Inmersiones en espacios generalizados de variación acotada

In this paper we show the validity of some embedding results on the space of  (\phi,α) -bounded variation, which is a generalization of the space of Riesz  p -variation.


Introduction
Two centuries ago, around 1880, C. Jordan (see [5]) introduced the notion of a function of bounded variation and established the relation between those functions and monotonic ones when he was studying convergence of Fourier series.Later on the concept of bounded variation was generalized in various directions by many mathematicians, such as F. Riesz, N. Wiener, R. E. Love, H. Ursell, L. C. Young, W. Orlicz, J. Musielak, L. Tonelli, L. Cesari, R. Caccioppoli, E. de Giorgi, O. Oleinik, E. Conway, J. Smoller, A. Vol'pert, S. Hudjaev, L. Ambrosio, G. Dal Maso, among many others.It is noteworthy to mention that many of these generalizations where motivated by problems in such areas as calculus of variations, convergence of Fourier series, geometric measure theory, mathematical physics, etc.For many applications of functions of bounded variation in mathematical physics see, e.g., the monograph [7].We just want to point out the recent generalization on bounded variation in the framework of variable spaces [2].
In his 1910 paper F. Riesz (see [6]) defined the concept of bounded pvariation (1 p < ∞) and proved that, for 1 < p < ∞, this class coincides with the class of functions f , absolutely continuous with derivative Although we will not make much references, for that see [1] and the references therein, we want to stress that there is a vast literature on the topic of bounded variation.
In [4] the first and third named authors of the present paper generalized the concept of bounded p-variation introducing a strictly increasing continuous function α : [a, b] → R and considering the bounded p-variation with respect to α.This new concept was called (p, α)-bounded variation and denoted by BV (p,α) [a, b].Recently the authors went a step further and generalized the (p, α)-variation to (φ, α)-variation and gave a characterization of this newly introduced spaces, see [3].
In this paper, we continue previous work [3,4] in the study of (φ, α)variation space and give some embedding results in it.

Preliminaries
In this section, we gather definitions, notations and results that will be used throughout the paper.Let α be any strictly increasing, continuous function defined on [a, b].Definition 2.1.A function f : [a, b] → R is said to be absolutely continuous with respect to α if for every ε > 0 there exists some δ > 0 such that if (a j , b j ) Thus, the collection α-AC[a, b] of all α-absolutely continuous functions on [a, b] is a function space and an algebra of functions.
By α-Lip we will denote the space of functions which are α-Lipschitz.If f ∈ α-Lip we define The space α-Lip[a, b] equipped with the norm is a Banach space.
We also introduce the space α-Lip 0 [a, b] as The following proposition is not hard to prove Then such a function is know as a φ-function.
where the supremum is taken over we say that f is a function of Riesz (φ, α)-bounded variation.The set of all these functions is denoted by we get back the concept of (p, α)bounded variation defined in [4].In the case φ = α = identity function, then we get back the classical concept of bounded variation, denoted by BV[a, b].Definition 2.6.Let φ be a convex φ-function.Then < +∞ is called the vector space of (φ, α)-bounded variation function in the sense of Riesz and we denote it by and f (a) = 0 is the vector space of Riesz (φ, α)-variation which vanishes at a. Let us now define the Minkowski functional The following results were shown in [3] Lemma 2.9.Let φ be a convex φ-function.
Using the ∞ 1 -condition we obtain several relations among some spaces.
Since φ satisfy the ∞ 1 -condition, there exists x 0 ∈ (0, ∞) such that φ(x) mx for x x 0 .Next, let us consider the following set Since φ satisfy the ∞ 1 -condition we have and thus From this last inequality we obtain

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Now, choose 0 < δ < ε/(2x 0 ).Thus, if Finally, collecting all of this information we conclude that, given ε > 0 there exists δ > 0 such that for all finite family of disjoint subintervals (a j , b j ) and thus for all x, y ∈ [a, b] and x = y.
Since φ is an increasing function, we have 1.By Lemma 2.9 ii) we obtain (1) and there exists m > 0 such that there exists m > 0 such that Then there exists m > 0 such that where BV 0 [a, b] is the BV[a, b] space of vanishing functions at the point a.
Proof.From Theorem 3.2 we know that f is of bounded variation.If f R (φ,α) = 0 then f = 0 and thus V(f ) = 0; then the inequality holds trivially.