Weighted Trudinger inequality associated with rough multilinear fractional type operators

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DOI: 10.1186/1029-242X-2012-179

Cite this article as:
Feng, H. & Xue, Q. J Inequal Appl (2012) 2012: 179. doi:10.1186/1029-242X-2012-179
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Abstract

Let I Ω , α Θ Open image in new window 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 Open image in new window. In this paper, we study the weighted estimates for the Trudinger inequality associated to I Ω , α Θ Open image in new window 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 Open image in new window, where d y = d y 1 d y m Open image in new window.

Keywords

Riesz potential multilinear fractional integral  A p Open image in new window weights  A p , q Open image in new window 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 Open image in new window, such that if u C 1 ( R n ) Open image in new window, normalized and supported in a domain D with finite measure in R n Open image in new window, such that D | u ( x ) | n d x 1 Open image in new window, then there is a constant c 0 Open image in new window depending only on n such that for all β β 0 = n w n 1 1 / ( n 1 ) Open image in new window, where w n 1 Open image in new window 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 | . Open image in new window
(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 Open image in new window (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 < ) . Open image in new window
(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, 16, 17, 18, 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 Open image in new window, 0 < α < n Open image in new window, Ω is homogeneous of degree zero, and Ω L s ( S n 1 ) Open image in new window ( s > 1 Open image in new window), where S n 1 Open image in new window denotes the unit sphere of R n Open image in new window. 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 Open image in new window
(1.3)
and the fractional maximal operator M Ω , α Open image in new window 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 . Open image in new window
(1.4)

Multilinear fractional integral I Ω , α Θ Open image in new window 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 Open image in new window, θ j = 1 Open image in new window, and Ω = 1 Open image in new window, then I Ω , α Θ Open image in new window is just the well-known classical fractional integral operator studied by Muckenhoupt and Wheeden in [15]. We denote it by I α Open image in new window. If Ω 1 Open image in new window, we simply denote I Ω , α Θ = I α Θ Open image in new window. 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 Open image in new window, s = n α Open image in new window, 1 s = 1 p 1 + 1 p 2 + + 1 p m Open image in new window, p j > 1 Open image in new window, j = 1 , 2 , , m Open image in new window, m 2 Open image in new window. DenoteBas a ball with a radiusRin R n Open image in new window. If f j L p j ( B ) Open image in new window, supp ( f j ) B Open image in new window, and Ω L n / ( n α ) ( S n 1 ) Open image in new window, then for any γ < 1 Open image in new window, there is a constantC, independent ofn, α, θ j Open image in new window, 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 , Open image in new window
where L = j = 1 m | θ j | n / p j Open image in new window, Θ = ( θ 1 , θ 2 , , θ m ) Open image in new window, f = ( f 1 , f 2 , , f m ) Open image in new windowand
Ω L n / ( n α ) = ( S n 1 | Ω ( x ) | n / ( n α ) d σ ( x ) ) ( n α ) / n . Open image in new window

The definition of multiple weights A p , q Open image in new window 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 Open image in new window as follows.

Theorem B ([16])

Let 0 < α < n Open image in new window, s = n α Open image in new window, 1 s = 1 p 1 + 1 p 2 + + 1 p m Open image in new window, p j > 1 Open image in new window, ω j ( x ) A p j Open image in new window, and ω j 1 Open image in new window, j = 1 , 2 , , m Open image in new window, m 2 Open image in new window, ν ω = j = 1 m ω j s / p j Open image in new window. DenoteBas a ball with the radiusRin R n Open image in new window, if f j L ω j p j ( B ) Open image in new window, supp ( f j ) B Open image in new window, j = 1 , 2 , , m Open image in new window, then for any γ < 1 Open image in new window, there is a constantC, independent ofn, α, θ j Open image in new window, 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 ) , Open image in new window

where L = j = 1 m | θ j | n / p j Open image in new window, Θ = ( θ 1 , θ 2 , , θ m ) Open image in new window, f = ( f 1 , f 2 , , f m ) Open image in new window.

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 , Open image in new window

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

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

Theorem 1.1Let 0 < α < n Open image in new window, s = n α Open image in new window, 1 s = 1 p 1 + 1 p 2 + + 1 p m Open image in new window, p j > 1 Open image in new window, j = 1 , 2 , , m Open image in new window, m 2 Open image in new window. DenoteBas a ball with radiusRin R n Open image in new window; if f j L ω j p j ( B ) Open image in new window, supp ( f j ) B Open image in new window ( j = 1 , 2 , , m Open image in new window), Ω L n / ( n α ) ( S n 1 ) Open image in new window, and ν ω = j = 1 m ω j s p j Open image in new window, where ω j A s Open image in new window, ω j 1 Open image in new window. Then for any γ < 1 Open image in new window, there is a constantC, independent ofn, α, θ j Open image in new window, 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 ) , Open image in new window

where L = j = 1 m | θ j | n / p j Open image in new window, Θ = ( θ 1 , θ 2 , , θ m ) Open image in new window, f = ( f 1 , f 2 , , f K ) Open image in new window.

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

Remark 1.2 Assume m = 1 Open image in new window, ω j = 1 Open image in new window. If α = 1 Open image in new window, Trudinger [20] proved exponential integrability of I α ( f ) Open image in new window, 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 Open image in new window, for α = 1 Open image in new window 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 Open image in new window, it is true. In 1985, Chang and Marshall [4] proved a similar sharp exponential inequality concerning the Dirichlet integral. Assume m 2 Open image in new window, w j = 1 Open image in new window, then the result was obtained by Grafakos [7] as we have already mentioned above.

Corollary 1.2LetB, f j Open image in new window, p j Open image in new window, s, and ν ω Open image in new windowbe the same as in Theorem  1.1, then I Ω , α Θ ( f ) Open image in new windowis in L q ( ν ω ( B ) ) Open image in new windowfor every q > 0 Open image in new window, 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 Open image in new window

for some constantCdepending only onqonnonαand on the θ j Open image in new window’s.

Theorem 1.3Let m 2 Open image in new window, 0 < α < m n Open image in new window, 1 / p = 1 / p 1 + 1 / p 2 + + 1 / p m = α / n Open image in new windowwith 1 < p i < Open image in new windowfor i = 1 , 2 , , m Open image in new window. LetBbe a ball with radiusRin R n Open image in new windowand let f j L p j ( B ) Open image in new windowbe supported inB, and if Ω j Open image in new windowis homogeneous of degree zero, and Ω j L p j ( S n 1 ) Open image in new window, where S n 1 Open image in new windowdenotes the sphere of R n Open image in new window, and ν ω ( y ) = j = 1 m ω j 1 / p j ( y j ) Open image in new window, where y = ( y 1 , y 2 , , y m ) Open image in new windowand ω j A s Open image in new window, ω j 1 Open image in new window. Then there exist constants k 1 Open image in new window, k 2 Open image in new windowdepending only onn, m, α, p, and the p j Open image in new windowsuch 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 ) . Open image in new window

Remark 1.3 If we take Ω = 1 Open image in new window, w j 1 Open image in new window for j = 1 , , m Open image in new window, 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 Open image in new window, one cannot obtain the conclusion that F 2 C 2 [ log 2 m R δ ] ( m n α ) / n Open image in new window. Thus, our proof gives an alternative correction of Theorem 1.3 in [20].

Corollary 1.4LetB, f j Open image in new window, p j Open image in new window, s, and ν ω Open image in new windowbe the same as in Theorem  1.3. Then I ¯ Ω , α ( f ) Open image in new windowis in L q ( ν ω ( B ) ) Open image in new windowfor every q > 0 Open image in new window, 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 Open image in new window

for some constantCdepending only onqonnonα.

Corollary 1.2 and Corollary 1.4 follow since exponential integrability of I ¯ Ω , α ( f ) Open image in new window 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 1.5If 1 < p j < Open image in new window, 1 s = j = 1 m 1 p j Open image in new window, 1 r = 1 s α n Open image in new window, ω j p j s A ( s , s r j p j ) Open image in new window, 1 / r j = 1 / p j ( 1 α s / n ) Open image in new window, j = 1 , 2 , , m Open image in new window, ν ω = j = 1 m ω j Open image in new window, then there is a constantC, independent f j Open image in new window, 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 , Open image in new window

where f = ( f 1 , f 2 , , f m ) Open image in new window, f j L ω j p j ( R n ) Open image in new window.

2 The proof of Theorem 1.1

In this section, we will prove Theorem 1.1.

Proof For any δ > 0 Open image in new window,
| 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 . Open image in new window
Set P = 2 min { 1 θ j : j = 1 , 2 , , K } Open image in new window. For any R > 0 Open image in new window, denote B ( R ) Open image in new window as a ball with radius R in R n Open image in new window, then for any x B ( R ) Open image in new window, when | x θ j y | < R Open image in new window, | θ j y | < 2 R Open image in new window for j = 1 , , m Open image in new window. Therefore, | y | < R P Open image in new window. 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 . Open image in new window
According to the relationship between s and p j Open image in new window: 1 p 1 + 1 p 2 + + 1 p m + 1 n / ( n α ) = 1 Open image in new window, from the Hölder’s inequality and ν ω 1 Open image in new window, 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 . Open image in new window
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 . Open image in new window
Set δ = ε ( | I Ω , α Θ ( f ) ( x ) | / C M Ω ( f ) ( x ) ) 1 / α Open image in new window, 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 . Open image in new window
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 } Open image in new window, B 2 = B B 1 Open image in new window, 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 . Open image in new window
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 . Open image in new window
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 . Open image in new window

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

From ω j 1 Open image in new window ( j = 1 , 2 , , m Open image in new window), we get
R n = c B d x c B ω j ( x ) d x = c ω j ( B ) . Open image in new window
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 ) . Open image in new window
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 ) . Open image in new window
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 ) . Open image in new window

 □

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 . Open image in new window
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 . Open image in new window
In addition, from the condition ω j p j / s ( x ) A ( s , s r j p j ) Open image in new window, 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 . Open image in new window
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 . Open image in new window
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 , Open image in new window

where Θ = ( θ 1 , θ 2 , , θ m ) Open image in new window, θ j R Open image in new window holds, also. □

4 The proof of Theorem 1.3

In this section, we will prove Theorem 1.3.

Proof For any δ > 0 Open image in new window and x B Open image in new window,
| 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 . Open image in new window
For F 1 Open image in new window, let α = j = 1 m α j Open image in new window with α j = n / p j Open image in new window for j = 1 , 2 , , m Open image in new window. 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 ) , Open image in new window

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

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

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 . Open image in new window
Define that f j 0 = f j χ B ( x , δ / m ) Open image in new window and f j = f f j 0 Open image in new window for j = 1 , 2 , , m Open image in new window. By the condition of ν ω Open image in new window, 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 , Open image in new window
where r = ( r 1 , r 2 , , r m ) Open image in new window. In the case that r 1 = r 2 = = r m 1 = 0 Open image in new window, 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 , Open image in new window
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 . Open image in new window
Consider the case where exactly l of the r j Open image in new window are ∞ for some 1 l m Open image in new window. Without losing the generalization, we only give the argument for r j = Open image in new window, j = 1 , 2 , , l Open image in new window, 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 . Open image in new window
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 . Open image in new window
Thus, by the estimates for F 1 Open image in new window, F 2 Open image in new window, 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 . Open image in new window
In particular, we chose δ = 2 m R Open image in new window for all x B Open image in new window, then
I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) C 1 δ α j = 1 m M Ω j ( f j ) ( x ) . Open image in new window
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 , Open image in new window

where ε < 1 Open image in new window.

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 . Open image in new window
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 . Open image in new window
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 } Open image in new window and B 2 = B B 1 Open image in new window, 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 ) . Open image in new window
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 ) . Open image in new window
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 ) , Open image in new window

where k 1 Open image in new window, k 2 Open image in new window are constants depending only on n, m, α, p, and the p j Open image in new window. □

Authors’ information

  1. H.

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

     

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.

Copyright information

© Feng and Xue; licensee Springer 2012

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems, School of Mathematical SciencesBeijing Normal University, Ministry of EducationBeijingPeople’s Republic of China

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