Journal of Function Spaces

Journal of Function Spaces / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 658527 | https://doi.org/10.1155/2013/658527

Fernando Farroni, Raffaella Giova, "The Distance to in the Grand Orlicz Spaces", Journal of Function Spaces, vol. 2013, Article ID 658527, 7 pages, 2013. https://doi.org/10.1155/2013/658527

The Distance to in the Grand Orlicz Spaces

Academic Editor: Carlo Sbordone
Received30 May 2013
Accepted07 Jul 2013
Published09 Sep 2013

Abstract

We establish a formula for the distance to from the grand Orlicz space introduced in Capone et al. (2008). A new formula for the distance to from the grand Lebesgue space introduced in Iwaniec and Sbordone (1992) is also provided.

1. Introduction

Let be a bounded open subset of and let . The grand space, denoted by , consists of functions such that

where denotes the average over . Note that is a norm and is a Banach space. This space was introduced by Iwaniec and Sbordone in connection with the integrability of the Jacobian [1], and it comes into play in a various number of problems (see, e.g., [215]).

It is worth pointing out that is not a dense subspace of (see [9]); it is proved in [16] that the distance to in is given by

A generalization of the grand Lebesgue space is the grand Orlicz space , introduced by Capone et al. in [17]. Let us recall that is called an Orlicz function if it is continuous, strictly increasing, and satisfies and . The Orlicz space associated with consists of all measurable functions for which there exists such that

Let us introduce the Luxemburg functional defined as

Because of the monotonicity of we have

and among Orlicz functions we will consider the ones satisfying the following condition: for some constant such that as . This will be done in order to ensure that the functional in (4) is a quasinorm. In what follows, we will lose no generality in assuming that

Suppose that and let be the increasing weight defined as

Following a definition given in [18], we suppose that is tempered; that is, for some .

An example of function satisfying (6)–(10) is for , and in this case as when and as when (see Section 5 for details).

The grand Orlicz space consists of all measurable functions for which there exists such that

where is the decreasing rearrangement of and is the distribution function of

The quasinorm denoted by is defined as follows:

We address that if we take also the grand Orlicz space reduces to the grand Lebesgue space (see [17, Proposition ], [6]).

Our main result provides a formula for the distance of a function to , defined by

Theorem 1. Let be a bounded open set of . Assume that is an Orlicz function verifying (6)–(10). For every function , one has

Our theorem is in the framework of the results of paper [19], which cannot be directly applied to our context, without a preliminary check that the grand Orlicz spaces can be characterized as interpolation or extrapolation spaces. We also refer to [5, 16, 2024] for the problem of finding formulae for the distance to a subspace in a given function space.

Theorem 1 gives, as byproduct, a characterization of the closure of in with respect to the norm, which will be denoted by .

Theorem 2. A function belongs to if and only if

For the special choice , Theorem 1 also provides new formula for the distance to in (see Theorem 5).

2. The Main Result

We start this section recalling few basic properties of the decreasing rearrangement of a measurable function defined in a bounded open set of . We refer the reader to [25, Propositions and ] for details.

Lemma 3. Let be measurable functions defined in a bounded open set of :

We need a technical result providing a useful property of the quantity

We recall that the goal of Theorem 1 consists in proving that is equal to . We notice that if , then , because from (8) and (10) it is as and the average remains bounded for every .

Lemma 4. Let be an Orlicz function satisfying the assumptions of Theorem 1. Assume that and . Then

Proof. Let and let so that
We use (18) with and and (20), and we get
We use (5) to get
We multiply by and we integrate over to get
We multiply by , and since as , we have
From (23) we get
We apply the definition of , and we have and then, passing to the limit as , we have
By replacing with and with in (30), we obtain the converse inequality
Equality (22) is finally proved.

Now, we are in a position to prove Theorem 1.

Proof of Theorem 1. From Lemma 4 we know that for every . This clearly proves that since for every .
Now, we want to show that
In order to achieve the claimed inequality, we prove that if then Without loss of generality we may assume that . From (35) we find such that For each there exists such that Let be such that and let . From (38), we find some constant (depending on ), with , such that Using the monotonicity of weight , the fact that , and (39), we deduce from (40) that
We set if and, if , and we show that
Let us observe that while
Using the fact that the distribution function is decreasing, we easily see that
Therefore, if we let , we see that condition is verified for all . Thus for . On the other hand, if we let , we see that condition (46) is the same as requiring
Thus holds if , and (42) is proved.
It follows directly from (42) that for every . Hence, we make use of (38) if and of (41) if to conclude that
In particular,
Since (50) holds for every , we obtain that its left-hand side is smaller than 1, and therefore . We get Hence (36) is established. Since is any arbitrary number for which (35) holds, we may pass to the limit as approaches in (36) to get
Combining (52) with (33) we obtain (16) as desired.

Proof of Theorem 2. As a consequence of Theorem 1, it is clear that if and only if
We fix an arbitrary and we set . Using (6) we have
Hence, using (53) we have
and (17) follows since as .

3. The Case of the Grand Lebesgue Space

We denote by the functional as in (21) when . In this case, takes the form

Our next result proves that the distance given by formula (2) reduces to .

Theorem 5. Let be a bounded open set of . For every function , one has

Proof. First we prove that
To this aim, we consider and such that
Using Hölder's inequality we have which in turn implies that Since as , we deduce from (61) that
Since is any number strictly greater than 1, (62) immediately implies (58).
We wish to prove the converse inequality
For each , we have
Thus
We consider such that
Then which proves (63).

4. Few Properties of the Distance

In this concluding section we provide certain properties of the functional .

Lemma 6. Let be an Orlicz function satisfying the assumptions of Theorem 1 and let . Assume that for some constants positive and . Then, there exists a positive constant depending only on such that

Proof. Let be the constant appearing in (6). We may take such that since as . We use (6) to get
We take the as and use (68) to get
Therefore, from (72) and (70) we have
The desired constant is obtained by setting . We address that is independent of , and thus the proof is completed.

Remark 7. It is clear from the definition of that we can pick if .

Our next lemma provides a sort of triangle inequality involving the functional .

Lemma 8. Let be an Orlicz function satisfying the assumptions of Theorem 1 and let . Then, there exists a constant depending only on such that

Proof. Take
Let . We use (18) with , the monotonicity of , to obtain
Fix . We multiply by and we integrate over to get
With the aid of two changes of variables in the integrals appearing at the right-hand side of (77) we have
which in turn implies
We multiply both sides of (79) by , and we take the as and use (75) to get
We appeal to Lemma 6 to conclude that there exists a constant such that
Finally, (74) follows letting and , respectively.

5. An Example

In this section we study the behaviour of weight as when with . We follow closely the lines of Example in [17].

Example 9. Let and let . We start by proving that
To see this, let . Then
We pick in such a way that
A similar argument leads to

Acknowledgment

The research of the first author has been supported by the 2008 ERC Advanced Grant 226234 “Analytic Techniques for Geometric and Functional Inequalities.”

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Copyright © 2013 Fernando Farroni and Raffaella Giova. 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.


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