The Space of Bounded p(⋅)-Variation in the Sense Wiener-Korenblum with Variable Exponent

In this paper we present the notion of the space of bounded p(⋅)-variation in the sense of Wiener-Korenblum with variable exponent. We prove some properties of this space and we show that the composition operator H, associated with h → :  , maps the ( ) [ ] ( ) W p BV a b ⋅ , κ into itself, if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by [ ] h a b × → : ,   maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable, i.e. satisfies the Matkowski’s weak condition.


Introduction
A number of generalizations and extensions of variation of a function have been given in many directions since Camile Jordan in 1881 gave a first notion of bounded variation in the paper [1] devoted to the convergence of Fourier series.Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis.Two well-known generalizations are the functions of bounded p-variation and the functions of bounded ϕ-variation, due to N. Wiener [2] and L. C. Young [3] respectively.In 1924 N.
Wiener [2] generalized the Jordan notion and introduced the notion of p-variation (variation in the sense of Wiener).Later, in 1937, L. Young [3] introduced the notion of ϕ-variation of a function.The p-variation of a function f is the supremum of the sums of the pth powers of absolute increments of f over no overlapping inter-vals.Wiener mainly focused on the case 2 p = , the 2-variation.For p-variations with 2 p ≠ , the first major work was done by Young [3], partly with Love [4].After a long hiatus following Young's work, p th -variations were reconsidered in a probabilistic context by R. Dudley [5] [6], in 1994 and 1997, respectively.Many basic properties of the variation in the sense of Wiener and a number of important applications of the concept can be found in [7] [8].Also, the paper by V. V. Chistyakov and O. E. Galkin [9], in 1998, is very important in the context of p-variation.They study properties of maps of bounded p-variation ( ) in the sense of Wiener, which are defined on a subset of the real line and take values in metric or normed spaces.In 1997 while studying Poisson integral representations of certain class of harmonic functions in the unit disc of the complex plan B. Korenblum [10] introduced the notion of bounded κ-variation and proved that a function f is of bounded κ-variation if ot can be written as the difference of two κ-decreasing functions.This concept differs from others due to the fact that it introduces a distortion function κ that measures intervals in the domain of the function and not in the range.In 1986, S. Ki Kim and J. Kim [11], gave the notion of the space of functions of κφ-bounded variation on [ ] , a b , which is a combination of the notion of bounded φ-variation in the sense of Schramm and bounded κ-variation in the sense of Korenblum, and J. Park et al. [12] [13] proved some properties in this space.Considering ( ) p n x x φ = for 1 p < < ∞ and 1 n ≥ , then it follows that this space generalized the space of functions of κp-bounded variation in the sense of Wiener- Korenblum.In 1990 S. Ki Kim and J. Yoon [14] showed the existence of the Riemann-Stieltjes integral of functions of bounded κ-variation and in 2011 W. Aziz, J. Guerrero, J. L. Sánchez and M. Sanoja, in [15], showed that the space of bounded κ-variation satisfies the Matkowski's weak condition.Also, in 2012, M. Castillo, M. Sanoja and I. Zea [16] presented the space of functions of bounded κ-variation in the sense of Riez-Korenblum, denoted by < < ∞ and bounded κ-variation in the sense of Korenblum.Recently, there has been an increasing interest in the study of various mathematical problems with variable exponents.With the emergency of nonlinear problems in applied sciences, standard Lebesgue and Sobolev spaces demonstrated their limitations in applications.The class of nonlinear problems with exponent growth is a new research field and it reflects a new kind of physical phenomena.In 2000 the field began to expand even further.Motivated by problems in the study of electrorheological fluids, L. Diening [17] raised the question of when the Hardy-Littlewood maximal operator and other classical operators in harmonic analysis are bounded on the variable Lebesgue spaces.These and related problems are the subject of active research nowadays.These problems are interesting in applications (see [18]- [21]) and give rise to a revival of the interest in Lebesgue and Sobolev spaces with variable exponent, the origins of which can be traced back to the work of W. Orlicz in the 1930's [22].In the 1950's, this study was carried on by H. Nakano [23] [24] who made the first systematic study of spaces with variable exponent.Later, Polish and Czechoslovak mathematicians investigated the modular function spaces (see for example J. Musielak [25] [26], O. Kovacik and J. Rakosnik [27]).We refer to books [21] for the detailed information on the theoretical approach to the Lebesgue and Sobolev spaces with variable exponents.In 2015, R. Castillo, N. Merentes and H. Rafeiro [28] studied a new space of functions of generalized bounded variation.There the authors introduced the notion of bounded variation in the Wiener sense with the exponent p(⋅)-variable.In the same year, O. Mejia, N. Merentes and J. L. Sánchez in [29], proved some properties in this space, for the composition operator and showed a structural theorem for mappings of bounded variation in the sense of Wiener with the exponent p(⋅)-variable.
The main purpose of this paper is threefold: First, we provide extension of the space of generalized bounded variation present in [28] and [29] in the sense Wiener-Korenblum and we give a detailed description of the new class formed by the functions of bounded variation in the sense of Wiener-Korenblum with the exponent p(⋅)variable.Second, we prove a necessary and sufficient condition for the acting of composition operator (Nemystskij) on the space and, third we show that any uniformly bounded composition operator that maps the space into itself necessarily satisfies the so called Matkowski's weak condition.

Preliminaries
We use throughout this paper the following notation: we will denote by The class of bounded variation functions exhibit many interesting properties that it makes them a suitable class of functions in a variety of contexts with wide applications in pure and applied mathematics (see [8] and [30]).Since C. Jordan in 1881 (see [1]) gave the complete characterization of functions of bounded variation as a difference of two increasing functions, the notion of bounded variation functions has been generalized in different ways.: ; , : sup , where the supremum is taken over all partitions π of the interval [ ] : ; , : sup in the range.On advantage of this alternative approach is that a function of bounded κ-variation may be decomposed into the difference of two simpler functions called κ-decreasing functions.: where the supremum is taken over all partitions π of the interval [ ] Some properties of κ-function cab be found in [12] [14] [16].
In 2013 R. Castillo, N. Merentes and H. Rafeiro [28] introduce the notation of bounded variation space in the Wiener sense with variable exponent on [ ] , a b and study some of its basic properties., , : sup 2) If ( ) In [29], O. Mejia, N. Merentes and J. L. Sánchez proved some properties in this space, for the composition operator and show a structural theorem for mappings of bounded variation in the sense of Wiener with the exponent p(⋅)-variable., , : sup ., , n  x x −  subject to the conditions that for each i, 2) If ( ) Example 2.9.
( ) . Then, from mean value theorem, we have

Properties of the Space
p a b → ∞ and κ be a distortion function then : Then, by the κ subadditivity, we have: Then considering the supremum of the left side we get , then for each partition 0 1 : for all j t , and we obtain On the other hand, since p(⋅) is bounded, there exists 0 L > such that and α ∈  .By Theorem 3.3 where , α ∈  .Then, we have that: Then by convexity of for all 0 λ > , and so Conversely, suppose that , we get without loss of generality, considering the partition : 0 and 0, 0 and 0.
 In the following, we show that f ∈ be a Cauchy sequence in , , , . and  , , .
; , 1, In consequence, the sequence { } 1 n n f ≥ , is a uniformly sequence of Cauchy, on the interval [ ]  We will show that n f converge on the norm .
From the fact that { } 1 n n f ≥ converge uniformly to the function f on the interval [ ] , a b , we get Therefore, the sequence { } 1 n n f ≥ converge to the function f on the norm such that ( ) t ϕ τ = and again by the monotonicity of ϕ : , , sup , , : , , , .
On the other hand, for any number there is a partition  In the next section we will be dealing with the composition operator (Nemitskij).(see [8]).Now, we define the composition operator.Given a function : h →   , the composition operator H, associated to a function f (autonomous case) maps each function

Composition Operator between the Space
This operator is also called superposition operator or susbtitution operator or Nemytskij operator.In what follows, will refer (9) as the autonomus case and to (10) as the non-autonomus case.
In order to obtain the main result of this section, we will use a function of the zig-zag type such as the employed by J. Appell et al. [8] [37] that the locally Lipschitz condition of the function h is a necessary and sufficient condition such that [ ] ( ) and that in this situation H is bounded.
One of our main goals is to prove a result in the case when h is locally Lipschitz if and only if the composition operator maps the space of functions of bounded p(⋅)-variation into itself.
The proof of the only if direction will be by contradiction, that is we assume and h is not locally Lipschitz.Since the identity function and therefore h is bounded in the interval [ ] 0,1 .Without loss of generality we may assume that Since h is not locally Lipschitz in  there is a closed interval I such that h does not satisfy any Lipschitz condition.In order to simplify the proof we can assume that [ ] 0,1 .

I =
In this way for any increasing sequence of positive real numbers { } 1 n n k ≥ that converge to infinite, that we will define later, we can choose sequences In addition choose , Considering subsequence if it necessary, we can assume without loss of generality that the sequence { } 1 n n a ≥ is monotone increasing.
Since [ ] 0,1 is compact, from inequality (13) we have that exist subsequences of { } 1 n n a ≥ and { } 1 n n b ≥ that we will denote in the same way, and that converge to Since the sequence { } 1 n n a ≥ is a Cauchy sequence we can assume (taking subsequence if it is necessary) that ( ) Again considering subsequences if needed and using the properties of the function κ we can assume that Consider the new sequence { } 1 , .
From of inequalities ( 12) and ( 13) it follows that 2 This sequence is strictly increasing and from the relations ( 14) and ( 15), we get , is sufficient to suppose that We define the continuous zig-zag function .
n n n m u t t t a t I In all these situations the slopes of these segments of lines is 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 From relations ( 16), ( 17) and ( 18) follows ( ) ( )  16) and (17) we have n n n i n i u t u s b a t s t t > + , again using the Lemma 4.1 and relations ( 16), ( 17) and ( 18) we obtain   [ ]( ) , which is a contradiction.

Uniformly Continuous Composition Operator
In a seminal article of 1982, J. Matkowski [39] showed that if the composition operator

,
Lip a b that verify this result [37].The spaces of Banach ( ) , ⋅  that fulfill this property are said to satisfy the Matkowski property [32].
In 1984, J. Matkowski and J. Miś [40] considered the same hypotheses on the operator H for the space [ ] , BV a b of the function of bounded variation and concluded that ( 19) is true for the regularization h − of the function h with respect of the first variable; that is, . The spaces that satisfy this condition are said to verify weak Matkowski property, [32].In this section, we give the other main result of this paper, namely, we show that any uniformly bounded composition operator that maps the space From the inequality (22) and Lemma 5.2, if ( ) Let a s t b ≤ < ≤ , and let be the equidistant partition defined by ( ) , : , if for some even , : , if for some odd , 2 linear, otherwise , if for some even , 2 : , if for some odd , linear, otherwise.

. j j j j h f t h f t h f t h f t u v u v h v h h h u u v h v h h u
the notions of bounded p-variation in the sense of Riesz ( )1 p (or the oscillation of f on [ ] , a b ) and by ts x a number between [ ] , t s .
→  be a function.For each partition 0 1 is called the Wiener variation (or p-variation in Wiener's sense) of f on [ ] , a b where the supremum is taken over all partitions π .b < ∞ , we say that f has bounded Wiener variation (or bounded p-variation in Wien- er's sense) on [ ] will denote the space of functions of bounded p-variation in Wiener's sense on [ ] , a b .Other generalized version was given by B. Korenblum in 1975 [10].He considered a new kind of variation, called κ-variation, and introduced a function κ for distorting the expression 1 j j t t − − in the partition if self, rather than the expression ( ) ( )

2 . 4 .
distortion function (κ-function) if κ satisfies the fol-(see [10]), introduced the definition of bounded κ-variation as follows.Definition Let κ be a distortion function, f a real function [ ] : , f a b →  , and says that f has bounded κ-variation on [ ] , a b and one will denote by

2 . 7 .
Now, we generalized the notion of bounded variation space in the sense of Wiener-Korenblum with variable exponent on [ ] , a b .For this, we defined bellow the bounded p(⋅)-variation in the sense of Wiener-Korenblum with exponent variable.Definition Given is called Wiener-Korenblum variation with variable exponent (or p(⋅)-variation in the sense of Wiener-Korenblum) of f on [ ] , a b where * π is a tagged partition of the interval [ ] , a b , i.e., a partition of the interval [ ] , a b together with a finite sequence of numbers 0 1 we say that f has bounded Wiener-Korenblum variation with variable exponent (or bounded p(⋅)-variation in the sense of Wiener-Korenblum) on [ ] space of functions of bounded p(⋅)-variation in the sense Wiener-Korenblum with variable exponent on [ ]

2 .
From this result we deduce that every function of bounded p(⋅)-variation in of Wiener's sense with variable exponent on the interval [ ] , a b is a bounded p(⋅)-variation in the Wiener-Korenblum sense on the interval [ ] , a b .Now we will see that the class of function of bounded p(⋅)-variation in the sense of Wiener-Korenblum has a structure of vector space.Theorem 3.3.Let [ ] ( ) variation in the sense of Wiener-Korenblum with variable exponent on [ ]

,
a b .Since  is complete, there exists a function f defined on [ ] necessary strictly) monotone function, 0 π a tagged partition of the interval [ ]

,,,
In any field of nonlinear analysis composition operators (Nemytskij), the superposition operators generated by appropriate functions, play a crucial role in the theory of differential, integral and functional equations.Their analytic properties depend on the postulated properties of the defining function and on the function space in which they are considered.A rich source of related questions is the monograph by J. Appell and P. P. Zabrejko[31] and J. Appell, J. Banas, N. Merentes[8].The composition operator problem refers to determining the conditions on a function : h →   , such that the composition operator, associated with the function h, maps a space  of functions[ ] : , u a b →  into itself [32][33].There are several spaces where the composition operator problem has been resolved.In 1961, A. A. Babaev[34] showed that the composition H, associated with the function :h →   ,maps the space [ ] Lip a b of the Lipschitz functions into itself if and only if h is locally Lipschitz; in 1967, K. S. Mukhtarov [35] obtained the same result for the space [ ] Lip a b of the Hölder functions of order α ( ) The first work on the composition operator problem in the space of functions of bounded variation [ ] , BV a bwas made by M. Josephy in 1981,[36].Other work of this type have been preformed over × →   , we consider the operator H, defined by as the union of the family of non-overlapping intervals By Lemma 4.1 and relations (

−→
satisfies the so called Matkowski's weak condition.First of all we will give the definition of left regularization of a function. of mapping f is the function given as

(
From the inequality(23) and the definition of p(⋅)-variation in the sense of Wiener-Korenblum we have

⋅
satisfies the Jensen equation in  (see[41], page 315).The continuity of h − with respect of the second variable implies that for every[42] introduced the notion of a uniformly bounded operator and proved that any uniformly bounded composition operator acting between general Lipschitz function normed spaces must be of the form(21).Definition 5.6.([42],Def.
we say that f has bounded variation.The collection of all functions of bounded variation on [ ] ) , then are the following possibilities for the location of s and t on [ ] .
Every uniformly continuous operator or Lipschitzian operator is uniformly bounded.
1) Let  and  be two metric (or normed) spaces.We say that a mapping : H →   is uniformly bounded if, for any 0 t > there exists a nonnegative real number ( )