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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 748645, 11 pages
Admissible Differential Bases in Banach Spaces
Yaroslavl State University, Sovetskaya Street 14, Yaroslavl 150000, Russia
Received 27 February 2013; Accepted 2 July 2013
Academic Editor: Marco Squassina
Copyright © 2013 Evgenii I. Berezhnoj. 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.
In the present work, a new method for constructing differential bases is presented. The bases constructed by this method allow us to distinguish symmetric spaces with different behaviour of fundamental functions at zero, as in the case of Hayes and Stokolos, and even allow us to distinguish Lorentz and Marzinkiewicz spaces or Lebesgue and Marzinkiewicz spaces whose fundamental functions are the same.
We recall certain definitions connected with differential bases . By a differential basis at a point is meant a family of bounded measurable sets with positive measure which contain and are such that there is at least one sequence satisfying the condition . A union of such families is called a differential basis in . We note that the bulk of the problems and assertions in the theory of differentiation of integrals in consists of testing the validity almost everywhere of some fact. Therefore, it is natural to consider differential bases which are defined, not for all points in , but only almost everywhere. In what follows, we will make use of these observations.
The classical examples of differential bases are the bases in , usually denoted by and consisting of all rectangular parallelepipeds of the form which satisfy the condition for and , .
A basis made up of open sets is called a Busemann-Feller basis (BF-basis) if it follows from the conditions and such that . The significance of the introduction of -bases lies in the fact that questions arising in the theory of differentiation with respect to bases can be easily resolved for bases. We now define the upper and lower derivatives of the integral of a locally integrable function at a point with respect to a basis by means of the identities
We note that the upper and lower derivatives are variants of the functionals and which are the subject of investigation in several sections of the chapter 1 in .
Following , we say that a basis differentiates the integral of if the identities hold almost everywhere. If differentiates the integral of any function in the space , then we say that the basis differentiates the space . If the basis differentiates , then it is said to be a density basis .
One of the fundamental problems of the theory of differentiation of integrals has the following form: given two function spaces , which are different in some sense, is it possible to distinguish these two spaces with the help of differential bases? In other words, does there exist a differential basis, which differentiates all integrals obtained from functions in , but a function can be found in whose integral is not differentiated by the given basis? This problem is still far from its final solution. The first result related to this problem is due to Hayes [1, page 155], who showed that it is possible to distinguish the spaces and for by constructing an appropriate differential basis. Then, Stokolos [3, 4] showed that it is possible to distinguish Orlicz spaces by constructing an appropriate differential basis. Apparently, everything that has been achieved on the problem of distinguishing spaces with the help of differential bases is exhausted by these two results.
The following classical results are fundamental for the theory of differentiation of integrals in : differentiates (the case is due to Lebesgue , the case to Jessen et al. , and the case of arbitrary to Zygmund ). In 1935, Saks  showed that does not differentiate . It was proved later in  that does not differentiate the space .
Let be the unit cube in with the usual Lebesgue measure. For , we will denote by the distribution function of : its rearrangement in nonincreasing order will be denoted by : A Banach space consisting of measurable functions is said to be ideal  if, given , measurability of and almost everywhere, it follows that and . An ideal space is said to be symmetric  if for , given that the inequality holds for all , it follows that . As usual we denote by the space of integral functionals on with norm
For each symmetric space , its fundamental function is defined by means of the identity where denotes the characteristic function of a set . This function is nondecreasing and does not increase. The spaces of Lebesgue, of Orlicz, of Marzinkiewicz, and of Lorentz are the classical examples of symmetric spaces. The theory of symmetric spaces of functions is described in detail in  and that of symmetric spaces of sequences in . Since Lorentz and Marzinkiewicz spaces will be used below, we will describe and in greater detail. Let us denote by the set of concave functions , each of which is continuous with and . Then, the Lorentz space (resp., the Marzinkiewicz space ) consists of all those for which the following expression Respectively, is a finite norm. It is clear that for the fundamental functions of the Lebesgue, Orlicz, Lorentz, and Marzinkiewicz spaces we have the identities
The duality between Lorentz and Marzinkiewicz spaces, is well known, as is the embedding theorem of Krejn et al. : let be a symmetric space and its fundamental function; then we have the continuous embeddings
If with , then in this case the Lorentz (Marzinkiewicz) space will be denoted by .
2. The Construction of Differential Bases
Now we pass to the construction of differential bases. For simplicity and clarity, we will demonstrate the construction for . We begin the construction of the corresponding bases from the construction of a certain collection of sets in the square ; we will refer to the construction process as the basic construction.
Suppose that we are given a sequence of triples of natural numbers , for which , , and . First we divide the square into equal squares . These we will call squares of the first level. Then again we divide each of the squares into equal squares and obtain the squares . These we will call squares of the second level, and so on. Each of the squares of the th level is divided in turn into equal rectangles , which we will take as open. For the sake of definiteness, we will agree that the vertical side of the square is divided into equal parts, while the horizontal side of the square is divided into equal parts. Now, we will construct a collection of sets relating to the th level.
We take the square of the th level with the first index. The sets of the collection which are contained in the square are formed in the following manner. By construction the square is split up into vertical strips and horizontal strips which form equal rectangles. We take the lower left rectangle , which lies at the intersection of the first horizontal and the first vertical. All the other rectangles from the square lying on the first horizontal do not take part in the continuation of the construction and can be discarded. The set consists of the rectangle to which have been added the rectangles from the first vertical. The set consists of the rectangle to which have been added the rectangles from the second vertical. Then, we again take the rectangle , adding to it the rectangles from the third vertical, and we will form the set . We continue this process and thus obtain the collection of sets
The sets of the collection have three important properties:
Now, we repeat this construction for all the remaining squares of the th level . We will obtain the collections of sets . The sets of the collections have the following important properties:
We now define the collection as the union over of all the collections . Now on combining the collections of sets constructed for each level, we obtain the desired collection . We note that the measure of those points of which are not included in the sets of the collection does not exceed . Therefore, from the definition and (15), we may take as a differential basis.
Corresponding to the collection , we construct the maximal operator and for the collection and , we construct its maximal operator
For , we will write simply in place of . We denote by the union of the boundaries of all the sets appearing in the collection . It is easy to see that the identity holds.
3. An Effective Bound for Maximal Operator
Our immediate goal is to obtain an effective bound for the maximal operator. The following lemma can be obtained easily from well-known facts of the metric theory of functions and the identity .
Lemma 1. Suppose that the collections of sets have been constructed relative to the triples of natural numbers . Then, for any , there exists a number such that for all the following condition is satisfied: if the square is divided into equal nonintersecting squares , then one has the inequality
In other words, the area of those squares which intersect the boundary of at least one set of the collections will be less than .
Now from all the differential bases obtained by means of the basic construction we extract a certain subclass for which it is possible to verify its differential properties.
Definition 2. We will say that a sequence of triples of natural numbers is admissible if the conditions are satisfied, where the number is taken from Lemma 1 for .
Definition 3. We will say that a differential basis is admissible if the families have been constructed by means of the basic construction applied with an admissible sequence of triples.
The following lemma is a key tool for the characterization of differential properties of admissible bases.
Lemma 4. Let with almost everywhere, and let be an admissible differential basis. Suppose that and are given and choose such that
Then, one has the following relation: where and the set consists of the union of nonintersecting sets with , each of which can be represented in the form for some , and moreover for each square the condition is satisfied.
Proof. We will establish the lemma for . We will indicate all the changes required for other at the end. Let
We introduce the set
Then, as it is not difficult to verify, we have the identity
Let us put (recall that (15) holds)
If , then from the inequality we obtain
Now the sets and are disjoint for according to (15). Therefore, we have the relation
Then, if , according to (15) and the construction, we will obtain and for the set we will have according to the construction
Next, among the sets we reject those which lie entirely in and put
We divide the set into two parts:
Let us denote by the operator, analogous to from (16), but with the upper bound taken only over those sets which are disjoint from . Then, as is not difficult to verify, we have the identity
Therefore, we can put
If , then analogous to (30) we obtain
Because the sets and are disjoint for , we have the relations
And for the set , we have in analogy to the following relations: if , then
The set is made up of those which intersect the boundaries of the sets contained in . Since the basis is admissible, we can write the bound
Now we reject those among the sets , which lie entirely in , and put
We divide the set into two parts: construct the sets , and the operator , and write analogues of (39), (40), (41), and so on. We continue this process to infinity and obtain as a result the collections of sets
Moreover, and the following relations hold: and if , then
To complete the proof, it is enough to put
Then by (47), all the relations (22) and (24) will be satisfied for and . But if , then all the arguments remain in force, only , and therefore (22) and (24) are always true. The lemma is proved.
It is immediately useful to note that for admissible differential bases as .
Lemma 4 has a series of important corollaries.
Corollary 5. Each admissible differential basis is a density basis; that is, it differentiates .
Proof. Let and fix . We first choose such that the inequality holds for . Then by (15), we have for such the inequality
Therefore, if we put , then it will follow from (49) the definition of and the definition of the sets that , and, consequently,
It follows from (50) that as and . According to the criterion given in Theorem 1.1 of the book , the last observation implies that is a density basis.
Corollary 6. Let . If the relation holds, where , then differentiates .
Proof. Let with almost everywhere. We will bound the measure of the set in accordance with Lemma 4. We obtain from duality and (22) for any symmetric space the following:
In the case , we have
To obtain the bound for , we can put and by (24), we can write the relation,
But in our case
Taking into account (15), (51), and (55) along with the fact that consists of nonintersecting sets , and consequently of squares , we obtain
If we put and apply the bounds (53) and (57) to , then we will obtain as and . For density bases, the last observation is equivalent by the criterion given in Theorem 2.2 of the book  to the differentiability of the integral . Since this result is true for any , almost everywhere, it follows that differentiates . The corollary is proved.
Corollary 7. Let and . If the relations are satisfied and does not depend on , then differentiates .
Proof. The proof of this corollary follows the scheme of the proof of Corollary 6. Let with almost everywhere. We find a bound for the measure of the set by using Lemma 4 and (52) and (55). From (52) and duality, we obtain
and we will determine directly the norm of in . According to (59) for , we have the inequality
and for , there is the bound
From this, we obtain
The final part of the proof repeats word for word the proof of the corresponding part in Corollary 6.
Corollary 8. Let and . If the conditions are satisfied and does not depend on , then differentiates .
Proof. Let with almost everywhere. Using Lemma 4 and (52), and (55) we find a bound for the measure of the set . From duality and (52), we obtain
and we will arrive at a bound for the norm of the function in by use of (61) and (62). If , then
According to the definition,
Therefore, from (61), (62), and (65), we obtain and, consequently,
Therefore, for , we will have
The final part of the proof is similar to that in the proof of the preceding corollary.
4. The Admissible Differential Bases
We now construct for each admissible differential basis a certain function which will play one of the key roles in the resolution of the problems formulated at the beginning of this section. Suppose that we are given a differential basis which has been constructed relative to an admissible sequence of triples . Suppose that we are also given a sequence of numbers . We define the function by means of the identity
It turns out that with a suitable choice of numbers the basis does not differentiate .
Lemma 9. Let be a differential basis constructed relative to . Suppose that the following relation holds for the sequence :
Then, the differential basis does not differentiate .
Proof. Choose a positive integer and put . Then, in accordance with the construction and condition (15), we will have for any set
Therefore, from (15) and (74), we have the identity
If we note that as , we can deduce from Theorem in  that the basis does not differentiate . The lemma is proved.
Now, we will produce certain conditions on the function (73) which should guarantee its membership of certain spaces.
Lemma 10. Let be an admissible collection of triples and put
Then, the following inequalities hold:
Proof. We have from the definition the inequality
Therefore, which also establishes (78). Let us verify (79). If , then
According to the definition of and (80)
Therefore, from the monotonicity of the function , we obtain
The lemma is proved.
Now, we have everything ready for the proof of the following theorem. We note that the fundamental functions of the spaces involved in Theorem 11 are the same.
Theorem 11. Let . Then, there exists an admissible differential basis which differentiates but not , .
Proof. We will construct an admissible sequence of triples in the following way. First, we put and choose a monotone increasing sequence of natural numbers such that for sufficiently large we have the inequality
This is possible because of the inequality . Then, we choose a sequence such that the sequence of triples is admissible. The possibility of such a choice of is also obvious. Now, we construct the differential basis corresponding to . It follows from (85) and Corollary 6 that differentiates . Let us show that does not differentiate the Marzinkiewicz space . With the help of formula (73), we construct the function , putting . It then follows from Lemma 9 that the basis does not differentiate . We show that . For this, we make use of Lemma 10. From the definition of and (85), we obtain
Thus, by Lemma 10, but does not differentiate , and, consequently, U does not differentiate the Marzinkiewicz space . The theorem is proved.
Now, we show that with the help of the bases which have been set up it is possible to distinguish Lorentz and Marzinkiewicz spaces with exactly the same quasiexponential fundamental functions.
Theorem 12. Let and suppose that the following conditions are satisfied:
Then, there exists a differential basis which differentiates but does not differentiate .
Proof. We define the sequences , by means of the equations , and put , , ( denotes the integer part of ). Now, we choose a sequence such that the sequence of triples is admissible and construct the corresponding basis . Then, in accordance with (88), we have the following relations:
Since , then
We therefore obtain from this inequality and (89) by Corollary 8 that differentiates . Let us verify that does not differentiate . For this, we construct the function according to (73). Since and , then by Lemma 9 does not differentiate . We show that , for which we make use of Lemma 10. Since , we have by (88) the following relations:
We therefore obtain from Lemma 10 and the fact that does not increase that
Thus, , and it follows from Lemma 9 that does not differentiate . The theorem is proved.
Now, we show that if there is a pair of symmetric spaces with nonequivalent fundamental functions, then there exists a differential basis which differentiates one but not the other. For this, we consider first the case of Lorentz and Marzinkiewicz spaces.
Theorem 13. Let , let be the Lorentz space and let be the Marzinkiewicz space. Suppose that
Then, there exists a differential basis which differentiates but not .
Proof. It follows from (93) that , and it can be assumed that (as ), and is continuous. We choose a sequence such that the conditions of the following two inequalities are satisfied:
Then from (94) and the concavity of follow the inequalities
Let us construct the function according to (73). We will verify that . For this, we make use of Lemma 10. According to (96), we will have
Thus, . Now, we choose a sequence in such a way that the sequence of triples is admissible. It follows from the definition of and in (95) that does not differentiate . Therefore, according to (98), does not differentiate the space .
It remains to be shown that differentiates the Marzinkiewicz space . According to the definitions (95) and (96),
On the other hand we obtain from the inequality , (94), and (95), that
Thus, condition (59) for Corollary 7 is satisfied, and so differentiates . The theorem is proved.
This theorem has an important corollary.
Corollary 14. Let , be a pair of symmetric spaces and , their fundamental functions. If the condition (93) is satisfied, then there exists a differential basis which differentiates the space but not the space .
Proof. We apply the embedding theorem (12) for the spaces and . Then,
Now, we apply Theorem 13 to the spaces and . As a result of this, we find a basis which differentiates , and consequently , but does not differentiate , and consequently does not differentiate .
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