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

We show that the composition operator H, associated with h : [a, b] → R, maps the spaces Lip[a, b] on to the space κBVφ[a, b] of functions of bounded variation in Schramm-Korenblum’s sense if and only if h is locally Lipschitz. Also, verify that if the composition operator generated by h : [a, b] × R → Rmaps this space into itself and is uniformly bounded, then regularization of h is affine in the second variable.


Introduction
The composition operator problem (or COP, for short) refers to determining the conditions on a function ℎ : R → R, such that the composition operator, associated with the function ℎ, maps a space X of functions  : [, ] → R 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 ℎ : R → R, maps the space Lip [𝑎, 𝑏] of the Lipschitz functions into itself if and only if ℎ is locally Lipschitz; in 1967, Mukhtarov [4] obtained the same result for the space Lip  [𝑎, 𝑏] of the Hölder functions of order  (0 <  < 1).
The first work on the COP in the space of functions of bounded variation BV [𝑎, 𝑏] 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 HBV [𝑎, 𝑏].In the years 1991 and 1995 Merentes showed a similar result for the spaces of absolutely continuous functions AC [𝑎, 𝑏] and the space of function of bounded -variation in Riesz's sense RV  [𝑎, 𝑏] (see [8,9]), and in 1998 Merentes and Rivas achieved the same result when the composition operator maps the space RV  [𝑎, 𝑏] of the functions of bounded -variation in Riesz's sense (1 <  < ∞) into the space BV[, ] [10].In 2003, Pierce and Waterman solved the COP for the spaces BV [𝑎, 𝑏] and ΛBV [𝑎, 𝑏] [11].More recently, in 2011, Appell et al. [1] conclude the same results verifying when the composition operator maps Lip [𝑎, 𝑏] into BV [𝑎, 𝑏].Finally, Appell and Merentes verify the same result for the space of functions of bounded -variation [12].
There exist spaces X of real functions defined on an interval [𝑎, 𝑏], such that  maps X into itself and ℎ is not locally Lipschitz.For example, in the case the space of continuous functions  [𝑎, 𝑏] it follows from the Tietze-Urysohn theorem that the composition operator  acts from  [𝑎, 𝑏] into itself and the function ℎ must be continuous; that is, ℎ 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 ℎ, maps the space Lip [𝑎, 𝑏] of the Lipschitz functions into the space BV  [𝑎, 𝑏] of functions of bounded variation in Schramm-Korenblum's sense or into the space BV [𝑎, 𝑏] of functions of bounded variation in Korenblum's sense if and only if ℎ is locally Lipschitz.We also extend this result to function spaces X, Y , such that Lip[, ] ⊂ X ⊂ Y , where Y ⊂ BV [𝑎, 𝑏] or Y ⊂ BV  [𝑎, 𝑏].
There are a variety of spaces besides Lip [𝑎, 𝑏] that verify this result [15].The spaces of Banach (X, ‖ ⋅ ‖) 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 BV [𝑎, 𝑏] of the function of bounded variation and concluded that (1) is true for the regularization ℎ − of the function ℎ with respect of the first variable; that is, ℎ − (, ) =  ()  +  () ,  ∈ [𝑎, 𝑏] ,  ∈ R, where ,  ∈ BV − [a, b].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 ℎ(, ⋅) is continuous in the second variable, for each  ∈ [, ], and the composition operator , associated with the function ℎ, is uniformly bounded, then ℎ satisfies (2).

Preliminaries
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  : [, ] → R 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.
where the supremum is taken over all partitions  of the interval [𝑎, 𝑏].
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][19][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 |  −  −1 | in the partition itself rather than the expression |(  ) − ( −1 )| 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  : [0, 1] → [0, 1] is said to be a -function or distortion function if it satisfies the following properties: (1)  is continuous with (0) = 0 and (1) = 1, (2)  is concave, increasing, and

Simple examples of distortion functions are
From Definition 2 we can see that  is subadditive; that is, and since lim  → 0 + (()/) = ∞, then without loss of generality we can assume that Furthermore Korenblum introduces the following concept of variation.
where the supremum is taken over all partitions  of the interval [ Some properties of the functions with bounded variation are summarized in the following theorem.
We denote by  N [𝑎, 𝑏]  The next lemma is useful for building the space generated by several classes of functions.Lemma 6.Let X be a vector space and  ⊂ X a nonempty and symmetric set.Then (1) 0 ∈ .
(2) The vector space generated for  is equal to ⟨⟩ = { ∈ X : ∃ > 0 such that  ∈ } = ⋃ >0 .(13) Some properties of functions of bounded -variation in Schramm's sense are given in the following theorem.
Theorem 7 (see [23]).Let  = {  } ≥1 be -sequence then (1) BV  [𝑎, 𝑏] is a Banach space endowed with the norm where () := inf >0 { > 0 :   (/) ≤ 1}. ( (  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.A particular case of -sequence is when all the functions   ,  ∈ N are equal to a fixed -function .In this situation the class   [𝑎, 𝑏] is the class of the functions that have bounded -variation in Wiener-Korenblum's sense.This class of functions is denoted by   [𝑎, 𝑏] and the vectorial space generated by this class of function is denoted by BV  [𝑎, 𝑏].
Some properties of functions of bounded -variation in Schramm-Korenblum's sense are given in the following theorem.
Part (4).We get Part (4) by the convexity of functions   ,  ∈ N and the definition of -variation.
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  1 ( − ) = 0, which is absurd.From each it follows that By a similar argument it follows that there exist lim ↓ * (),  * ∈ [, ).
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 BV +  [𝑎, 𝑏].Recently Castillo et al. [25] introduced the concept of variation in Riesz-Korenblum's sense in the following way.Some properties of these functions are exposed in the following theorem.
This concept was generalized by Castillo et al. [26] as stated in the following definition.
where The space of all functions that have bounded -variation on [, ] is denoted by RV  [𝑎, 𝑏].Some properties of these functions are exposed in the following theorem.

Composition Operator between
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 ℎ so that the operators  map the space X of real functions defined on [, ] into itself, that is, (X) ⊂ X, or in more general way that operator  maps the space X into space of functions Y ((X) ⊂ Y ).This problem is sometimes referred to as the composition operator problem (or COP).The solution to this problem for given X 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 ℎ is locally Lipschitz.Another interesting problem is to determine the smallest space of functions X and the bigger space Y such that (X) ⊂ Y .
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 ℎ is a necessary and sufficient condition such that (Lip[, ]) ⊂ BV [𝑎, 𝑏] and that in this situation  is bounded.
The following lemma will be useful in the proof of our main theorem (Theorem 17).
The proof of the only if direction will be by contradiction, that is, we assume (Lip[0, 1]) ⊂ BV  [0, 1] and ℎ is not locally Lipschitz.Since the identity function Without loss of generality we may assume that Since ℎ is not locally Lipschitz in R, there is a closed interval  such that ℎ does not satisfy any Lipschitz condition.In order to simplify the proof we can assume that  = [0, 1].In this way for any increasing sequence of positive real numbers {  } ≥1 that converge to infinite that we will define later, we can choose sequences In addition we can choose   ,   such that Considering subsequences if necessary, we can assume that the sequence {  } ≥1 is monotone.We can assume without loss of generality that sequence {  } ≥1 is increasing.
Since the sequence {  } ≥1 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 max { (  −   ) ,  (  −   )} < 1   , ( ∈ N,  ≥ ) . (50) Consider the new sequence {  } ≥1 defined by From inequalities ( 46) and (47) it follows that   > 2; therefore Consider the sequence defined recursively {  } ≥1 by This sequence is strictly increasing and from the relations (49) and (50), we get Then to ensure that In all these situations, the slopes of the segments of lines are equal to 1.
Hence, we have for  ∈ N, the absolute value of the slope of the line segments in these ranges are bounded by 1, as shown below We will show that  ∈ Lip[0, 1].Let 0 ≤  <  ≤ 1, and then there are the following possibilities for the location of  and  on [0, 1]. (62) If  =  + 1, proceed as ( 1 ).
In the following result we give a Lemma of invariance.
As consequence of Lemma 18 we have the following results

Lemma 19.
Let  be a distortion function,  = {  } ≥1 a -sequence, ℎ : R → R, and  the composition operator associated to the function ℎ.
(1) (Lip [𝑎, 𝑏]) ⊂ BV  [𝑎, 𝑏]  In many problems solving equation where the composition operator appears to guarantee the existence of solution it is necessary to apply a Fixed Point Theorem.To ensure the application this type of results is necessary to request the condition of global Lipschitz operator .In several works Matkowski and Mís have shown that this condition implies that the function ℎ has the form (1) or (2) (see, e.g., [16,27]).This means that we may apply the Banach contraction mapping principle only if the underlying problems are actually  and therefore are not interesting.
More recently, Matkowski and other researchers have replaced the condition of global Lipschitz by uniform continuity conditions or uniform boundedness composition operator (see e.g., [14]).
)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 BV [, ].