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Embeddings on Spaces of Generalized Bounded Variation

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Embeddings on Spaces of Generalized Bounded Variation Revista Colombiana de Matem´aticas Volumen 48(2014)1, p´aginas97-109 Embeddings on Spaces of Generalized Bounded Variation Inmersiones en espacios generalizados de variaci´onacotada Rene´ Erlin Castillo1;B, Humberto Rafeiro2, Eduard Trousselot3 1Universidad Nacional de Colombia, Bogot´a,Colombia 2Pontificia Universidad Javeriana, Bogot´a,Colombia 3Universidad de Oriente, Cuman´a,Venezuela Abstract. In this paper we show the validity of some embedding results on the space of (φ, α)-bounded variation, which is a generalization of the space of Riesz p-variation. Key words and phrases.(φ, α)-Bounded variation, Riesz p-variation, Embed- ding. 2010 Mathematics Subject Classification. 26A45, 26B30, 26A16, 26A24. Resumen. En este trabajo se muestra la validez de algunos resultados de in- mersi´onen el espacio de variaci´on(φ, α)-acotada, que es una generalizaci´on del espacio de Riesz de variaci´on p-acotada. Palabras y frases clave. Variaci´on(φ, α)-acotada, variaci´on p-acotada, inmersi´on. 1. 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 func- tions 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. Am- brosio, G. Dal Maso, among many others. It is noteworthy to mention that 97 98 RENE´ ERLIN CASTILLO, HUMBERTO RAFEIRO & EDUARD TROUSSELOT 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 vari- ation 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 p- variation (1 6 p < 1) and proved that, for 1 < p < 1, this class coincides 0 with the class of functions f, absolutely continuous with derivative f 2 Lp[a; b]. Moreover the p-variation of a function f on [a; b] is given by 0 p Vp(f; [a; b]) = Vp(f) = f Lp[a;b]: Although we will not make much references, for that see [1] and the refer- ences 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. 2. 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 " > 0 there exists some δ > 0 such that if n (aj; bj) j=1 are disjoint open subintervals of [a; b]; then n n X X α(bj) − α(aj) < δ implies f(bj) − f(aj) < ": j=1 j=1 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. Definition 2.2. A function f :[a; b] ! R is said to be α{Lipschitz if there exists a constant M > 0 such that Volumen 48, N´umero1, A~no2014 EMBEDDINGS ON SPACES OF GENERALIZED BOUNDED VARIATION 99 f(x) − f(y) 6 M α(x) − α(y) ; for all x; y 2 [a; b], x 6= y. By α{Lip we will denote the space of functions which are α{Lipschitz. If f 2 α{Lip we define Lipα(f) = inf M > 0 : f(x) − f(y) ≤ M α(x) − α(y) ; x 6= y 2 [a; b] ( ) f(x) − f(y) = sup : x 6= y 2 [a; b] : α(x) − α(y) The space α{Lip[a; b] equipped with the norm f = f(a) + Lipα(f) = f Lipα α{Lip[a;b] is a Banach space. We also introduce the space α{Lip0[a; b] as α{Lip0[a; b] = f 2 α{Lip[a; b]: f(a) = 0 ; in this case we sometimes use f α{Lip[a;b] for Lipα(f). The following proposition is not hard to prove Proposition 2.3. Let f; g 2 α{Lip[a; b], then i) Lipα(cf) = jcjLipα(f), ii) Lipα(f + g) 6 Lipα(f) + Lipα(g), iii) Lipα(f) = 0 if and only if f = const, iv) f 2 α-Lip[a; b] if and only if f − f(a) 2 α-Lip0[a; b]. Definition 2.4. Let φ : [0; 1) ! [0; 1) be a continuous, strictly increasing function with φ(0) = 0 and limt!1 φ(t) = 1. Then such a function is know as a φ-function. 2.1. Functions of (φ, α)-Bounded Variation Definition 2.5. Let f be a real{valued function on [a; b] and φ be a φ-function. Let Π = a = x0 < x1 < ··· < xn = b be a partition of [a; b]. We Consider n ! X f(xj) − f(xj−1) σR (f; Π) = φ α(x ) − α(x ) (φ,α) α(x ) − α(x ) j j−1 j=1 j j−1 Revista Colombiana de Matem´aticas 100 RENE´ ERLIN CASTILLO, HUMBERTO RAFEIRO & EDUARD TROUSSELOT and R R R V(φ,α) f;[a; b] = V(φ,α)(f) = sup σ(φ,α)(f; Π); Π R where the supremum is taken over all partitions Π of [a; b]. V(φ,α)(f) is called R the Riesz (φ, α){variation of f on [a; b]. If V(φ,α)(f) < 1, we say that f is a function of Riesz (φ, α)-bounded variation. The set of all these functions is denoted by R R BV(φ,α)[a; b] = f :[a; b] ! R j V(φ,α)(f) < 1 : p Note, if we set φ(t) = t (1 6 p < 1) 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 R f :[a; b] ! R j 9λ > 0 such that λf 2 BV(φ,α)[a; b] = R f :[a; b] ! R j 9λ > 0 such that V(φ,α)(λf) < +1 is called the vector space of (φ, α)-bounded variation function in the sense of Riesz and we denote it by RBV(φ,α)[a; b]. Definition 2.7. Let φ be a convex φ-function. Then 0 RBV(φ,α)[a; b] = f :[a; b] ! R j f 2 RBV(φ,α)[a; b] and f(a) = 0 is the vector space of Riesz (φ, α)-variation which vanishes at a. Let us now R 0 + define the Minkowski functional j·j(φ,α) : RBV(φ,α)[a; b] ! R by the formula R R jfj(φ,α) = inf " > 0 j V(φ,α)(f=") 6 1 . Definition 2.8. Let φ be a convex φ-function. We define a Luxemburg type norm by R k·k(φ,α) : RBV(φ,α)[a; b] ! R f with f 7! jf(a)j + jf − f(a)jR = jf(a)j + inf " > 0 VR 1 . (φ,α) (φ,α) " 6 The following results were shown in [3] 0 Lemma 2.9. Let φ be a convex φ-function. Let f 2 RBV(φ,α)[a; b]. Then R R R i) jfj(φ,α) 6= 0 =) V(φ,α) f=jfj(φ,α) 6 1; R R ii) jfj(φ,α) < k () V(φ,α) f=k 6 1; k > 0; R R R iii) 0 6 jfj(φ,α) 6 1 =) V(φ,α)(f) 6 jfj(φ,α); Lemma 2.10. Let φ be a φ-function, then f 2 RBV(φ,α)[a; b] if and only if 0 f − f(a) 2 RBV(φ,α)[a; b]. Volumen 48, N´umero1, A~no2014 EMBEDDINGS ON SPACES OF GENERALIZED BOUNDED VARIATION 101 3. Embedding Results φ(x) Definition 3.1. Let φ be a convex φ-function. If limn!1 x = +1, then we say that φ satisfies the 11-condition. Using the 11-condition we obtain several relations among some spaces. Theorem 3.2. Let φ be a convex φ-function which satisfy the 11-condition. Let f 2 RBV(φ,α)[a; b], then f is absolutely continuous with respect to α on [a; b], that is, RBV(φ,α)[a; b] ⊂ α-AC[a; b]. Proof. Let f 2 RBV(φ,α)[a; b]. Given " > 0, let us consider (aj; bj), j = 1; 2; : : : ; n a finite collection of disjoint subintervals contained in [a; b].
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