Functions of Bounded κφ-Variation in the Sense of Riesz-Korenblum

We present the space of functions of bounded κφ-variation in the sense of Riesz-Korenblum, denoted by κBV φ [a,b], which is a combination of the notions of bounded φ-variation in the sense of Riesz and bounded κ-variation in the sense of Korenblum. Moreover, we prove that the space generated by this class of functions is a Banach space with a given norm and we prove that the uniformly bounded composition operator satisfies Matkowski’s weak condition.


Introduction
The concept of functions of bounded variation has been well known since Jordan [1] gave the complete characterization of functions of a bounded variation as the difference of two increasing functions in 1881.This class of functions immediately proved to be important in connection with the rectification of curves and with Dirichlet's theorem on the convergence of Fourier series.Functions of a bounded variation exhibit many interesting properties that make them a suitable class of functions in a variety of contexts with wide applications in pure and applied mathematics (see [2][3][4]).
Riesz [5] in 1910 generalized the notion of Jordan and introduced the concept of bounded -variation (1 <  < ∞) and showed that, for 1 <  < ∞, this class coincides with the class of functions absolutely continuous with the derivative in the space   .On the other hand, this notion of bounded -variation was generalized by Medvedev [6] in 1953 who introduced the concept of bounded -variation in the sense of Riesz and also showed a Riesz's lemma for this class of functions.
Korenblum [7] in 1975 introduced the notion of bounded -variation.This concept differs from others due to the fact that it introduces a distortion function  that measures intervals in the domain of the function and not in the range.In 1985, Cyphert and Kelingos [8] showed that a function  is of bounded -variation if it can be written as the difference of two -decreasing functions.In 1986, S. K. Kim and J. Kim [9] and Park [10], in 2010, introduced the notion of functions of -bounded variation on compact interval [, ] ⊂ R which is a combination of concepts of bounded -variation and bounded -variation in the sense of Schramm [11], and in 2011 Aziz et al. [12] showed that the space of bounded variation satisfies Matkowski's weak condition.
Recently in [13] Castillo et al. introduce the notion of bounded -variation in the sense of Riesz-Korenblum, which is a combination of the notions of bounded -variation in the sense of Riesz and bounded -variation in the sense of Korenblum.The purpose of this paper is twofold.First, to introduce the concept of bounded -variation in the sense of Riesz-Korenblum, which is a combination of the notions of bounded -variation in the sense of Riesz and bounded variation in the sense of Korenblum.We prove some properties of this class of functions and its relation with the functions of bounded -variation and bounded -variation in the sense of Riesz.Second we prove that the space generated by this class of functions is a Banach space with a given norm and that the uniformly bounded composition operator satisfies Matkowski's weak condition in this space.The Matkowski property has been studied by several authors (see [14][15][16]), and for Matkowski's weak property, see also [3,[17][18][19][20][21].In [22][23][24] Matkowski, Merentes, and others authors have been studying a weaker condition on the composition operator Definition 2. A function  : [0, ∞) → [0, ∞) is said to be a -function if it satisfies the following properties.
The notion of bounded variation due to Jordan (Definition 1) was generalized by Medvedev (see [6]) as follows.
Definition 4. Let  be a -function and  : [, ] → R be a function.For each partition  :  =  0 <  1 < ⋅ ⋅ ⋅ <   =  of the interval [, ], we define ( ( Other generalization of the notion of bounded variation was introduced by Korenblum The set of all -functions will be denoted by K.Note that, every -function  is subadditive; that is, Then, for all partition  :  =  0 <  1 < ⋅ ⋅ ⋅ <   =  of [, ], we have Korenblum (see [7]) introduces the definition of bounded -variation as follows.
where the supremum is taken over all partitions  of the interval [, ].We denote by [, ] the collection of all functions of bounded -variation on [, ].
Next, some properties of the space [, ] are exposed (see [8]). ( (4) A function  has bounded -variation in an interval [, ] if and only if it can be decomposed as a difference of -decreasing functions.
(5) Every function of bounded -variation has left-and right-hand limits at each point of its domain.

Main Results
In this section we present the principal results of this paper.
Taking the supremum over all partitions  of the interval [, ], the greater value of the right side of the above expression is obtain for the partition  :  =  0 <  1 =  and in this case we get Therefore, In the following proposition, we prove two important properties of the space   [, ].
Since  is a convex -function and (0) = 0, we have then Then Considering the supremum over all partitions  of the interval [, ] in the above expression, we get therefore Now, we will show part (b).If then there exist  0 > 0 and  > 0, such that Let us consider the partition  : and then thus, Then by considering the supremum over all partitions  of the interval [, ] of the left side, we get that is, Therefore, from part (a) and ( 28 considering the supremum over all partitions  of the interval [, ], we get The class of functions of a bounded -variation has many interesting properties as the following proposition showes.
Proposition 11.Let  be a -function,  ∈ K and  : [, ] → R be a function, then is convex.
As a consequence of Lemma 15 and since    is a convex and symmetric set, we have the following corollary.(57) Thus, Then, considering the supremum of the left side, we get

Journal of Function Spaces and Applications
Proof.First we show the convexity.Let ,  ∈ [0, 1] such that  +  = 1 and , V ∈ Λ.Then, by Proposition 11 we get thus  + V ∈ Λ.Now let  ∈ Λ and 0 < || ≤ 1.If 0 <  ≤ 1 by Proposition 11, we have For the case −1 ≤  < 0, by the symmetric and convexity of the functional    (⋅) given in Proposition 11 we get that Hence, we have shown that Λ is balance.Now we will show that Λ is absorbent.Let  ∈   [, ] then there exist  > 0 such that    () < ∞.If    () ≤ 1 then  ∈ Λ.On the other hand, if    () > 1 we have is a normed space.
that is, Then, by Lemma 21 and the last inequality, we have Then, for a partition  :  ≤  <  ≤  we get that Here R [,] denotes the family of all functions  : [, ] → R.
Remark 24.Since   [, ] ⊂ [, ], then every function of bounded -variation in the sense of Riesz-Korenblum has left-and right-hand limits at each point of its domain (see [8]).Now, we will give the definition of left regularization of a function.