## Geometry of Banach Spaces, Operator Theory, and Their Applications

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# The Composition Operator and the Space of the Functions of Bounded Variation in Schramm-Korenblum's Sense

**Academic Editor:**Gen-Qi Xu

#### Abstract

We show that the composition operator , associated with , maps the spaces on to the space of functions of bounded variation in Schramm-Korenblum's sense if and only if is locally Lipschitz. Also, verify that if the composition operator generated by maps this space into itself and is uniformly bounded, then regularization of is affine in the second variable.

#### 1. Introduction

The composition operator problem (or COP, for short) refers to determining the conditions on a function , such that the composition operator, associated with the function , maps a space of functions into itself [1, 2]. There are several spaces where the COP has been resolved. For example, in 1961, Babaev [3] showed that the composition operator , associated with the function , maps the space of the Lipschitz functions into itself if and only if is locally Lipschitz; in 1967, Mukhtarov [4] obtained the same result for the space of the Hölder functions of order .

The first work on the COP in the space of functions of bounded variation was made by Josephy in 1981 [5]. In 1986, Ciemnoczołowski and Orlicz [6] got the same result for the space of the functions of bounded -variation in Wiener’s sense. In 1974, Chaika and Waterman [7] reached a similar result for the space of functions of bounded harmonic variation . In the years 1991 and 1995 Merentes showed a similar result for the spaces of absolutely continuous functions and the space of function of bounded -variation in Riesz’s sense (see [8, 9]), and in 1998 Merentes and Rivas achieved the same result when the composition operator maps the space of the functions of bounded -variation in Riesz’s sense () into the space [10]. In 2003, Pierce and Waterman solved the COP for the spaces and [11]. More recently, in 2011, Appell et al. [1] conclude the same results verifying when the composition operator maps into . Finally, Appell and Merentes verify the same result for the space of functions of bounded -variation [12].

There exist spaces of real functions defined on an interval , such that maps into itself and is not locally Lipschitz. For example, in the case the space of continuous functions it follows from the Tietze-Urysohn theorem that the composition operator acts from into itself and the function must be continuous; that is, does not need to be Lipschitz. A similar result was obtained in the space of regulated functions [13].

A first objective of this work is to demonstrate that the composition operator, associated with the function , maps the space of the Lipschitz functions into the space of functions of bounded variation in Schramm-Korenblum’s sense or into the space of functions of bounded variation in Korenblum’s sense if and only if is locally Lipschitz. We also extend this result to function spaces , such that , where or .

In a seminal article of 1982, Matkowski [14] showed that if the composition operator , associated with the function , maps the space of the Lipschitzian functions into itself and is a globally Lipschitzian map, then the function has the form for some .

There are a variety of spaces besides that verify this result [15]. The spaces of Banach that fulfill this property are said to satisfy the *Matkowski property* [1].

In 1984, Matkowski and Miś [16] considered the same hypotheses on the operator for the space of the function of bounded variation and concluded that (1) is true for the regularization of the function with respect of the first variable; that is,
where . Spaces that satisfy this conditions said to be verified *Weak Matkowski Property* [1].

A second objective of this paper is to show that if function is continuous in the second variable, for each , and the composition operator , associated with the function , is uniformly bounded, then satisfies (2).

#### 2. Preliminaries

Let be a closed interval of the real line . From now on, for a function denote by the Lipschitz constant of ; that is,

By we will denote the space of the Lipschitz functions. It is well known that the space is a Banach space endowed with the norm

Ever since the notion of a function of bounded variation appeared, it has led to an incredible number of generalizations. In 1881, Jordan [17] introduced the definition of function of bounded variation for a function and showed that these kinds of function can be decomposed as the difference of two monotone functions. As a consequence of this result we have that those functions satisfy the Dirichlet criterion, that is, the functions that have pointwise convergent Fourier series.

Jordan defined such functions in the following way.

*Definition 1. *Let and be a partition of the interval . Consider
where the supremum is taken over all partitions of the interval .

If , then has bounded variation on the interval and this number is called the variation in Jordan’s sense on . This space of function is denoted by .

The concept of bounded variation has been the subject of intensive research, and many applications, generalizations, and improvements of them can be found in the literature (see for instance [18–20]). Some generalizations have been introduced by De La Vallée Poussin, F. Riesz, N. Wiener, L. C. Young, Yu. T. Medvedev, D. Waterman, B. Korenblum and M. Schramm.

In 1975, Korenblum in [21] considered a new kind of variation, called -variation, introducing a function for distorting the expression in the partition itself rather than the expression in the range. Subsequently, this class of functions has been studied in detail by Cyphert and Kelingos [22]. One advantage of this alternate approach is that a function of bounded -variation may be decomposed into the difference of two simpler functions called -decreasing functions (for the precise definition see the following).

*Definition 2. *A function is said to be a -function or distortion function if it satisfies the following properties:(1)is continuous with and , (2) is concave, increasing, and (3).

Simple examples of distortion functions are

From Definition 2 we can see that is subadditive; that is, and since , then without loss of generality we can assume that

Furthermore Korenblum introduces the following concept of variation.

*Definition 3. *Let be a distortion function and and a partition of the interval . Consider
where the supremum is taken over all partitions of the interval , called the -variation of on . In the case, we say that has -variation on and we will denote by the space of functions of -variation on .

In the case that the equality (9) becomes

Some properties of the functions with bounded -variation are summarized in the following theorem.

Theorem 4 (see [22]). *Let be a distortion function. Then*(1)* is a Banach space endowed with the norm
*(2)*If the function is monotone, then . *(3)*If , then is bounded and .*(4)*, where denotes the space of regulated functions. *(5)*If , then can be decomposed as difference of two -decreasing functions, that is, there exist functions such that .*

It is easy to show that if is finite, then . In [1] it is shown that inclusions (4) of Theorem 4 are strict.

Throughout this paper a -function is a continuous increasing function , such that and .

A -sequence is a sequence of decreasing of convex -function that satisfies diverge for .

We denote by the collection of finite or numerable family of nonoverlapping interval , such that .

In 1985, Schramm [23] introduced a new concept of variation as follows.

*Definition 5. *Let be a -sequence, , and . We define
If , we say that has bounded -variation in the interval and this number denotes the -variation of in Schramm’s sense in . The class of functions that have bounded -variation in the interval is denoted by . The vectorial space generated by this class is denoted by .

The next lemma is useful for building the space generated by several classes of functions.

Lemma 6. *Let be a vector space and a nonempty and symmetric set. Then *(1)*. *(2)*The vector space generated for is equal to
*

Some properties of functions of bounded -variation in Schramm’s sense are given in the following theorem.

Theorem 7 (see [23]). *Let be -sequence then *(1)* is a Banach space endowed with the norm
* *where .*(2) *If is monotone, then .*(3)*If , then is bounded and .*(4) * is a symmetrical and convex set. *(5)*. *(6) *, then has lateral limits at each point of . *

In 1986, S. K. Kim and J. Kim [24] combined the concepts of -variation and -variation introduced by Korenblum and Schramm to create the concept of -variation or variation in Schramm-Korenblum’s sense.

*Definition 8. *Let be a distortion function, a -sequence, , and . We define
If , we say that has bounded -variation in the interval and this number denotes the -variation of in Schramm-Korenblum’s sense in . The class of functions that have bounded -variation in the interval is denoted by . The vectorial space generated by this class is denoted by .

A particular case of -sequence is when all the functions are equal to a fixed -function . In this situation the class is the class of the functions that have bounded -variation in Wiener-Korenblum’s sense. This class of functions is denoted by and the vectorial space generated by this class of function is denoted by .

Some properties of functions of bounded -variation in Schramm-Korenblum’s sense are given in the following theorem.

Theorem 9. *Let be a distortion function and a -sequence, then *(1) * is a Banach space endowed with the norm
* *where .*(2) *If is monotone, then .*(3) *If , then is bounded and .*(4) * is a symmetrical and convex set. *(5) *such that.*(6) *, and therefore . *(7) *. *(8) *, then has lateral limits at each point of . *

*Proof*

*Part *(1). See [24].

*Part *(2). We take , then since the functions are increasing, we obtain

from which is obtained

From the Definition of we have the reciprocal inequality.

*Part* (3). We consider , then
Then and from this inequality we have the required relation.

*Part* (4). We get Part (4) by the convexity of functions and the definition of -variation.

*Part* (5). It follows from part (4) and Lemma 6.

*Part* (6). Let and consider . Define

Then

The last sum has at most terms, where . Because otherwise it has at least summands.

Accordingly

Which is a contradiction. Therefore

This concludes that .

Let us show that . In Fact let and , then

Hence, we get that .

*Part* (7). Let , then by part (3) is bounded in . Let us fix , such that . Let , then from the convexity of the functions , we have

Thus Lemma 6 concludes that .

*Part* (8). Suppose that there is such that does not exist.

By part (3) is bounded then

For each integer (large enough) we can choose such that

Using the definition of -variation, we have

Therefore,

By taking limit when , we obtain , which is absurd. From each it follows that

By a similar argument it follows that there exist .

Assuming that is finite, then . For the last part of Theorem 9, we can give the definition of left and right regularizations of the function .

*Definition 10. *Let , then

The function is called the left regularization of the function and the function the right regularization of the function .

Applying the previous definition and the last part of Theorem 9, we can define

Similarly, we defined .

Recently Castillo et al. [25] introduced the concept of -variation in Riesz-Korenblum’s sense in the following way.

*Definition 11. *Let , be a distortion function and and a partition of the interval . We define
where
and the supremum is taken on the set of all partitions of . If , we say that has bounded -variation in the interval . The number denoted the -variation of in Riesz-Korenblum’s sense in . The space of functions that have bounded -variation in the interval is denoted by .

Some properties of these functions are exposed in the following theorem.

Theorem 12 (see [25]). *Let and let be a distortion function, then *(1)* is a Banach space endowed with the norm
*(2)*, where denote the space of the functions that have bounded -variation in Riesz’s sense [20].*(3)* is an algebra.*

This concept was generalized by Castillo et al. [26] as stated in the following definition.

*Definition 13. *Let be a -function, be a distortion function, and and a partition of the interval . We define
where
and the supremum is taken over all partitions of . If , we say that has bounded -variation in the interval and this number denotes the -variation of in Riesz-Korenblum’s sense in . The class of functions that have bounded -variation in the interval is denoted by . The vectorial space generate by this class is denoted by .

The space of all functions that have bounded -variation on is denoted by . Some properties of these functions are exposed in the following theorem.

Theorem 14 (see [26]). *Let be a convex -function and a distortion function, then *(1)*. *(2)*, where denote the space generated by the class of functions of bounded -variation in Riesz’s sense [20]. *(3)* If , then .*(4)* If , then is bounded. *(5)* is a convex and symmetric set. *(6)* such that .*(7)* is a Banach space endowed with the norm
* *where .*

#### 3. Composition Operator between and or

Given a function , the composition operator , associated to the function (case autonomous), maps each function into the composition function defined by

More generally, given , we consider operator defined by

This operator is also called *superposition operator* or *substitution operator* or *Nemytskii operator*. In what follows, will refer to (39) as the *autonomous case* and to (40) as the *nonautonomous case.*

A problem related with this operator is to establish necessary and sufficient conditions of function so that the operators map the space of real functions defined on into itself, that is, , or in more general way that operator maps the space into space of functions . This problem is sometimes referred to as the composition operator problem (or COP). The solution to this problem for given is sometimes very easy and sometimes highly nontrivial. As we mentioned in the introduction of this paper in a variety of spaces the required condition is that function is locally Lipschitz. Another interesting problem is to determine the smallest space of functions and the bigger space such that .

In order to obtain the main result of this section, we will use a function of the zig-zig type such as the employed by Appell et al. in [1, 15]. In this section we will show that the locally Lipschitz condition of the function is a necessary and sufficient condition such that and that in this situation is bounded.

The following lemma will be useful in the proof of our main theorem (Theorem 17).

Lemma 15. *Let , , then
*

*Proof. *Let . Then

Lemma 16. *Let be a distortion function, a -sequence, , and . Then if and only if .*

* Proof. *Let . Suppose that ; then, by definition of there exists such that and . Hence, by the convexity of the functions , we have

Conversely, assume , then ; hence .

Theorem 17. *Let be a distortion function, a -sequence, , and the composition operator associated to . maps the space into the space or if and only if is locally Lipschitz. Furthermore operator is bounded.*

* Proof. *Let , and suppose that is locally Lipschitz, then there exist , such that

Let and , such that , then

Then by Lemma 15 and Theorem 9, we have .

The proof of the only if direction will be by contradiction, that is, we assume and is not locally Lipschitz. Since the identity function belongs to , then and therefore is bounded in the interval . Without loss of generality we may assume that

Since is not locally Lipschitz in , there is a closed interval such that does not satisfy any Lipschitz condition. In order to simplify the proof we can assume that . In this way for any increasing sequence of positive real numbers that converge to infinite that we will define later, we can choose sequences , , such that

In addition we can choose such that

Considering subsequences if necessary, we can assume that the sequence is monotone. We can assume without loss of generality that sequence is increasing.

Since is compact, from inequality (47) we have that there exist subsequences of and that we will denote in the same way, and that converge to .

Since the sequence is a Cauchy sequence, we can assume (taking subsequence if necessary) that

Again considering subsequences if needed using the properties of the function we can assume that

Consider the new sequence defined by

From inequalities (46) and (47) it follows that ; therefore

Consider the sequence defined recursively by

This sequence is strictly increasing and from the relations (49) and (50), we get

Then to ensure that , it is sufficient to suppose that .

We define the continuous zig-zag function , as shown in the following:

Put

We can write each interval , as the union of the family of nonoverlapping intervals

And function is defined on as follows:

In all these situations, the slopes of the segments of lines are equal to 1.

Hence, we have for , the absolute value of the slope of the line segments in these ranges are bounded by 1, as shown below

We will show that .

Let , and then there are the following possibilities for the location of and on .*Case **1.* If and are in the same interval .

From relations (58), (59), and (60) it follows that .*Case **2*. If and are in two different intervals .

There are several possibilities. (a). . By Lemma 15 and relations (58) and (59) we have
. Then
(b). If , proceed as . If , again using the Lemma 15 and relations (58), (59), and (60) we obtain
*Case **3*. If .

From Lemma 15 and Case 2, we conclude that
*Case **4*. If .

Then from Lemma 15*Case **5*. If .

From Lemma 15 and Case 4,
*Case **6*. If .

In this circumstance and the situation is trivial. Therefore we have that

So is Lipschitz in . Moreover, for each partition of the interval of the form
and , using the inequality (47), convexity of the function , and definition of , we have
As the series diverge, , which is a contradiction.

Let us see that . In fact, as in the case of -variation we ha