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

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

In this paper we study functions on which possess the generalized partial derivatives

(1.1)

Our main goal is to obtain some norm estimates for the differences

(1.2)

( being the unit coordinate vector).

The classic Sobolev embedding theorem asserts that for any function in Sobolev space

(1.3)

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

(1.4)

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

(1.5)

Actually, this means that there holds the continuous embedding to the Besov space

(1.6)

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

(1.7)

where and the inequality is valid if or Using (1.7), we get the following continuous embedding:

(1.8)

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 ,

(1.9)

Let and for Denote

(1.10)

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

(1.11)

take arbitrary such that

(1.12)

and denote

(1.13)

then for any function which has the weak derivatives there holds the inequality

(1.14)

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

(2.1)

where

(2.2)

If then the following relation holds [16, Chapter 2]:

(2.3)

Set

(2.4)

Assume that A function belongs to the Lorentz space if

(2.5)

For , the space is defined as the class of all such that

(2.6)

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

(2.7)

If , denote It is well known that

(2.8)

and if then

(2.9)

where

(2.10)

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:

(2.11)

The space is the cone of all nonnegative nonincreasing functions in . We denote if

(2.12)

is bounded and denote if

(2.13)

is bounded.

Lemma 2.1 (Generalized Hardy's inequalities).

Let be nonnegative, measurable on and suppose and is a weight in , then one has

(2.14)

(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

(2.15)

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

(2.16)

Then decreases and

(2.17)

Using the conditions which satisfy, it gives

(2.18)

Furthermore, noticing is nonincreasing and applying Lemma 2.1, we get that

(2.19)

now set

(2.20)

Then increases on , and

(2.21)

Furthermore,

(2.22)

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

(2.23)

Then and

(2.24)

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

(2.25)

Let

(2.26)

and suppose that are positive continuously differentiable functions with on such that decreases and increases on . Set for

(2.27)

(2.28)

Then

(i)there holds the inequality

(2.29)

(ii)there exist continuously differentiable functions on such that

(2.30)

(iii)for any such that

(2.31)

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 Loomis-Whitney inequality [17, Chapter 4]

(2.32)

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

(2.33)

Lemma 2.4.

Let be nonincreasing, and when where . Function has weak derivatives Then for all and one has

(2.34)

where and is a constant depending on and .

Proof.

Let then

(2.35)

Due to the conditions of and (2.33), we can get

(2.36)

In [2, 12, 15], we have

(2.37)

So we immediately get (2.34).

Lemma 2.5.

If and , then

Proof.

Let Since so by [9, Chapter 1] we get

(2.38)

Then

(2.39)

where

(2.40)

So

Lemma 2.6.

Let for Assume that weight on satisfies the following conditions:

(i)it is nonincreasing, continuous, and ,

(ii)exists such that

(2.41)

Set

(2.42)

Assume that a locally integrable function has weak derivatives Then for any

(2.43)

where the constants depends only on , and

(2.44)

Proof.

For every fixed we take

(2.45)

Thanks to Lemma 2.5, and (for is nonincreasing), we know

(2.46)

Thus

(2.47)

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

(2.48)

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

(2.49)

and there holds

(2.50)

Assume that a locally integrable function has weak derivatives and for some with such that

(2.51)

Let and

(2.52)

Then and

(2.53)

Proof.

Let , with and . Applying Hölder's inequality and noticing and is nonincreasing, we obtain

(2.54)

So

(2.55)

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

(2.56)

By (2.55), Furthermore, from (2.49), we can get

(2.57)

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

(2.58)

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

(3.1)

and there holds

(3.2)

For every satisfying the condition

(3.3)

take arbitrary such that and

(3.4)

and denote

(3.5)

Then for any function with the weak derivatives there holds the inequality

(3.6)

where is a constant that does not depend on .

Proof.

First we can get by our conditions. denote

(3.7)

Further, assume that and set for

(3.8)

For almost all we have [1, Volume 1, page 101]

(3.9)

Thus,

(3.10)

Indeed, for any subset with

(3.11)

(3.10) then follows.

For is nonincreasing (), we get by Lemma 2.5. Thus from (3.10)

(3.12)

It follows Furthermore

(3.13)

Then due to Hardy lemma [16, page 56]

(3.14)

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

Denote for

(3.15)

Set ( are two constants in (3.1) for ), and

(3.16)

where is the constant in Lemma 2.5. Then by (3.1)

(3.17)

Therefore ,

(3.18)

Let

(3.19)

Now for every by applying Lemma 2.2 with We obtain on such that

(3.20)

(3.21)

(3.22)

For , it follows that

(3.23)

Thus

(3.24)

We will estimate for fixed and By Lemma 2.4, (3.21), we have that for each

(3.25)

where Applying Lemma 2.3, we obtain that there exist a nonnegative function and positive continuously differentiable functions on satisfying the following conditions:

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

Denote

(3.31)

We will prove that for any and any

(3.32)

where

(3.33)

By (3.24)

(3.34)

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

(3.35)

where

(3.36)

and is defined by (3.33).

Further, we have (see (3.18)

(3.37)

By (3.30), the function increases on . It follows easily that exists on and satisfies and

(3.38)

Furthermore, we have

(3.39)

Using Minkowsi's inequality, we obtain

(3.40)

Further, using Hölder's inequality and (3.38), we get when (the case is obvious)

(3.41)

Thus, by Fubini's theorem and (3.33)

(3.42)

The same argument gives that

(3.43)

By (3.33) the last integral is the same as one on the right side of (3.42). So, we have that

(3.44)

Now we apply Hölder's inequality with the exponents and Observe that

(3.45)

Therefore, we get, applying (3.27) and (3.34)

(3.46)

Since

(3.47)

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]):

(3.48)

where

(3.49)

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

Then we have the following.

Corollary 3.2.

Let for and

(3.50)

Let the weight be the same as that in Theorem 3.1. Take arbitrary such that

(3.51)

and denote

(3.52)

Then for any function which has the weak derivatives there hold

(3.53)

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.