Journal of Inequalities and Applications

, 2012:179

Weighted Trudinger inequality associated with rough multilinear fractional type operators

Authors

  • Han Feng
    • Laboratory of Mathematics and Complex Systems, School of Mathematical SciencesBeijing Normal University, Ministry of Education
    • Laboratory of Mathematics and Complex Systems, School of Mathematical SciencesBeijing Normal University, Ministry of Education
Open AccessResearch

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

Abstract

Let I Ω , α Θ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq1_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq2_HTML.gif. In this paper, we study the weighted estimates for the Trudinger inequality associated to I Ω , α Θ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq1_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq3_HTML.gif, where d y = d y 1 d y m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq4_HTML.gif.

Keywords

Riesz potential multilinear fractional integral A p https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq5_HTML.gif weights A p , q https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq6_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq7_HTML.gif, such that if u C 1 ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq8_HTML.gif, normalized and supported in a domain D with finite measure in R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq9_HTML.gif, such that D | u ( x ) | n d x 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq10_HTML.gif, then there is a constant c 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq11_HTML.gif depending only on n such that for all β β 0 = n w n 1 1 / ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq12_HTML.gif, where w n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq13_HTML.gif 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 | . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equ1_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq14_HTML.gif (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 < ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equ2_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq15_HTML.gif, 0 < α < n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq16_HTML.gif, Ω is homogeneous of degree zero, and Ω L s ( S n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq17_HTML.gif ( s > 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq18_HTML.gif), where S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq19_HTML.gif denotes the unit sphere of R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq9_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equ3_HTML.gif
(1.3)
and the fractional maximal operator M Ω , α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq20_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equ4_HTML.gif
(1.4)

Multilinear fractional integral I Ω , α Θ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq1_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq21_HTML.gif, θ j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq22_HTML.gif, and Ω = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq23_HTML.gif, then I Ω , α Θ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq1_HTML.gif is just the well-known classical fractional integral operator studied by Muckenhoupt and Wheeden in [15]. We denote it by I α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq14_HTML.gif. If Ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq24_HTML.gif, we simply denote I Ω , α Θ = I α Θ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq25_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq26_HTML.gif, s = n α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq27_HTML.gif, 1 s = 1 p 1 + 1 p 2 + + 1 p m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq28_HTML.gif, p j > 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq29_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif, m 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq31_HTML.gif. Denote B as a ball with a radius R in R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq9_HTML.gif. If f j L p j ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq32_HTML.gif, supp ( f j ) B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq33_HTML.gif, and Ω L n / ( n α ) ( S n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq34_HTML.gif, then for any γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq35_HTML.gif, there is a constant C, independent of n, α, θ j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq36_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equa_HTML.gif
where L = j = 1 m | θ j | n / p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq37_HTML.gif, Θ = ( θ 1 , θ 2 , , θ m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq38_HTML.gif, f = ( f 1 , f 2 , , f m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq39_HTML.gif and
Ω L n / ( n α ) = ( S n 1 | Ω ( x ) | n / ( n α ) d σ ( x ) ) ( n α ) / n . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equb_HTML.gif

The definition of multiple weights A p , q https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq40_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq41_HTML.gif as follows.

Theorem B ([16])

Let 0 < α < n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq26_HTML.gif, s = n α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq42_HTML.gif, 1 s = 1 p 1 + 1 p 2 + + 1 p m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq28_HTML.gif, p j > 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq29_HTML.gif, ω j ( x ) A p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq43_HTML.gif, and ω j 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq44_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif, m 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq31_HTML.gif, ν ω = j = 1 m ω j s / p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq45_HTML.gif. Denote B as a ball with the radius R in R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq9_HTML.gif, if f j L ω j p j ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq46_HTML.gif, supp ( f j ) B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq47_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif, then for any γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq35_HTML.gif, there is a constant C, independent of n, α, θ j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq36_HTML.gif, 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equc_HTML.gif

where L = j = 1 m | θ j | n / p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq48_HTML.gif, Θ = ( θ 1 , θ 2 , , θ m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq38_HTML.gif, f = ( f 1 , f 2 , , f m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq49_HTML.gif.

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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equd_HTML.gif

where i https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq50_HTML.gif is a linear combination of y j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq51_HTML.gifs and x depending on the matrix A. They showed that I α , A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq52_HTML.gif was of strong type ( L p 1 × × L p m , L q ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq53_HTML.gif and weak type ( L p 1 × × L p m , L q , ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq54_HTML.gif. When i ( y 1 , , y m , x ) = x y i https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq55_HTML.gif, we denote this multilinear fractional type operator by I ¯ α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq56_HTML.gif. In 2008, L. Tang [20] obtained the estimation of the exponential integrability of the above operator I ¯ α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq56_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq1_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Eque_HTML.gif

Inspired by the works above, in this paper, we study the Trudinger inequality associated to multilinear fractional integral operators I Ω , α Θ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq1_HTML.gif and I ¯ Ω , α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq57_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq26_HTML.gif, s = n α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq27_HTML.gif, 1 s = 1 p 1 + 1 p 2 + + 1 p m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq58_HTML.gif, p j > 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq29_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif, m 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq31_HTML.gif. Denote B as a ball with radius R in R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq9_HTML.gif; if f j L ω j p j ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq46_HTML.gif, supp ( f j ) B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq59_HTML.gif ( j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif), Ω L n / ( n α ) ( S n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq34_HTML.gif, and ν ω = j = 1 m ω j s p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq60_HTML.gif, where ω j A s https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq61_HTML.gif, ω j 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq44_HTML.gif. Then for any γ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq62_HTML.gif, there is a constant C, independent of n, α, θ j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq36_HTML.gif, 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equf_HTML.gif

where L = j = 1 m | θ j | n / p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq48_HTML.gif, Θ = ( θ 1 , θ 2 , , θ m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq38_HTML.gif, f = ( f 1 , f 2 , , f K ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq63_HTML.gif.

Remark 1.1 If we take Ω = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq23_HTML.gif, then Theorem 1.1 coincides with Theorem B. If w j 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq64_HTML.gif for j = 1 , , K https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq65_HTML.gif, then Theorem 1.1 is just Theorem A that appeared in [6]. We give an example of ν ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq66_HTML.gif as follows: Let ω j ( x ) = ( 1 + | x | ) α j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq67_HTML.gif ( α j 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq68_HTML.gif for each j), then ν ω ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq69_HTML.gif satisfy the conditions of the above Theorem 1.1.

Remark 1.2 Assume m = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq70_HTML.gif, ω j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq71_HTML.gif. If α = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq72_HTML.gif, Trudinger [20] proved exponential integrability of I α ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq73_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq74_HTML.gif, for α = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq72_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq75_HTML.gif, it is true. In 1985, Chang and Marshall [4] proved a similar sharp exponential inequality concerning the Dirichlet integral. Assume m 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq76_HTML.gif, w j = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq77_HTML.gif, then the result was obtained by Grafakos [7] as we have already mentioned above.

Corollary 1.2 Let B, f j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq78_HTML.gif, p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq79_HTML.gif, s, and ν ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq66_HTML.gif be the same as in Theorem  1.1, then I Ω , α Θ ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq80_HTML.gif is in L q ( ν ω ( B ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq81_HTML.gif for every q > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq82_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equg_HTML.gif

for some constant C depending only on q on n on α and on the θ j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq36_HTML.gif ’s.

Theorem 1.3 Let m 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq83_HTML.gif, 0 < α < m n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq84_HTML.gif, 1 / p = 1 / p 1 + 1 / p 2 + + 1 / p m = α / n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq85_HTML.gif with 1 < p i < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq86_HTML.gif for i = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq87_HTML.gif. Let B be a ball with radius R in R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq9_HTML.gif and let f j L p j ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq32_HTML.gif be supported in B, and if Ω j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq88_HTML.gif is homogeneous of degree zero, and Ω j L p j ( S n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq89_HTML.gif, where S n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq19_HTML.gif denotes the sphere of R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq9_HTML.gif, and ν ω ( y ) = j = 1 m ω j 1 / p j ( y j ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq90_HTML.gif, where y = ( y 1 , y 2 , , y m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq91_HTML.gif and ω j A s https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq61_HTML.gif, ω j 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq44_HTML.gif. Then there exist constants k 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq92_HTML.gif, k 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq93_HTML.gif depending only on n, m, α, p, and the p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq79_HTML.gif 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equh_HTML.gif

Remark 1.3 If we take Ω = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq23_HTML.gif, w j 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq64_HTML.gif for j = 1 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq94_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq95_HTML.gif, one cannot obtain the conclusion that F 2 C 2 [ log 2 m R δ ] ( m n α ) / n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq96_HTML.gif. Thus, our proof gives an alternative correction of Theorem 1.3 in [20].

Corollary 1.4 Let B, f j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq78_HTML.gif, p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq79_HTML.gif, s, and ν ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq66_HTML.gif be the same as in Theorem  1.3. Then I ¯ Ω , α ( f ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq97_HTML.gif is in L q ( ν ω ( B ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq98_HTML.gif for every q > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq82_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equi_HTML.gif

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq97_HTML.gif 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 < https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq99_HTML.gif, 1 s = j = 1 m 1 p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq100_HTML.gif, 1 r = 1 s α n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq101_HTML.gif, ω j p j s A ( s , s r j p j ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq102_HTML.gif, 1 / r j = 1 / p j ( 1 α s / n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq103_HTML.gif, j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif, ν ω = j = 1 m ω j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq104_HTML.gif, then there is a constant C, independent f j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq78_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equj_HTML.gif

where f = ( f 1 , f 2 , , f m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq49_HTML.gif, f j L ω j p j ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq105_HTML.gif.

2 The proof of Theorem 1.1

In this section, we will prove Theorem 1.1.

Proof For any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq106_HTML.gif,
| 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equk_HTML.gif
Set P = 2 min { 1 θ j : j = 1 , 2 , , K } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq107_HTML.gif. For any R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq108_HTML.gif, denote B ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq109_HTML.gif as a ball with radius R in R n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq9_HTML.gif, then for any x B ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq110_HTML.gif, when | x θ j y | < R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq111_HTML.gif, | θ j y | < 2 R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq112_HTML.gif for j = 1 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq113_HTML.gif. Therefore, | y | < R P https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq114_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equl_HTML.gif
According to the relationship between s and p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq79_HTML.gif: 1 p 1 + 1 p 2 + + 1 p m + 1 n / ( n α ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq115_HTML.gif, from the Hölder’s inequality and ν ω 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq116_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equm_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equn_HTML.gif
Set δ = ε ( | I Ω , α Θ ( f ) ( x ) | / C M Ω ( f ) ( x ) ) 1 / α https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq117_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equo_HTML.gif
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 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq118_HTML.gif, B 2 = B B 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq119_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equp_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equq_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equr_HTML.gif

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

From ω j 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq44_HTML.gif ( j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif), we get
R n = c B d x c B ω j ( x ) d x = c ω j ( B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equs_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equt_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equu_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equv_HTML.gif

 □

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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equw_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equx_HTML.gif
In addition, from the condition ω j p j / s ( x ) A ( s , s r j p j ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq120_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equy_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equz_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equaa_HTML.gif

where Θ = ( θ 1 , θ 2 , , θ m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq38_HTML.gif, θ j R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq121_HTML.gif holds, also. □

4 The proof of Theorem 1.3

In this section, we will prove Theorem 1.3.

Proof For any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq106_HTML.gif and x B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq122_HTML.gif,
| 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equab_HTML.gif
For F 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq123_HTML.gif, let α = j = 1 m α j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq124_HTML.gif with α j = n / p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq125_HTML.gif for j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif. 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equac_HTML.gif

where M Ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq126_HTML.gif denotes as M Ω ( f ) ( x ) = sup r > 0 1 r n | x y | < r | Ω ( y ) f ( y ) | d y https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq127_HTML.gif.

For F 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq128_HTML.gif, if ( y 1 , y 2 , , y m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq129_HTML.gif satisfies | ( x y 1 , x y 2 , , x y m ) | δ https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq130_HTML.gif, then for some j 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq131_HTML.gif, | x y j | δ m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq132_HTML.gif. Without losing the generalization, we set j = m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq133_HTML.gif.

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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equad_HTML.gif
Define that f j 0 = f j χ B ( x , δ / m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq134_HTML.gif and f j = f f j 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq135_HTML.gif for j = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq30_HTML.gif. By the condition of ν ω https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq136_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equae_HTML.gif
where r = ( r 1 , r 2 , , r m ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq137_HTML.gif. In the case that r 1 = r 2 = = r m 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq95_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equaf_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equag_HTML.gif
Consider the case where exactly l of the r j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq138_HTML.gif are ∞ for some 1 l m https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq139_HTML.gif. Without losing the generalization, we only give the argument for r j = https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq140_HTML.gif, j = 1 , 2 , , l https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq141_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equah_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equai_HTML.gif
Thus, by the estimates for F 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq123_HTML.gif, F 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq128_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equaj_HTML.gif
In particular, we chose δ = 2 m R https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq142_HTML.gif for all x B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq122_HTML.gif, then
I ¯ Ω , α ( f 1 , f 2 , , f m ) ( x ) C 1 δ α j = 1 m M Ω j ( f j ) ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equak_HTML.gif
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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equal_HTML.gif

where ε < 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq143_HTML.gif.

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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equam_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equan_HTML.gif
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 } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq144_HTML.gif and B 2 = B B 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq119_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equao_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equap_HTML.gif
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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_Equaq_HTML.gif

where k 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq92_HTML.gif, k 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq93_HTML.gif are constants depending only on n, m, α, p, and the p j https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-179/MediaObjects/13660_2012_Article_302_IEq79_HTML.gif. □

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.