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A Kind of Estimate of Difference Norms in Anisotropic Weighted SobolevLorentz Spaces
Journal of Inequalities and Applications volume 2009, Article number: 161405 (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 SobolevLorentz 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
In this paper we study functions on which possess the generalized partial derivatives
Our main goal is to obtain some norm estimates for the differences
( being the unit coordinate vector).
The classic Sobolev embedding theorem asserts that for any function in Sobolev space
Sobolev proved this inequality in 1938 for . His method, based on integral representations, did not work in the case . Only at the end of fifties Gagliardo and Nirenberg gave simple proofs of inequality (1.3) for all 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
For the result follows by interpolation (see [7, 8]). In the case 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 Il'in's result reads as follows: If and then
Actually, this means that there holds the continuous embedding to the Besov space
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 .
The generalization of the inequality (1.5) to the spaces was given in [12]. That is
where and the inequality is valid if or Using (1.7), we get the following continuous embedding:
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 .
In [15], there was the sharp estimates of the type (1.7) when the derivatives belong to different Lorentz spaces Before stating the theorem, we give some notations. Let be the class of all measurable and almost everywhere finite functions on such that for each ,
Let and for Denote
Now we state the main theorem in [15].
Theorem 1.1.
Let , and if . Let , and be the numbers defined by (1.10). For every satisfying the condition
take arbitrary such that
and denote
then for any function which has the weak derivatives there holds the inequality
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
Let be the class of all measurable and almost everywhere finite functions on . For , a nonincreasing rearrangement of is a nonincreasing function on that is, equimeasurable with . The rearrangement can be defined by the equality
where
If then the following relation holds [16, Chapter 2]:
Set
Assume that A function belongs to the Lorentz space if
For , the space is defined as the class of all such that
We also let . Let be a weight in (nonnegative locally integrable functions in ).
If , we replace with . For , or and , the weighted Lorentz space is defined in [9, Chapter 2] by
If , denote It is well known that
and if then
where
In following part of this paper, we will always denote .
The weighted Lorentz spaces have close connection with weights of for (see [9, Chapter 1]). Let be the Hardy operator as follows:
The space is the cone of all nonnegative nonincreasing functions in . We denote if
is bounded and denote if
is bounded.
Lemma 2.1 (Generalized Hardy's inequalities).
Let be nonnegative, measurable on and suppose and is a weight in , then one has
(with the obvious modification if ).
Proof.
It is easy to obtain this result applying Hardy's inequality [16].
Lemma 2.2.
Let be a nonnegative nonincreasing function on , be a nonincreasing weight on and there exists , such that
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.
Without loss of generality, we may suppose that . Set
Then decreases and
Using the conditions which satisfy, it gives
Furthermore, noticing is nonincreasing and applying Lemma 2.1, we get that
now set
Then increases on , and
Furthermore,
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.
Let and for Denote
Then and
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.
Let for and is continuous weight on . Set
Let
and suppose that are positive continuously differentiable functions with on such that decreases and increases on . Set for
Then
(i)there holds the inequality
(ii)there exist continuously differentiable functions on such that
(iii)for any such that
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 .
For any set denote by the orthogonal projection of onto the coordinate hyperplane . By the LoomisWhitney inequality [17, Chapter 4]
Let , and let be a set of type and measure such that for all . Denote by the dimensional measure of the projection By (2.32), we have that
Lemma 2.4.
Let be nonincreasing, and when where . Function has weak derivatives Then for all and one has
where and is a constant depending on and .
Proof.
Let then
Due to the conditions of and (2.33), we can get
So we immediately get (2.34).
Lemma 2.5.
If and , then
Proof.
Let Since so by [9, Chapter 1] we get
Then
where
So
Lemma 2.6.
Let for Assume that weight on satisfies the following conditions:
(i)it is nonincreasing, continuous, and ,
(ii)exists such that
Set
Assume that a locally integrable function has weak derivatives Then for any
where the constants depends only on , and
Proof.
For every fixed we take
Thanks to Lemma 2.5, and (for is nonincreasing), we know
Thus
Next we apply Lemma 2.2 with defined as in Lemma 2.3. In this way we obtain the functions which we denote by . Further, with these functions we define the function by (2.28). By Lemma 2.3, we have the inequality (2.44). Using Lemma 2.4 with we obtain
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 ,
(ii)there exist two constants with such that
and there holds
Assume that a locally integrable function has weak derivatives and for some with such that
Let and
Then and
Proof.
Let , with and . Applying Hölder's inequality and noticing and is nonincreasing, we obtain
So
Let . Using (2.43) with , which satisfies ( are two constants in (2.49) for ), combining (2.49), (2.52), and Hölder's inequality, we get
By (2.55), Furthermore, from (2.49), we can get
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.
Let for . Let , and be the numbers defined by (2.42). Assume that , and satisfies the conditions of Corollary 2.7 with . Then for any function with compact support we have
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,
(i) where

(ii)
(2.59)
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
(ii)there exist two constants with such that
and there holds
For every satisfying the condition
take arbitrary such that and
and denote
Then for any function with the weak derivatives there holds the inequality
where is a constant that does not depend on .
Proof.
First we can get by our conditions. denote
Further, assume that and set for
For almost all we have [1, Volume 1, page 101]
Thus,
Indeed, for any subset with
(3.10) then follows.
For is nonincreasing (), we get by Lemma 2.5. Thus from (3.10)
It follows Furthermore
Then due to Hardy lemma [16, page 56]
It follows Analogically we get Thus by Corollary 2.7 we have .
Denote for
Set ( are two constants in (3.1) for ), and
where is the constant in Lemma 2.5. Then by (3.1)
Therefore ,
Let
Now for every by applying Lemma 2.2 with We obtain on such that
For , it follows that
Thus
We will estimate for fixed and By Lemma 2.4, (3.21), we have that for each
where Applying Lemma 2.3, we obtain that there exist a nonnegative function and positive continuously differentiable functions on satisfying the following conditions:
Denote
We will prove that for any and any
where
By (3.24)
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).
Now, taking into account (3.26) and (3.32), we obtain that for and any
where
and is defined by (3.33).
Further, we have (see (3.18)
By (3.30), the function increases on . It follows easily that exists on and satisfies and
Furthermore, we have
Using Minkowsi's inequality, we obtain
Further, using Hölder's inequality and (3.38), we get when (the case is obvious)
Thus, by Fubini's theorem and (3.33)
The same argument gives that
By (3.33) the last integral is the same as one on the right side of (3.42). So, we have that
Now we apply Hölder's inequality with the exponents and Observe that
Therefore, we get, applying (3.27) and (3.34)
Since
we get the inequality (3.6). The theorem is proved.
Let be a rearrangement invariant space (r.i. space), be an r.i. space over and . Set integral part of ). The Besov space is defined as follows (see [18, 19]):
where
and denotes the fundamental function of with being any measurable subset of with .
Then we have the following.
Corollary 3.2.
Let for and
Let the weight be the same as that in Theorem 3.1. Take arbitrary such that
and denote
Then for any function which has the weak derivatives there hold
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.
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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).
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Chen, J., Li, H. A Kind of Estimate of Difference Norms in Anisotropic Weighted SobolevLorentz Spaces. J Inequal Appl 2009, 161405 (2009). https://doi.org/10.1155/2009/161405
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Keywords
 Besov Space
 Lorentz Space
 Sharp Estimate
 Hardy Operator
 Continuous Embedding