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

- Jiecheng Chen
^{1}and - Hongliang Li
^{1, 2}Email author

**2009**:161405

https://doi.org/10.1155/2009/161405

© Chen and H. Li. 2009

**Received: **27 April 2009

**Accepted: **2 July 2009

**Published: **4 August 2009

## Abstract

We investigate the functions spaces on for which the generalized partial derivatives exist and belong to different Lorentz spaces , where and 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 , weights.

## 1. Introduction

( being the unit coordinate vector).

For the result follows by interpolation (see [7, 8]). In the case some geometric inequalities were applied to prove (1.4) (see [9–13]).

It is easy to see that inequality (1.5) fails to hold for , but, it was proved in [14] that (1.5) is true for and .

For 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 .

Now we state the main theorem in [15].

Theorem 1.1.

where is a constant that does not depend on .

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 take place of , where is a weight on which satisfies some special conditions.

## 2. Auxiliary Proposition

We also let . Let be a weight in (nonnegative locally integrable functions in ).

In following part of this paper, we will always denote .

is bounded.

Lemma 2.1 (Generalized Hardy's inequalities).

(with the obvious modification if ).

Proof.

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

Lemma 2.2.

Then for there exists a continuously differentiable on such that

(ii) decreases and increases on ,

where is a constant depends only on , and .

Proof.

where that is, . Since is decreasing function on , thus is decreasing and is also decreasing on .

Finally, using Lemma 2.1 and (2.19), we get (iii). The Lemma 2.2 is proved.

To prove our main results we use the estimates of the rearrangement of a given function in term of its derivatives

We will use the notations (2.23).

Lemma 2.3.

Then

Proof.

The proof is similar to [15, Lemma ]. All the argument holds true when we substitute the weight in this lemma for .

The Lebesgue measure of a measurable set will be denoted by .

Lemma 2.4.

where and is a constant depending on and .

Proof.

So we immediately get (2.34).

Lemma 2.5.

Proof.

Lemma 2.6.

Let for Assume that weight on satisfies the following conditions:

(i)it is nonincreasing, continuous, and ,

Proof.

where . Taking into account (2.28), we get (2.43).

Corollary 2.7.

Let for and be the numbers defined by (2.42). Assume weight on satisfies the following conditions:

(i)it is nonincreasing, continuous, and ,

Proof.

Inequality (2.53) now follows from (2.44) and (2.55).

Remark 2.8.

If in Corollary 2.7, then it is easy to get .

Remark 2.9.

This statement can be easily got from Lemma 2.6. Inequality (2.58) gives a generalization of Remark of [15] when because satisfies the preceding conditions.

Remark 2.10.

Beyond constant weights, there are many weights satisfying conditions of Corollary 2.7. For example,

For weight in (i) or (ii), it is easy to see the weighted Lorentz space for does not coincide with any Lorentz space .

## 3. The Main Theorem

Theorem 3.1.

Let for Let , and be the numbers defined by (2.42). Suppose weight on satisfies the following conditions:

(i)it is nonincreasing, continuous, and

where is a constant that does not depend on .

Proof.

(3.10) then follows.

It follows Analogically we get Thus by Corollary 2.7 we have .

For the inequality (3.32) follows directly from (3.26) and (3.33). If then (3.32) is the immediate consequence of (3.10), (3.21), and (3.33).

we get the inequality (3.6). The theorem is proved.

and denotes the fundamental function of with being any measurable subset of with .

Then we have the following.

Corollary 3.2.

where is a constant that does not depend on .

Proof.

We can easily obtain the similar result to Lemma in [20] by substituting for there. Now the corollary is obvious using the Hardy's inequality and Theorem 3.1.

Remark 3.3.

If there exists with , whether Theorem 3.1 remains true is still a question now.

## Declarations

### Acknowledgments

This work is supported by NSFC (no. 10571156, 10871173), Natural Science Foundation of Zhejiang Province (no. Y606117), Foundation of Zhejiang Province Education Department (no. Y200803879) and 2008 Excellent Youth Foundation of College of Zhejiang Province (no. 01132047).

## Authors’ Affiliations

## References

- Besov OV, Il'in VP, Nikol'skiĭ SM:
*Integral Representations of Functions and Imbedding Theorems. Vol. I*. V. H. Winston & Sons, Washington, DC, USA; 1978:viii+345.MATHGoogle Scholar - Kolyada VI:
**Rearrangements of functions and embedding of anisotropic spaces of Sobolev type.***East Journal on Approximations*1998,**4**(2):111–199.MathSciNetMATHGoogle Scholar - Kudryavtsev LD, Nikol'skiĭ SM:
**Spaces of differentiable functions of several variables and imbedding theorems.**In*Analysis, Encyclopaedia Math. Sci.*.*Volume 26*. Springer, Berlin, Germany; 1991:1–140.Google Scholar - Nikol'skií SM:
*Approximation of Functions of Several Variables and Imbedding Theorems*. Springer, New York, NY, USA; 1975:viii+418.View ArticleGoogle Scholar - Triebel H:
*Theory of Function Spaces, Monographs in Mathematics*.*Volume 78*. Birkhäuser, Basel, Switzerland; 1983:284.View ArticleGoogle Scholar - Triebel H:
*Theory of Function Spaces. II, Monographs in Mathematics*.*Volume 84*. Birkhäuser, Basel, Switzerland; 1992:viii+370.View ArticleMATHGoogle Scholar - Peetre J:
**Espaces d'interpolation et théorème de Soboleff.***Université de Grenoble. Annales de l'Institut Fourier*1966,**16**(1):279–317. 10.5802/aif.232MathSciNetView ArticleMATHGoogle Scholar - Strichartz RS:
**Multipliers on fractional Sobolev spaces.***Journal of Mathematics and Mechanics*1967,**16:**1031–1060.MathSciNetMATHGoogle Scholar - Carro MJ, Raposo JA, Soria J:
**Recent developments in the theory of Lorentz spaces and weighted inequalities.***Memoirs of the American Mathematical Society*2007.,**187**(877):MathSciNetView ArticleMATHGoogle Scholar - Faris WG:
**Weak Lebesgue spaces and quantum mechanical binding.***Duke Mathematical Journal*1976,**43**(2):365–373. 10.1215/S0012-7094-76-04332-5MathSciNetView ArticleMATHGoogle Scholar - Kolyada VI:
**Rearrangements of functions, and embedding theorems.***Uspekhi Matematicheskikh Nauk*1989,**44**(5):61–95. [English translation: Russian Math. Surveys, vol. 44, no. 5, pp. 73–118, 1989] [English translation: Russian Math. Surveys, vol. 44, no. 5, pp. 73–118, 1989]MathSciNetMATHGoogle Scholar - Kolyada VI:
**On the embedding of Sobolev spaces.***Matematicheskie Zametki*1993,**54**(3):48–71. [English translation: Math. Notes, vol. 54, no. 3, pp. 908–922, 1993] [English translation: Math. Notes, vol. 54, no. 3, pp. 908–922, 1993]MathSciNetMATHGoogle Scholar - Poornima S:
**An embedding theorem for the Sobolev space**.*Bulletin des Sciences Mathématiques*1983,**107**(3):253–259.MathSciNetMATHGoogle Scholar - Kolyada VI:
**On the relations between moduli of continuity in various metrics.***Trudy Matematicheskogo Instituta imeni V. A. Steklova*1988,**181:**117–136. [English translation: Proc. Steklov Inst. Math., vol. 4, pp. 127–148, 1989] [English translation: Proc. Steklov Inst. Math., vol. 4, pp. 127–148, 1989]MathSciNetGoogle Scholar - Kolyada VI, Pérez FJ:
**Estimates of difference norms for functions in anisotropic Sobolev spaces.***Mathematische Nachrichten*2004,**267:**46–64. 10.1002/mana.200310152MathSciNetView ArticleMATHGoogle Scholar - Bennett C, Sharpley R:
*Interpolation of Operators, Pure and Applied Mathematics*.*Volume 129*. Academic Press, Boston, Mass, USA; 1988:xiv+469.MATHGoogle Scholar - Hadwiger H:
*Vorlesungen über Inhalt, Oberfläche und Isoperimetrie*. Springer, Berlin, Germany; 1957:xiii+312.View ArticleMATHGoogle Scholar - Martín J, Milman M:
**Symmetrization inequalities and Sobolev embeddings.***Proceedings of the American Mathematical Society*2006,**134**(8):2335–2347. 10.1090/S0002-9939-06-08277-3MathSciNetView ArticleMATHGoogle Scholar - Martín J, Milman M:
**Higher-order symmetrization inequalities and applications.***Journal of Mathematical Analysis and Applications*2007,**330**(1):91–113. 10.1016/j.jmaa.2006.07.033MathSciNetView ArticleMATHGoogle Scholar - Kolyada VI:
**Inequalities of Gagliardo-Nirenberg type and estimates for the moduli of continuity.***Uspekhi Matematicheskikh Nauk*2005,**60**(6):139–156. [English translation: Russian Math. Surveys, vol. 60, no. 6, pp. 1147–1164] [English translation: Russian Math. Surveys, vol. 60, no. 6, pp. 1147–1164]MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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