Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 161405, 22 pages
doi:10.1155/2009/161405
Research Article

A Kind of Estimate of Difference Norms in Anisotropic Weighted Sobolev-Lorentz Spaces

Jiecheng Chen1 and Hongliang Li1,2

1Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2Department of Mathematics, Zhejiang Education Institute, Hangzhou 310012, China

Received 27 April 2009; Accepted 2 July 2009

Academic Editor: Shusen Ding

Copyright © 2009 Jiecheng Chen and Hongliang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the functions spaces on ℝ 𝑛 for which the generalized partial derivatives 𝐷 𝑟 𝑘 𝑘 𝑓 exist and belong to different Lorentz spaces Λ 𝑝 𝑘 , 𝑠 𝑘 ( 𝑤 ) , where 𝑝 𝑘 > 1 and 𝑤 is nonincreasing and satisfies some special conditions. For the functions in these weighted Sobolev-Lorentz spaces, the estimates of the Besov type norms are found. The methods used in the paper are based on some estimates of nonincreasing rearrangements and the application of 𝐵 𝑝 , 𝐵 𝑝 , ∞ weights.

1. Introduction

In this paper we study functions 𝑓 on ℝ 𝑛 which possess the generalized partial derivatives 𝐷 𝑟 𝑘 𝑘 𝜕 𝑓 ≡ 𝑟 𝑘 𝑓 𝜕 𝑥 𝑟 𝑘 𝑘 𝑟 𝑘 ∈ ℕ . ( 1 . 1 ) Our main goal is to obtain some norm estimates for the differences Δ 𝑟 𝑘 𝑘 ( ℎ ) 𝑓 ( 𝑥 ) ≡ 𝑟 𝑘 𝑗 = 0 ( − 1 ) 𝑟 𝑘 − 𝑗 𝑟 𝑘 𝑗 𝑓 𝑥 + 𝑗 ℎ 𝑒 𝑘 ( ℎ ∈ ℝ ) ( 1 . 2 ) ( 𝑒 𝑘 being the unit coordinate vector).

The classic Sobolev embedding theorem asserts that for any function 𝑓 in Sobolev space 𝑊 1 𝑝 ( ℝ 𝑛 ) ( 1 ≤ 𝑝 < 𝑛 ) ‖ 𝑓 ‖ 𝑞 ∗ ≤ 𝐶 𝑛 𝑘 = 1 ‖ ‖ ‖ 𝜕 𝑓 𝜕 𝑥 𝑘 ‖ ‖ ‖ 𝑝 , 𝑞 ∗ = 𝑛 𝑝 𝑛 − 𝑝 . ( 1 . 3 ) Sobolev proved this inequality in 1938 for 𝑝 > 1 . His method, based on integral representations, did not work in the case 𝑝 = 1 . Only at the end of fifties Gagliardo and Nirenberg gave simple proofs of inequality (1.3) for all 1 ≤ 𝑝 < 𝑛 . Inequality (1.3) has been generalized in various directions (see [1–6] for details). It was proved that the left hand side in (1.3) can be replaced by the stronger Lorentz norm, that is, there holds the inequality ‖ 𝑓 ‖ 𝑞 ∗ , 𝑝 ≤ 𝐶 𝑛 𝑘 = 1 ‖ ‖ ‖ 𝜕 𝑓 𝜕 𝑥 𝑘 ‖ ‖ ‖ 𝑝 , 1 ≤ 𝑝 < 𝑛 . ( 1 . 4 ) For 𝑝 > 1 the result follows by interpolation (see [7, 8]). In the case 𝑝 = 1 some geometric inequalities were applied to prove (1.4) (see [9–13]).

The sharp estimates of the norms of differences for the functions in Sobolev spaces have firstly been proved by Besov et al. [1, Volume 2, page 72]. For the space 𝑊 1 𝑝 ( ℝ 𝑛 ) ( 1 ≤ 𝑝 < 𝑛 ) Il'in's result reads as follows: If 𝑛 ∈ ℕ , 1 < 𝑝 < 𝑞 < ∞ and 𝛼 ≡ 1 − 𝑛 ( 1 / 𝑝 − 1 / 𝑞 ) > 0 , then 𝑛 𝑘 = 1 ∞ 0 ℎ − 𝛼 ‖ Δ 1 𝑘 ( ℎ ) 𝑓 ‖ 𝑞 𝑝 𝑑 ℎ ℎ 1 / 𝑝 ≤ 𝐶 𝑛 𝑘 = 1 ‖ ‖ ‖ 𝜕 𝑓 𝜕 𝑥 𝑘 ‖ ‖ ‖ 𝑝 . ( 1 . 5 ) Actually, this means that there holds the continuous embedding to the Besov space 𝑊 1 𝑝 ( ℝ 𝑛 ) ↪ 𝐵 𝛼 𝑝 , 𝑞 ( ℝ 𝑛 ) . ( 1 . 6 ) It is easy to see that inequality (1.5) fails to hold for 𝑝 = 𝑛 = 1 , but, it was proved in [14] that (1.5) is true for 𝑝 = 1 and 𝑛 ≥ 2 .

The generalization of the inequality (1.5) to the spaces 𝑊 𝑟 1 , … , 𝑟 𝑛 𝑝 was given in [12]. That is 𝑛 𝑘 = 1 ∞ 0 ℎ − 𝛼 𝑘 ‖ Δ 𝑟 𝑘 𝑘 ( ℎ ) 𝑓 ‖ 𝑞 , 𝑝 𝑝 𝑑 ℎ ℎ 1 / 𝑝 ≤ 𝐶 𝑛 𝑘 = 1 ‖ 𝐷 𝑟 𝑘 𝑘 𝑓 ‖ 𝑝 , ( 1 . 7 ) where ∑ 0 < 1 / 𝑝 − 1 / 𝑞 < 𝑟 / 𝑛 , 𝑟 = 𝑛 ( 𝑛 𝑖 = 1 𝑟 𝑖 − 1 ) − 1 , and 𝛼 𝑘 = 𝑟 𝑘 [ 1 − ( 𝑟 / 𝑛 ) ( 1 / 𝑝 − 1 / 𝑞 ) ] ; the inequality is valid if 𝑝 > 1 , 𝑛 ≥ 1 or 𝑝 = 1 , 𝑛 ≥ 2 . Using (1.7), we get the following continuous embedding: 𝑊 𝑟 1 , … , 𝑟 𝑛 𝑝 ( ℝ 𝑛 ) ↪ 𝐵 𝛼 1 , … , 𝛼 𝑛 𝑞 , 𝑝 ( ℝ 𝑛 ) . ( 1 . 8 ) For 𝑝 > 1 this embedding was proved by Besov et al. [1, Volume 2, page 72]. The main result in [12] is the proof of (1.7) for 𝑝 = 1 , 𝑛 ≥ 2 .

In [15], there was the sharp estimates of the type (1.7) when the derivatives 𝐷 𝑟 𝑘 𝑘 𝑓 belong to different Lorentz spaces 𝐿 𝑝 𝑘 , 𝑠 𝑘 . Before stating the theorem, we give some notations. Let 𝑆 0 ( ℝ 𝑛 ) be the class of all measurable and almost everywhere finite functions 𝑓 on ℝ 𝑛 such that for each 𝑦 > 0 , 𝜆 𝑓 | | ( 𝑦 ) = 𝑥 ∈ ℝ 𝑛 ∶ | | 𝑓 | | | | ( 𝑥 ) > 𝑦 < ∞ . ( 1 . 9 ) Let 𝑟 𝑘 ∈ ℕ and 1 ≤ 𝑝 𝑘 , 𝑠 𝑘 < ∞ for 𝑘 = 1 , … , 𝑛 ( 𝑛 ≥ 2 ) . Denote 𝑟 = 𝑛 𝑛 𝑘 = 1 1 𝑟 𝑘 − 1 𝑛 , 𝑝 = 𝑟 𝑛 𝑘 = 1 1 𝑝 𝑘 𝑟 𝑘 − 1 , 𝑛 𝑠 = 𝑟 𝑛 𝑘 = 1 1 𝑠 𝑘 𝑟 𝑘 − 1 . ( 1 . 1 0 )

Now we state the main theorem in [15].

Theorem 1.1. Let 𝑛 ≥ 2 , 𝑟 𝑘 ∈ ℕ , 1 ≤ 𝑝 𝑘 , 𝑠 𝑘 < ∞ , and 𝑠 𝑘 = 1 if 𝑝 𝑘 = 1 . Let 𝑟 , 𝑝 , and 𝑠 be the numbers defined by (1.10). For every 𝑝 𝑗 ( 1 ≤ 𝑗 ≤ 𝑛 ) satisfying the condition 𝜌 𝑗 ≡ 𝑟 𝑛 + 1 𝑝 𝑗 − 1 𝑝 > 0 , ( 1 . 1 1 ) take arbitrary 𝑞 𝑗 > 𝑝 𝑗 such that 1 𝑞 𝑗 > 1 𝑝 − 𝑟 𝑛 , ( 1 . 1 2 ) and denote 𝐻 𝑗 1 = 1 − 𝜌 𝑗 1 𝑝 𝑗 − 1 𝑞 𝑗 , 𝛼 𝑗 = 𝐻 𝑗 𝑟 𝑗 , 1 𝜃 𝑗 = 1 − 𝐻 𝑗 𝑠 + 𝐻 𝑗 𝑠 𝑗 , ( 1 . 1 3 ) then for any function 𝑓 ∈ 𝑆 0 ( ℝ 𝑛 ) which has the weak derivatives 𝐷 𝑟 𝑘 𝑘 𝑓 ∈ 𝐿 𝑝 𝑘 , 𝑠 𝑘 ( ℝ 𝑛 ) ( 𝑘 = 1 , … , 𝑛 ) there holds the inequality ∞ 0 ℎ − 𝛼 𝑗 ‖ ‖ Δ 𝑟 𝑗 𝑗 ‖ ‖ ( ℎ ) 𝑓 𝑞 𝑗 , 1 𝜃 𝑑 ℎ ℎ 1 / 𝜃 𝑗 ≤ 𝐶 𝑛 𝑘 = 1 ‖ 𝐷 𝑟 𝑘 𝑘 𝑓 ‖ 𝑝 𝑘 , 𝑠 𝑘 , ( 1 . 1 4 ) where 𝐶 is a constant that does not depend on 𝑓 .

In many cases, the Lorentz space should be substituted by more general space, the weighted Lorentz space. In this paper, we will generalize the above result when the weighted Lorentz spaces Λ 𝑝 𝑘 , 𝑠 𝑘 ( 𝑤 ) take place of 𝐿 𝑝 𝑘 , 𝑠 𝑘 , where 𝑤 is a weight on ℝ + which satisfies some special conditions.

2. Auxiliary Proposition

Let ℳ ( 𝑋 , 𝜇 ) be the class of all measurable and almost everywhere finite functions on 𝑋 . For 𝑓 ∈ ℳ ( 𝑋 , 𝜇 ) , a nonincreasing rearrangement of 𝑓 is a nonincreasing function 𝑓 ∗ on ℝ + ≡ ( 0 , + ∞ ) , that is, equimeasurable with | 𝑓 | . The rearrangement 𝑓 ∗ can be defined by the equality 𝑓 ∗ ( 𝑡 ) = i n f 𝜆 ∶ 𝜇 𝑓 ( 𝜆 ) ≤ 𝑡 , 0 < 𝑡 < ∞ , ( 2 . 1 ) where 𝜇 𝑓 | | 𝑓 | | ( 𝜆 ) = 𝜇 𝑥 ∈ 𝑋 ∶ ( 𝑥 ) > 𝜆 , 𝜆 ≥ 0 . ( 2 . 2 ) If 𝑋 = ℝ 𝑛 , 𝜇 ( 𝐸 ) = | 𝐸 | , then the following relation holds [16, Chapter 2]: s u p | 𝐸 | = 𝑡 𝐸 | | | | 𝑓 ( 𝑥 ) 𝑑 𝑥 = 𝑡 0 𝑓 ∗ ( 𝑢 ) 𝑑 𝑢 . ( 2 . 3 ) Set 𝑓 ∗ ∗ ( 1 𝑡 ) = 𝑡 𝑡 0 𝑓 ∗ ( 𝑠 ) 𝑑 𝑠 . ( 2 . 4 ) Assume that 0 < 𝑞 , 𝑝 < ∞ . A function 𝑓 ∈ ℳ ( 𝑋 , 𝜇 ) belongs to the Lorentz space 𝐿 𝑞 , 𝑝 ( 𝑋 ) if ‖ 𝑓 ‖ 𝑞 , 𝑝 = ∞ 0 ( 𝑡 1 / 𝑞 𝑓 ∗ ( 𝑡 ) ) 𝑝 𝑑 𝑡 𝑡 1 / 𝑝 < ∞ . ( 2 . 5 ) For 0 < 𝑝 < ∞ , the space 𝐿 𝑝 , ∞ ( 𝑋 ) is defined as the class of all 𝑓 ∈ ℳ ( 𝑋 , 𝜇 ) such that ‖ 𝑓 ‖ 𝑝 , ∞ = s u p 𝑡 > 0 𝑡 1 / 𝑝 𝑓 ∗ ( 𝑡 ) < ∞ . ( 2 . 6 ) We also let 𝐿 ∞ , ∞ ( 𝑋 ) = 𝐿 ∞ ( 𝑋 ) . Let 𝑤 be a weight in ℝ + (nonnegative locally integrable functions in ℝ + ).

If ( 𝑋 , 𝜇 ) = ( ℝ + , 𝑤 ( 𝑡 ) 𝑑 𝑡 ) , we replace 𝐿 𝑞 , 𝑝 ( 𝑋 ) with 𝐿 𝑞 , 𝑝 ( 𝑤 ) . For 0 < 𝑝 , 𝑞 < ∞ , or 0 < 𝑝 ≤ ∞ and 𝑞 = ∞ , the weighted Lorentz space Λ ℝ 𝑝 , 𝑞 𝑛 ( 𝑤 ) = Λ 𝑝 , 𝑞 ( 𝑤 ) is defined in [9, Chapter 2] by Λ 𝑝 , 𝑞 ( 𝑤 ) = 𝑓 ∈ ℳ ( ℝ 𝑛 ) ∶ ‖ 𝑓 ‖ Λ 𝑝 , 𝑞 ( 𝑤 ) = ‖ 𝑓 ∗ ‖ 𝐿 𝑝 , 𝑞 ( 𝑤 ) < ∞ . ( 2 . 7 ) If 𝑝 = 𝑞 , denote Λ 𝑝 ( 𝑤 ) = Λ 𝑝 , 𝑝 ( 𝑤 ) . It is well known that Λ 𝑝 , 𝑞 ( 1 ) = 𝐿 𝑝 , 𝑞 ( ℝ 𝑛 ) , ( 2 . 8 ) and if 0 < 𝑝 , 𝑞 < ∞ , then Λ 𝑝 , 𝑞 ( 𝑤 ) = Λ 𝑞 𝑤 , ( 2 . 9 ) where 𝑤 ( 𝑡 ) = 𝑊 𝑞 / 𝑝 − 1 ( 𝑡 ) 𝑤 ( 𝑡 ) , 𝑊 ( 𝑡 ) = 𝑡 0 𝑤 ( 𝑠 ) 𝑑 𝑠 . ( 2 . 1 0 )

In following part of this paper, we will always denote ∫ 𝑊 ( 𝑡 ) = 𝑡 0 𝑤 ( 𝑠 ) 𝑑 𝑠 .

The weighted Lorentz spaces have close connection with weights of 𝐵 𝑝 , 𝐵 𝑝 , ∞ for 0 < 𝑝 < ∞ (see [9, Chapter 1]). Let 𝐴 be the Hardy operator as follows: 1 𝐴 𝑓 ( 𝑡 ) = 𝑡 𝑡 0 𝑓 ( 𝑠 ) 𝑑 𝑠 , 𝑡 > 0 . ( 2 . 1 1 ) The space 𝐿 𝑝 d e c is the cone of all nonnegative nonincreasing functions in 𝐿 𝑝 . We denote 𝑤 ∈ 𝐵 𝑝 if 𝐴 ∶ 𝐿 𝑝 d e c ( 𝑤 ) ⟶ 𝐿 𝑝 ( 𝑤 ) ( 2 . 1 2 ) is bounded and denote 𝑤 ∈ 𝐵 𝑝 , ∞ if 𝐴 ∶ 𝐿 𝑝 d e c ( 𝑤 ) ⟶ 𝐿 𝑝 , ∞ ( 𝑤 ) ( 2 . 1 3 ) is bounded.

Lemma 2.1 (Generalized Hardy's inequalities). Let 𝜓 be nonnegative, measurable on ( 0 , ∞ ) and suppose − ∞ < 𝜆 < 1 , 1 ≤ 𝑞 ≤ ∞ , and 𝑤 is a weight in ℝ + , 𝑊 ( ∞ ) = ∞ , then one has ∞ 0 𝑊 ( 𝑡 ) 𝜆 1 𝑊 ( 𝑡 ) 𝑡 0 𝜓 ( 𝑠 ) 𝑤 ( 𝑠 ) 𝑑 𝑠 𝑞 𝑤 ( 𝑡 ) 𝑊 ( 𝑡 ) 𝑑 𝑡 1 / 𝑞 ≤ 1 1 − 𝜆 ∞ 0 𝑊 ( 𝑡 ) 𝜆 𝜓 ( 𝑡 ) 𝑞 𝑤 ( 𝑡 ) 𝑊 ( 𝑡 ) 𝑑 𝑡 1 / 𝑞 , ∞ 0 𝑊 ( 𝑡 ) 1 − 𝜆 ∞ 𝑡 𝜓 ( 𝑠 ) 𝑤 ( 𝑠 ) 𝑊 ( 𝑠 ) 𝑑 𝑠 𝑞 𝑤 ( 𝑡 ) 𝑊 ( 𝑡 ) 𝑑 𝑡 1 / 𝑞 ≤ 1 1 − 𝜆 ∞ 0 𝑊 ( 𝑡 ) 1 − 𝜆 𝜓 ( 𝑡 ) 𝑞 𝑤 ( 𝑡 ) 𝑊 ( 𝑡 ) 𝑑 𝑡 1 / 𝑞 ( 2 . 1 4 ) (with the obvious modification if 𝑞 = ∞ ).

Proof. It is easy to obtain this result applying Hardy's inequality [16].

Lemma 2.2. Let 𝜓 ∈ Λ 𝑝 , 𝑠 ( 𝑤 ) ( 1 ≤ 𝑝 , 𝑠 < ∞ ) be a nonnegative nonincreasing function on ℝ + , 𝑤 be a nonincreasing weight on ℝ + and there exists 𝐴 > 0 , such that 𝑊 ( 𝜉 𝑡 ) ≥ 𝜉 𝐴 𝑊 ( 𝑡 ) , ∀ 𝜉 > 1 , ∀ 𝑡 > 0 , ( 2 . 1 5 ) Then for 𝛿 > 0 there exists a continuously differentiable 𝜙 on ℝ + such that
(i) 𝜓 ( 𝑡 ) ≤ 𝐶 𝜙 ( 𝑡 ) , 𝑡 ∈ ℝ + , (ii) 𝜙 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 − 𝛿 decreases and 𝜙 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 + 𝛿 increases on ℝ + ,(iii) ‖ 𝜙 ‖ Λ 𝑝 , 𝑠 ( 𝑤 ) ≤ 𝐶 ‖ 𝜓 ‖ Λ 𝑝 , 𝑠 ( 𝑤 ) ,
where 𝐶 is a constant depends only on 𝑝 , 𝛿 , and 𝐴 .

Proof. Without loss of generality, we may suppose that 𝛿 < 1 / 𝑝 . Set 𝜙 1 ( 𝑡 ) = 𝑊 ( 𝑡 ) 𝛿 − 1 / 𝑝 ∞ 𝑡 / 2 𝜓 ( 𝑢 ) 𝑊 ( 𝑢 ) 1 / 𝑝 − 𝛿 𝑤 ( 𝑢 ) 𝑊 ( 𝑢 ) 𝑑 𝑢 . ( 2 . 1 6 ) Then 𝜙 1 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 − 𝛿 decreases and 𝜙 1 ( 𝑡 ) ≥ 𝑊 ( 𝑡 ) 𝛿 − 1 / 𝑝 𝑡 𝑡 / 2 𝜓 ( 𝑢 ) 𝑊 ( 𝑢 ) 1 / 𝑝 − 𝛿 𝑤 ( 𝑢 ) 𝑊 ( 𝑢 ) 𝑑 𝑢 ≥ 𝑊 ( 𝑡 ) 𝛿 − 1 / 𝑝 𝜓 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 − 𝛿 − 𝑊 ( 𝑡 / 2 ) 1 / 𝑝 − 𝛿 . 1 / 𝑝 − 𝛿 ( 2 . 1 7 ) Using the conditions which 𝑤 satisfy, it gives 𝜙 1 ( 𝑡 ) ≥ 𝐶 𝜓 ( 𝑡 ) . ( 2 . 1 8 ) Furthermore, noticing 𝑤 is nonincreasing and applying Lemma 2.1, we get that ‖ 𝜙 1 ‖ Λ 𝑝 , 𝑠 ( 𝑤 ) = 2 ∞ 0 𝑊 ( 2 ℎ ) 𝛿 ∞ ℎ 𝑊 ( 𝑢 ) 1 / 𝑝 − 𝛿 𝜓 ( 𝑢 ) 𝑤 ( 𝑢 ) 𝑊 ( 𝑢 ) 𝑑 𝑢 𝑠 𝑤 ( 2 ℎ ) 𝑊 ( 2 ℎ ) 𝑑 ℎ 1 / 𝑠 ≤ 2 1 / 𝑠 + 𝛿 ∞ 0 𝑊 ( ℎ ) 𝛿 ∞ ℎ 𝑊 ( 𝑢 ) 1 / 𝑝 − 𝛿 𝜓 ( 𝑢 ) 𝑤 ( 𝑢 ) 𝑊 ( 𝑢 ) 𝑑 𝑢 𝑠 𝑤 ( ℎ ) 𝑊 ( ℎ ) 𝑑 ℎ 1 / 𝑠 ≤ 𝐶 ∞ 0 𝑊 ( ℎ ) 1 / 𝑝 𝜓 ( ℎ ) 𝑠 𝑤 ( ℎ ) 𝑊 ( ℎ ) 𝑑 ℎ 1 / 𝑠 = 𝐶 ‖ 𝜓 ‖ Λ 𝑝 , 𝑠 ( 𝑤 ) . ( 2 . 1 9 ) now set 1 𝜙 ( 𝑡 ) = 𝛿 + 𝑝 𝑊 ( 𝑡 ) − 1 / 𝑝 − 𝛿 𝑡 0 𝜙 1 ( 𝑢 ) 𝑊 ( 𝑢 ) 𝛿 + 1 / 𝑝 𝑤 ( 𝑢 ) 𝑊 ( 𝑢 ) 𝑑 𝑢 . ( 2 . 2 0 ) Then 𝜙 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 + 𝛿 increases on ℝ + , and 𝜙 ( 𝑡 ) ≥ 𝜙 1 ( 𝑡 ) ≥ 𝐶 𝜓 ( 𝑡 ) . ( 2 . 2 1 ) Furthermore, 𝜙 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 − 𝛿 = 1 𝛿 + 𝑝 𝑊 ( 𝑡 ) − 2 𝛿 𝑡 0 𝜙 1 ( 𝑢 ) 𝑊 ( 𝑢 ) 𝛿 + 1 / 𝑝 𝑤 ( 𝑢 ) 𝑊 ( 𝑢 ) 𝑑 𝑢 = 𝑊 ( 𝑡 ) − 2 𝛿 𝑡 0 𝜙 1 ( 𝑢 ) 𝑑 𝑊 ( 𝑢 ) 𝛿 + 1 / 𝑝 = 𝑊 ( 𝑡 ) − 2 𝛿 𝑊 ( 𝑡 ) 2 𝛿 0 𝜙 1 ( ℎ ( 𝑣 ) ) 𝑣 ( 1 / 𝑝 − 𝛿 ) / ( 2 𝛿 ) 𝑑 𝑣 , ( 2 . 2 2 ) where 𝑣 = 𝑊 ( 𝑢 ) 2 𝛿 , ℎ ( 𝑣 ) = 𝑢 , that is, ℎ ( 𝑣 ) = 𝑊 − 1 ( 𝑣 1 / ( 2 𝛿 ) ) . Since 𝜙 1 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 − 𝛿 is decreasing function on ℝ + , thus 𝜙 1 ( ℎ ( 𝑣 ) ) 𝑣 ( 1 / 𝑝 − 𝛿 ) / ( 2 𝛿 ) is decreasing and 𝜙 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 − 𝛿 is also decreasing on ℝ + .
Finally, using Lemma 2.1 and (2.19), we get (iii). The Lemma 2.2 is proved.

Let 𝑟 𝑘 ∈ ℕ and 1 < 𝑝 𝑘 < ∞ for 𝑘 = 1 , … , 𝑛 ( 𝑛 ≥ 2 ) . Denote 𝑟 = 𝑛 𝑛 𝑗 = 1 1 𝑟 𝑗 − 1 𝑛 , 𝑝 = 𝑟 𝑛 𝑗 = 1 1 𝑝 𝑗 𝑟 𝑗 − 1 , 𝛾 𝑘 1 = 1 − 𝑟 𝑘 𝑟 𝑛 + 1 𝑝 𝑘 − 1 𝑝 . ( 2 . 2 3 ) Then 𝛾 𝑘 > 0 and 𝑛 𝑘 = 1 𝛾 𝑘 = 𝑛 − 1 . ( 2 . 2 4 )

To prove our main results we use the estimates of the rearrangement of a given function in term of its derivatives 𝐷 𝑟 𝑘 𝑘 𝑓 ( 𝑘 = 1 , … , 𝑛 ) .

We will use the notations (2.23).

Lemma 2.3. Let 𝑟 𝑘 ∈ ℕ , 1 < 𝑝 𝑘 < ∞ , 1 ≤ 𝑠 𝑘 < ∞ for 𝑘 = 1 , … , 𝑛 ( 𝑛 ≥ 2 ) and 𝑤 is continuous weight on ℝ + . Set 𝑛 𝑠 = 𝑟 𝑛 𝑗 = 1 1 𝑠 𝑗 𝑟 𝑗 − 1 . ( 2 . 2 5 ) Let 1 0 < 𝛿 < 4 m i n 𝛾 𝑗 < 1 1 − 𝛾 𝑗 , ( 2 . 2 6 ) and suppose that 𝜙 𝑘 ∈ Λ 𝑝 𝑘 , 𝑠 𝑘 ( 𝑤 ) ( 𝑘 = 1 , … , 𝑛 ) are positive continuously differentiable functions with 𝜙 𝑘 ( 𝑡 ) < 0 on ℝ + such that 𝜙 𝑘 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 𝑘 − 𝛿 decreases and 𝜙 𝑘 ( 𝑡 ) 𝑊 ( 𝑡 ) 1 / 𝑝 𝑘 + 𝛿 increases on ℝ + . Set for 𝑢 , 𝑡 > 0 , 𝜂 𝑘 ( 𝑢 , 𝑡 ) = 𝑊 ( 𝑡 ) 𝑢 𝑟 𝑘 𝜙 𝑘 ( 𝑡 ) , ( 2 . 2 7 ) 𝜎 ( 𝑡 ) = s u p m i n 1 ≤ 𝑘 ≤ 𝑛 𝜂 𝑘 𝑢 𝑘 ∶ , 𝑡 𝑛 𝑘 = 1 𝑢 𝑘 = 𝑊 ( 𝑡 ) 𝑛 − 1 , 𝑢 𝑘 . > 0 ( 2 . 2 8 ) Then
(i)there holds the inequality ∞ 0 𝑊 ( 𝑡 ) 𝑠 ( 1 / 𝑝 − 𝑟 / 𝑛 ) − 1 𝜎 ( 𝑡 ) 𝑠 𝑤 ( 𝑡 ) 𝑑 𝑡 1 / 𝑠 ≤ 𝐶 𝑛 𝑘 = 1 ‖ 𝜙 𝑘 ‖ 𝑟 / ( 𝑛 𝑟 𝑘 ) Λ 𝑝 𝑘 𝑘 , 𝑠 ( 𝑤 ) ; ( 2 . 2 9 ) (ii)there exist continuously differentiable functions 𝑢 𝑘 ( 𝑡 ) on ℝ + such that 𝑛 𝑘 = 1 𝑢 𝑘 ( 𝑡 ) = 𝑊 ( 𝑡 ) 𝑛 − 1 , 𝜎 ( 𝑡 ) = 𝜂 𝑘 𝑢 𝑘 ( 𝑡 ) , 𝑡 𝑡 ∈ ℝ + ; , 𝑘 = 1 , … , 𝑛 ( 2 . 3 0 ) (iii)for any 𝑘 such that 1 𝑝 𝑘 > 1 𝑝 − 𝑟 𝑛 ( 2 . 3 1 )
the function 𝑢 𝑘 ( 𝑡 ) 𝑊 ( 𝑡 ) 𝛿 − 1 decreases on ℝ + .

Proof. The proof is similar to [15, Lemma 2 . 2 ]. All the argument holds true when we substitute the weight 𝑤 ( 𝑡 ) in this lemma for 𝑤 ( 𝑡 ) = 1 .

The Lebesgue measure of a measurable set 𝐴 ⊂ ℝ 𝑘 will be denoted by m e s 𝑘 𝐴 .

For any 𝐹 𝜎 − set 𝐸 ⊂ ℝ 𝑛 denote by 𝐸 𝑗 the orthogonal projection of 𝐸 onto the coordinate hyperplane 𝑥 𝑗 = 0 . By the Loomis-Whitney inequality [17, Chapter 4] ( m e s 𝑛 𝐸 ) 𝑛 − 1 ≤ 𝑛 𝑗 = 1 m e s 𝑛 − 1 𝐸 𝑗 . ( 2 . 3 2 )

Let 𝑓 ∈ 𝑆 0 ( ℝ 𝑛 ) , 𝑡 > 0 , and let 𝐸 𝑡 be a set of type 𝐹 𝜎 and measure 𝑡 such that | 𝑓 ( 𝑥 ) | ≥ 𝑓 ∗ ( 𝑡 ) for all 𝑥 ∈ 𝐸 𝑡 . Denote by 𝜆 𝑗 ( 𝑡 ) the ( 𝑛 − 1 ) -dimensional measure of the projection 𝐸 𝑗 𝑡 ( 𝑗 = 1 , … , 𝑛 ) . By (2.32), we have that 𝑛 𝑗 = 1 𝜆 𝑗 ( 𝑡 ) ≥ 𝑡 𝑛 − 1 . ( 2 . 3 3 )

Lemma 2.4. Let 𝑛 ≥ 2 , 𝑟 𝑘 ∈ ℕ ( 𝑘 = 1 , … , 𝑛 ) , 𝑤 be nonincreasing, and 𝑤 ( 𝑡 ) → 𝑎 when 𝑡 → ∞ where 𝑎 > 0 . Function 𝑓 ∈ 𝑆 0 ( ℝ 𝑛 ) has weak derivatives 𝐷 𝑟 𝑘 𝑘 𝑓 ∈ 𝐿 l o c ( ℝ 𝑛 ) ( 𝑘 = 1 , … , 𝑛 ) . Then for all 0 < 𝑡 < 𝜏 < ∞ and 𝑘 = 1 , … , 𝑛 one has 𝑓 ∗ 𝑓 ( 𝑡 ) ≤ 𝐾 ∗ 𝜏 ( 𝜏 ) + 𝑡 𝑟 𝑘 𝑊 ( 𝑡 ) 𝜆 𝑘 ( 𝑡 ) 𝑟 𝑘 ( 𝐷 𝑟 𝑘 𝑘 𝑓 ) ∗ ∗ ( 𝜏 ) , ( 2 . 3 4 ) where ∏ 𝑛 𝑘 = 1 𝜆 𝑘 ( 𝑡 ) ≥ 𝑊 ( 𝑡 ) 𝑛 − 1 and 𝐾 is a constant depending on 𝑟 1 , … , 𝑟 𝑛 and 𝑎 .

Proof. Let 𝜆 𝑘 ( 𝑡 ) = ( 1 / 𝑛 √ 𝑎 ) ( 𝑊 ( 𝑡 ) / 𝑡 ) 𝜆 𝑘 ( 𝑡 ) , then 𝑛 𝑘 = 1 𝜆 𝑘 1 ( 𝑡 ) = 𝑎 𝑊 ( 𝑡 ) 𝑡 𝑛 𝑛 𝑘 = 1 𝜆 𝑘 ( 𝑡 ) . ( 2 . 3 5 ) Due to the conditions of 𝑤 and (2.33), we can get 𝑛 𝑘 = 1 𝜆 𝑘 ( 𝑡 ) ≥ 𝑊 ( 𝑡 ) 𝑛 − 1 . ( 2 . 3 6 ) In [2, 12, 15], we have 𝑓 ∗ 𝑓 ( 𝑡 ) ≤ 𝐾 ∗ 𝜏 ( 𝜏 ) + 𝜆 𝑘 ( 𝑡 ) 𝑟 𝑘 𝐷 𝑟 𝑘 𝑘 𝑓 ∗ ∗ ( 𝜏 ) . ( 2 . 3 7 ) So we immediately get (2.34).

Lemma 2.5. If 𝑤 ∈ 𝐵 1 , ∞ , 1 < 𝑝 0 < ∞ and 1 ≤ 𝑠 0 < ∞ , then 𝑣 ≡ 𝑊 ( 𝑡 ) 𝑠 0 / 𝑝 0 − 1 𝑤 ( 𝑡 ) ∈ 𝐵 𝑠 0 .

Proof. Let 𝑤 ∈ 𝐵 1 , ∞ . Since 𝐵 1 , ∞ ⊂ 𝐵 𝑝 0 , so by [9, Chapter 1] we get 𝑟 0 1 𝑊 ( 𝑡 ) 1 / 𝑝 0 𝑟 𝑑 𝑡 ≤ 𝐶 𝑊 ( 𝑟 ) 1 / 𝑝 0 , ∀ 𝑟 > 0 . ( 2 . 3 8 ) Then 𝑟 0 1 𝑉 ( 𝑡 ) 1 / 𝑠 0 𝑟 𝑑 𝑡 ≤ 𝐶 𝑉 ( 𝑟 ) 1 / 𝑠 0 , ∀ 𝑟 > 0 , ( 2 . 3 9 ) where 𝑉 ( 𝑡 ) = 𝑡 0 𝑣 ( 𝑡 ) 𝑑 𝑡 . ( 2 . 4 0 ) So 𝑣 ∈ 𝐵 𝑠 0 .

Lemma 2.6. Let 𝑛 ≥ 2 , 𝑟 𝑘 ∈ ℕ , 1 < 𝑝 𝑘 < ∞ , 1 ≤ 𝑠 𝑘 < ∞ for 𝑘 = 1 , … , 𝑛 . Assume that weight 𝑤 on ℝ + satisfies the following conditions:
(i)it is nonincreasing, continuous, and l i m 𝑡 → ∞ 𝑤 ( 𝑡 ) = 𝑎 , 𝑎 > 0 ,(ii)exists 𝐴 > 0 , such that 𝑊 ( 𝜉 𝑡 ) ≥ 𝜉 𝐴 𝑊 ( 𝑡 ) , ∀ 𝜉 > 1 , ∀ 𝑡 > 0 . ( 2 . 4 1 )
Set 𝑟 = 𝑛 𝑛 𝑘 = 1 1 𝑟 𝑘 − 1 𝑛 , 𝑝 = 𝑟 𝑛 𝑘 = 1 1 𝑝 𝑘 𝑟 𝑘 − 1 , 𝑛 𝑠 = 𝑟 𝑛 𝑘 = 1 1 𝑠 𝑘 𝑟 𝑘 − 1 . ( 2 . 4 2 ) Assume that a locally integrable function 𝑓 ∈ 𝑆 0 ( ℝ 𝑛 ) has weak derivatives 𝐷 𝑟 𝑘 𝑘 𝑓 ∈ Λ 𝑝 𝑘 , 𝑠 𝑘 ( 𝑤 ) ( 𝑘 = 1 , … , 𝑛 ) . Then for any 𝜉 > 1 𝑓 ∗ 𝑓 ( 𝑡 ) ≤ 𝐾 ∗ ( 𝜉 𝑡 ) + 𝜉 𝑟 𝜎 ( 𝑡 ) , ( 2 . 4 3 ) where 𝑟 = m a x 𝑟 𝑘 , the constants 𝐾 depends only on 𝑟 1 , … , 𝑟 𝑛 , 𝑤 , and ∞ 0 𝑊 ( 𝑡 ) 𝑠 ( 1 / 𝑝 − 𝑟 / 𝑛 ) − 1 𝑤 ( 𝑡 ) 𝜎 ( 𝑡 ) 𝑠 𝑑 𝑡 1 / 𝑠 ≤ 𝐶 𝑛 𝑘 = 1 ‖ 𝐷 𝑟 𝑘 𝑘 𝑓 ‖ 𝑟 / ( 𝑛 𝑟 𝑘 ) Λ 𝑝 𝑘 𝑘 , 𝑠 ( 𝑤 ) . ( 2 . 4 4 )

Proof. For every fixed 𝑘 = 1 , … , 𝑛 we take 𝜓 𝑘 𝐷 ( 𝑡 ) = 𝑟 𝑘 𝑘 𝑓 ∗ ∗ ( 𝑡 ) . ( 2 . 4 5 ) Thanks to Lemma 2.5, and 𝑤 ∈ 𝐵 1 , ∞ (for 𝑤 is nonincreasing), we know 𝑣 = 𝑊 ( 𝑡 ) 𝑠 𝑘 / 𝑝 𝑘 − 1 𝑤 ( 𝑡 ) ∈ 𝐵 𝑠 𝑘 . ( 2 . 4 6 ) Thus ‖ 𝜓 𝑘 ‖ Λ 𝑝 𝑘 𝑘 , 𝑠 ( 𝑤 ) = ‖ ‖ 𝐷 𝑟 𝑘 𝑘 𝑓 ∗ ∗ ‖ ‖ 𝐿 𝑠 𝑘 ( 𝑣 ) ‖ ‖ 𝐷 ≤ 𝐶 𝑟 𝑘 𝑘 𝑓 ∗ ‖ ‖ 𝐿 𝑠 𝑘 ( 𝑣 ) ‖ ‖ 𝐷 = 𝐶 𝑟 𝑘 𝑘 𝑓 ‖ ‖ Λ 𝑝 𝑘 𝑘 , 𝑠 ( 𝑤 ) . ( 2 . 4 7 ) Next we apply Lemma 2.2 with 𝛿 defined as in Lemma 2.3. In this way we obtain the functions which we denote by 𝜙 𝑘 ( 𝑡 ) ( 𝑘 = 1 , … , 𝑛 ) . Further, with these functions 𝜙 𝑘 ( 𝑡 ) we define the function 𝜎 ( 𝑡 ) by (2.28). By Lemma 2.3, we have the inequality (2.44). Using Lemma 2.4 with 𝜏 = 𝜉 𝑡 , we obtain 𝑓 𝑓 ( 𝑡 ) ≤ 𝐾 ∗ ( 𝜉 𝑡 ) + 𝜉 𝑟 𝑊 ( 𝑡 ) 𝜆 𝑘 ( 𝑡 ) 𝑟 𝑘 𝜙 𝑘 , ( 2 . 4 8 ) where ∏ 𝑛 𝑘 = 1 𝜆 𝑘 ( 𝑡 ) ≥ 𝑊 ( 𝑡 ) 𝑛 − 1 . Taking into account (2.28), we get (2.43).

Corollary 2.7. Let 0 < 𝜃 ≤ 1 , 𝑛 ≥ 2 , 𝑟 𝑘 ∈ ℕ , 1 < 𝑝 𝑘 < ∞ , 1 ≤ 𝑠 𝑘 < ∞ for 𝑘 = 1 , … , 𝑛 , and 𝑟 , 𝑝 , 𝑠 be the numbers defined by (2.42). Assume weight 𝑤 on ℝ + satisfies the following conditions:
(i)it is nonincreasing, continuous, and l i m 𝑡 → ∞ 𝑤 ( 𝑡 ) = 𝑎 , 𝑎 > 0 ,(ii)there exist two constants 𝜂 , 𝛽 with 𝛽 <