- Research Article
- Open Access

# 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.

## Keywords

- Besov Space
- Lorentz Space
- Sharp Estimate
- Hardy Operator
- Continuous Embedding

## 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

(i)

(ii) decreases and increases on ,

(iii)

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

the function decreases on .

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.

If and , then

Proof.

So

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,

where

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).

and is defined by (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

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