ω ( x ) | n / ( n α ) d σ ( x ) ) ( n α ) / n .

The definition of multiple weights A p , q was given in [5] and [13] independently, including some weighted estimates for a class of multilinear fractional type operators. These results together with [12] answered an open problem in [8], namely the existence of the multiple weights.

In 2010, W. Li, Q. Xue, and K. Yabuta [16] obtained the weighted estimates for the Trudinger inequality associated to I α Θ as follows.

Theorem B ([16])

Let 0 < α < n , s = n α , 1 s = 1 p 1 + 1 p 2 + + 1 p m , p j > 1 , ω j ( x ) A p j , and ω j 1 , j = 1 , 2 , , m , m 2 , ν ω = j = 1 m ω j s / p j . Denote B as a ball with the radius R in R n , if f j L ω j p j ( B ) , supp ( f j ) B , j = 1 , 2 , , m , then for any γ < 1 , there is a constant C, independent of n, α, θ j , γ, such that
B exp ( n ω n 1 γ ( L I α Θ ( f ) ( x ) j = 1 m f j L ω j p j ) n / ( n α ) ) ν ω d x C j = 1 m ω j ( B ) ,

where L = j = 1 m | θ j | n / p j , Θ = ( θ 1 , θ 2 , , θ m ) , f = ( f 1 , f 2 , , f m ) .

On the other hand, in 1999, Kenig and Stein [11] considered another more general type of multilinear fractional integral which was defined by
I α , A ( f ) ( x ) = ( R n ) m 1 | ( y 1 , , y m ) | m n α i = 1 m f i ( i ( y 1 , , y m , x ) ) d y i ,

where i is a linear combination of y j s and x depending on the matrix A. They showed that I α , A was of strong type ( L p 1 × × L p m , L q ) and weak type ( L p 1 × × L p m , L q , ) . When i ( y 1 , , y m , x ) = x y i , we denote this multilinear fractional type operator by I ¯ α . In 2008, L. Tang [20] obtained the estimation of the exponential integrability of the above operator I ¯ α , which is quite similar to Theorem B.

Thus, it is natural to ask whether Theorem B is true or not for I Ω , α Θ with rough kernels. Moreover, one may ask if Theorem B still holds or not for the operator with rough kernels defined by
I ¯ Ω , α ( f ) ( x ) = ( R n ) m j = 1 m | f j ( y j ) | | Ω j ( x y j ) | | ( x y 1 , x y 2 , , x y m ) | m n α d y .

Inspired by the works above, in this paper, we study the Trudinger inequality associated to multilinear fractional integral operators I Ω , α Θ and I ¯ Ω , α with rough homogeneous kernels. Precisely, we obtain the following theorems, which give a positive answer to the above questions.

Theorem 1.1 Let 0 < α < n , s = n α , 1 s = 1 p 1 + 1 p 2 + + 1 p m , p j > 1 , j = 1 , 2 , , m , m 2 . Denote B as a ball with radius R in R n ; if f j L ω j p j ( B ) , supp ( f j ) B ( j = 1 , 2 , , m ), Ω L n / ( n α ) ( S n 1 ) , and ν ω = j = 1 m ω j s p j , where ω j A s , ω j 1 . Then for any γ < 1 , there is a constant C, independent of n, α, θ j , γ, such that
B exp ( n γ ( L I Ω , α Θ ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L ω j p j ) n / ( n α ) ) ν ω d x C j = 1 m ω j ( B ) ,

where L = j = 1 m | θ j | n / p j , Θ = ( θ 1 , θ 2 , , θ m ) , f = ( f 1 , f 2 , , f K ) .

Remark 1.1 If we take Ω = 1 , then Theorem 1.1 coincides with Theorem B. If w j 1 for j = 1 , , K , then Theorem 1.1 is just Theorem A that appeared in [6]. We give an example of ν ω as follows: Let ω j ( x ) = ( 1 + | x | ) α j ( α j 0 for each j), then ν ω ( x ) satisfy the conditions of the above Theorem 1.1.

Remark 1.2 Assume m = 1 , ω j = 1 . If α = 1 , Trudinger [20] proved exponential integrability of I α ( f ) , and Strichartz [19] for other α. In 1972, Hedberg [9] gave a simpler proof for all α. In 1970, Hempel-Morris-Trudinger [10] showed that if γ > 1 , for α = 1 the inequality in Theorem 1.1 cannot hold, and later Adams [1] obtained the same conclusion for all α; meanwhile, in the endpoint case γ = 1 , it is true. In 1985, Chang and Marshall [4] proved a similar sharp exponential inequality concerning the Dirichlet integral. Assume m 2 , w j = 1 , then the result was obtained by Grafakos [7] as we have already mentioned above.

Corollary 1.2 Let B, f j , p j , s, and ν ω be the same as in Theorem  1.1, then I Ω , α Θ ( f ) is in L q ( ν ω ( B ) ) for every q > 0 , that is,
I Ω , α Θ ( f ) L q ( ν ω ( B ) ) C Ω L n / ( n α ) ( S n 1 ) j = 1 m f j L ω j p j

for some constant C depending only on q on n on α and on the θ j ’s.

Theorem 1.3 Let m 2 , 0 < α < m n , 1 / p = 1 / p 1 + 1 / p 2 + + 1 / p m = α / n with 1 < p i < for i = 1 , 2 , , m . Let B be a ball with radius R in R n and let f j L p j ( B ) be supported in B, and if Ω j is homogeneous of degree zero, and Ω j L p j ( S n 1 ) , where S n 1 denotes the sphere of R n , and ν ω ( y ) = j = 1 m ω j 1 / p j ( y j ) , where y = ( y 1 , y 2 , , y m ) and ω j A s , ω j 1 . Then there exist constants k 1 , k 2 depending only on n, m, α, p, and the p j such that
B exp ( k 1 ( | I ¯ Ω , α ( f ) ( x ) | j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j ) n / ( m n α ) ) ν ω ( x ) d x k 2 j = 1 m ω j ( B ) .

Remark 1.3 If we take Ω = 1 , w j 1 for j = 1 , , m , then Theorem 1.3 is just as Theorem 1.3 appeared in [20]. But there is something that needs to be changed in the proof of Theorem 1.3 in [20]. In the case r 1 = r 2 = = r m 1 = 0 , one cannot obtain the conclusion that F 2 C 2 [ log 2 m R δ ] ( m n α ) / n . Thus, our proof gives an alternative correction of Theorem 1.3 in [20].

Corollary 1.4 Let B, f j , p j , s, and ν ω be the same as in Theorem  1.3. Then I ¯ Ω , α ( f ) is in L q ( ν ω ( B ) ) for every q > 0 , that is,
I ¯ Ω , α ( f ) L q ( ν ω ( B ) ) C j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j

for some constant C depending only on q on n on α.

Corollary 1.2 and Corollary 1.4 follow since exponential integrability of I ¯ Ω , α ( f ) implies integrability to any power q.

On the other hand, we shall study the boundedness of the multilinear fractional maximal operator with a weighted norm. It follows the following theorem.

Theorem 1.5 If 1 < p j < , 1 s = j = 1 m 1 p j , 1 r = 1 s α n , ω j p j s A ( s , s r j p j ) , 1 / r j = 1 / p j ( 1 α s / n ) , j = 1 , 2 , , m , ν ω = j = 1 m ω j , then there is a constant C, independent f j , such that
( R n ( M 1 , α Θ ( f ) ( x ) ν ω ( x ) ) r d x ) 1 r C j = 1 m ( R n | f j ( x ) ω j ( x ) | p j d x ) 1 p j ,

where f = ( f 1 , f 2 , , f m ) , f j L ω j p j ( R n ) .

2 The proof of Theorem 1.1

In this section, we will prove Theorem 1.1.

Proof For any δ > 0 ,
| I Ω , α Θ ( f ) ( x ) | C δ α M Ω ( f ) ( x ) + | y | δ | Ω ( y ) | | y | n α j = 1 m f j ( x θ j y ) d y .
Set P = 2 min { 1 θ j : j = 1 , 2 , , K } . For any R > 0 , denote B ( R ) as a ball with radius R in R n , then for any x B ( R ) , when | x θ j y | < R , | θ j y | < 2 R for j = 1 , , m . Therefore, | y | < R P . So,
| y | δ j = 1 m f j ( x θ j y ) | y | α n d y = δ | y | < P R j = 1 m f j ( x θ j y ) | y | α n d y .
According to the relationship between s and p j : 1 p 1 + 1 p 2 + + 1 p m + 1 n / ( n α ) = 1 , from the Hölder’s inequality and ν ω 1 , it follows that
δ | y | < P R Ω ( y ) j = 1 m f j ( x θ j y ) | y | α n d y ( δ | y | P R ( j = 1 m f j ( x θ j y ) ) s d y ) 1 / s ( δ | y | P R ( | Ω ( y ) | | y | n α ) s d y ) 1 / s ( δ | y | P R j = 1 m f j ( x θ j y ) s ν ω ( x θ j y ) d y ) 1 / s Ω L s ( ln P R δ ) n α n j = 1 m ( δ | y | P R | f j ( x θ j y ) | p j ω j ( x θ j y ) d y ) 1 p j Ω L s ( 1 n ln ( P R δ ) n ) n α n L 1 j = 1 m f j L ω j p j Ω L s ( 1 n ln ( P R δ ) n ) n α n .
Hence, we obtain that
| I Ω , α Θ ( f ) ( x ) | C δ α M Ω f ( x ) + L 1 j = 1 m f j L ω j p j Ω L s ( 1 n ln ( P R δ ) n ) n α n .
Set δ = ε ( | I Ω , α Θ ( f ) ( x ) | / C M Ω ( f ) ( x ) ) 1 / α , then
exp { n γ ( L I Ω , α Θ ( f ) ( x ) Ω L s j = 1 m f j L ω j p j ) n n α } ln C R n ( M Ω ( f ) ( x ) I Ω , α Θ ( f ) ( x ) ) n / α .
Now we put B 1 = { x B : I Ω , α Θ ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L ω j p j 1 } , B 2 = B B 1 , thus
B 1 exp ( n γ ( L I Ω , α Θ ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L ω j p j ) n / ( n α ) ) ν ω ( x ) d x C R n B 1 ( M Ω ( f ) ( x ) I Ω , α Θ ( f ) ( x ) ) n / α ν ω ( x ) d x C R n B 1 ( M Ω ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L ω j p j ) n / α ν ω ( x ) d x .
By the fact that
M Ω ( f ) ( x ) = sup r > 0 | y | < r | Ω ( y ) | j = 1 m s p j j = 1 m f j ( x θ j y ) d y sup r > 0 j = 1 m ( 1 r n | y | < r | Ω ( y ) | f j p j s ( x θ j y ) d y ) s p j j = 1 m ( M Ω ( f p j s ) ( x ) ) s p j .
Therefore, we get
B 1 exp ( n γ ( L I Ω , α Θ ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L ω j p j ) n / ( n α ) ) ν ω ( x ) d x C R n Ω L n / ( n α ) j = 1 m f j L ω j p j s B 1 j = 1 m ( M Ω ( f j p j s ( x ) ) ) s 2 p j ν ω ( x ) d x C R n Ω L n / ( n α ) j = 1 m f j L ω j p j s j = 1 m ( B 1 ( M Ω ( f j p j s ( x ) ) ) s ω j ( x ) d x ) 1 s s 2 p j C R n Ω L n / ( n α ) j = 1 m f j L ω j p j s j = 1 m f j p j s L ω j s s 2 p j C R n .

Here, in the above third inequality, we have used the well-known weighted result of Hardy-Littlewood maximal function.

From ω j 1 ( j = 1 , 2 , , m ), we get
R n = c B d x c B ω j ( x ) d x = c ω j ( B ) .
Hence,
B 1 exp ( n γ ( L I Ω , α Θ ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L ω j p j ) n / ( n α ) ) ν ω ( x ) d x C j = 1 m ω j ( B ) .
On the other hand,
B 2 exp ( n γ ( L I Ω , α Θ ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L ω j p j ) n / ( n α ) ) ν ω ( x ) d x exp ( n γ ) ( L Ω L s ) n n α B 2 ν ω ( x ) d x C j = 1 m ω j ( B ) .
From the above all, we obtain that
B exp ( n γ ( L I Ω , α Θ ( f ) ( x ) Ω L n / ( n α ) j = 1 m f j L ω j p j ) n / ( n α ) ) ν ω ( x ) d x C j = 1 m ω j ( B ) .

 □

3 The proof of Theorem 1.5

In this section, we will prove Theorem 1.5.

Proof By the well-known Hölder’s inequality, we get
M 1 , α ( f ) ( x ) = sup r > 0 1 | r | n α | y | < r j = 1 m f j ( x y ) d y sup r > 0 1 | r | n α j = 1 m ( | y | < r f j p j s ( x y ) d y ) s p j j = 1 m ( sup r > 0 1 | r | n α | y | < r f j p j s ( x y ) d y ) s p j = j = 1 m ( M 1 , α ( f p j / s ) ( x ) ) s p j .
Hence,
( R n ( M 1 , α ( f ) ( x ) ν ω ( x ) ) r d x ) 1 / r [ R n ( j = 1 m [ M 1 , α ( f p j / s ) ( x ) ω j p j / s ( s ) ] s p j ) r d x ] 1 / r j = 1 m [ R n ( M 1 , α ( f j p j / s ) ( x ) ω p j / s ( x ) ) s r j / p j d x ] p j s r j s p j .
In addition, from the condition ω j p j / s ( x ) A ( s , s r j p j ) , it follows that
[ R n ( M 1 , α ( f j p j / s ) ( x ) ω p j / s ( x ) ) s r j / p j d x ] p j s r j s p j C j [ R n ( f j p j / s ( x ) ω j p j / s ( x ) ) s d x ] 1 / p j .
According to the above, we obtain that
( R n ( M 1 , α ( f ) ( x ) ν ω ( x ) ) r d x ) 1 / r = C j = 1 m ( R n ( f j ( x ) ω j ( x ) ) p j d x ) 1 / p j .
It is easy to see that
M 1 , α Θ ( f ) ( x ) = sup r > 0 1 r n α | y | < r j = 1 m | f j ( x θ j y ) | d y ,

where Θ = ( θ 1 , θ 2 , , θ m ) , θ j R holds, also. □

4 The proof of Theorem 1.3

In this section, we will prove Theorem 1.3.

Proof For any δ > 0 and x B ,
| I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | | ( x y 1 , x y 2 , , x y m ) | < δ j = 1 m | Ω j ( y j ) f j ( y j ) | | ( x y 1 , x y 2 , , x y m ) | m n α d y + | ( x y 1 , x y 2 , , x y m ) | δ j = 1 m | Ω j ( y j ) f j ( y j ) | | ( x y 1 , x y 2 , , x y m ) | m n α d y : = F 1 + F 2 .
For F 1 , let α = j = 1 m α j with α j = n / p j for j = 1 , 2 , , m . Then
F 1 | ( x y 1 , x y 2 , , x y m ) | < δ | Ω j ( y j ) f j ( y j ) | j = 1 m | x y j | n α j d y j = 1 m | x y j | < δ | Ω j ( y j ) f j ( y j ) | | x y j | n α j d y j C j = 1 m δ α j M Ω j ( f j ) ( x ) : = C 1 δ α j = 1 m M Ω j ( f j ) ( x ) ,

where M Ω denotes as M Ω ( f ) ( x ) = sup r > 0 1 r n | x y | < r | Ω ( y ) f ( y ) | d y .

For F 2 , if ( y 1 , y 2 , , y m ) satisfies | ( x y 1 , x y 2 , , x y m ) | δ , then for some j 1 , 2 , , m , | x y j | δ m . Without losing the generalization, we set j = m .

Thus,
F 2 δ / m | x y m | 2 R ( R n ) m 1 j = 1 m | Ω j ( y j ) f j ( y j ) | | ( x y 1 , x y 2 , , x y m ) | m n α d y .
Define that f j 0 = f j χ B ( x , δ / m ) and f j = f f j 0 for j = 1 , 2 , , m . By the condition of ν ω , we have
F 2 r { 0 , } m δ / m | x y m | 2 R ( R n ) m 1 j = 1 m 1 | Ω j ( y j ) f j r j ( y j ) | | Ω m ( y m ) f m ( y m ) | | ( x y 1 , x y 2 , , x y m ) | m n α d y r { 0 , } m δ / m | x y m | 2 R ( R n ) m 1 j = 1 m 1 | Ω j ( y j ) f j r j ( y j ) | | Ω m ( y m ) f m ( y m ) | | ( x y 1 , x y 2 , , x y m ) | m n α ν ω ( y ) d y ,
where r = ( r 1 , r 2 , , r m ) . In the case that r 1 = r 2 = = r m 1 = 0 , by the fact that
| ( x y 1 , x y 2 , , x y m ) | m n α | x y m | m n α = | x y m | n α m | x y m | j = 1 m 1 n / p j | x y m | n α m ( δ m ) j = 1 m 1 n / p j ,
we have
δ / m | x y m | 2 R ( R n ) m 1 j = 1 m 1 | Ω j ( y j ) f j 0 ( y j ) | | Ω ( y m ) f m ( y m ) | | ( x y 1 , x y 2 , , x y m ) | m n α ν ω ( y ) d y j = 1 m 1 δ n p j δ m | x y m | 2 R | Ω m ( y m ) f m ( y m ) | | x y m | n α m ω m 1 / p m ( y m ) d y m × j = 1 m 1 | x y j | < δ / m | Ω j ( y j ) f j ( y j ) | ω j 1 / p j ( y j ) d y j C j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j ( log 2 R m δ ) 1 / p m C j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j ( log 2 R m δ ) ( m n α ) / n .
Consider the case where exactly l of the r j are ∞ for some 1 l m . Without losing the generalization, we only give the argument for r j = , j = 1 , 2 , , l , then
δ / m | x y m | 2 R ( R n ) m 1 j = 1 m Ω j ( y j ) j = 1 l | f j ( y j ) k = l + 1 m 1 f k 0 ( y k ) f m ( y m ) | | ( x y 1 , x y 2 , , x y m ) | m n α ν ω d y k = l + 1 m 1 | x y k | < δ / m | Ω k ( y k ) f k ( y k ) | ω k 1 / p m ( y k ) d y k × j = 1 l δ / m | x y j | 2 R | Ω j ( y j ) f j ( y j ) | | x y j | n α j ω j 1 / p j ( y j ) d y j × δ / m | x y m | 2 R | Ω m ( y m ) f m ( y m ) | | x y m | ( m l ) n k = l + 1 m α k ω m 1 / p m ( y m ) d y m C [ log 2 m R δ ] k = 1 l 1 p m j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j C j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j [ log 2 m R δ ] ( m n α ) / n .
Combining the above cases, we obtain
F 2 C 2 j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j [ log 2 m R δ ] ( m n α ) / n .
Thus, by the estimates for F 1 , F 2 , we have
I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) C 1 δ α j = 1 m M Ω j ( f j ) ( x ) + C 2 j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j [ log 2 m R δ ] ( m n α ) / n .
In particular, we chose δ = 2 m R for all x B , then
I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) C 1 δ α j = 1 m M Ω j ( f j ) ( x ) .
Now, we set
δ = δ ( x ) = ε [ | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | / C 1 j = 1 m M Ω j ( f j ) ( x ) ] 1 / α ,

where ε < 1 .

Then
| I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | ε α | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | + C 2 j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j [ 1 n log ( ( 2 m R ) n [ C 1 j = 1 m M Ω j ( f j ) ( x ) ] n / α ε n | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | n / α ) ] ( m n α ) / n .
Hence,
exp ( k 1 ( | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j ) n / ( m n α ) ) C [ j = 1 m M Ω j ( f j ) ( x ) ] n / α | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | n / α .
Let B 1 = { x B : | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | j = 1 m Ω j L p j ( S n 1 ) f j L p j 1 } and B 2 = B B 1 , then
B 1 exp ( k 1 ( | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j ) n / ( m n α ) ) ν ω d x C R n B 1 ( j = 1 m M Ω j ( f j ) ( x ) j = 1 m Ω j L p j ( S n 1 ) f j L ω j p j ) n / α ν ω d x C R n ( j = 1 m M Ω j ( f j ) L ω j p j Ω j L p j ( S n 1 ) f j L ω j p j ) n / α C R n C j m ω j ( B ) .
On the other hand,
B 2 exp ( k 1 ( | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | j = 1 m Ω j L p j ( S n 1 ) f j L p j ) n / ( m n α ) ) ν ω ( x ) d x exp ( k 1 ) j = 1 m B 2 ω j ( x ) d x C j = 1 m ω j ( B ) .
Combining the above results, we obtain
B exp ( k 1 ( | I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) | j = 1 m Ω j L p j ( S n 1 ) f j L p j ) n / ( m n α ) ) ν ω ( x ) d x k 2 j = 1 m ω j ( B ) ,

where k 1 , k 2 are constants depending only on n, m, α, p, and the p j . □

Authors’ information

  1. H.

    Feng’s current address: Department of Mathematical and Statistical Sciences, University of Alberta, Canada.

     

Declarations

Acknowledgement

The second author was supported partly by NSFC (Grant No. 10701010), NSFC (Key program Grant No. 10931001), Beijing Natural Science Foundation (Grant: 1102023), Program for Changjiang Scholars and Innovative Research Team in University.

Authors’ Affiliations

(1)
Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Ministry of Education, Beijing, 100875, People’s Republic of China

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