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 < ∞) and proved that, for 1 < p < ∞, this class coincides 0 with the class of functions f, absolutely continuous with derivative f ∈ 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 ε > 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) < ε. 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 ∈ [a, b], x 6= y.

By α–Lip we will denote the space of functions which are α–Lipschitz. If f ∈ α–Lip we define  Lipα(f) = inf M > 0 : f(x) − f(y) ≤ M α(x) − α(y) , x 6= y ∈ [a, b] ( ) f(x) − f(y) = sup : x 6= y ∈ [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 ∈ 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 ∈ α–Lip[a, b], then

i) Lipα(cf) = |c|Lipα(f),

ii) Lipα(f + g) 6 Lipα(f) + Lipα(g), iii) Lipα(f) = 0 if and only if f = const, iv) f ∈ α-Lip[a, b] if and only if f − f(a) ∈ α-Lip0[a, b].

Definition 2.4. Let φ : [0, ∞) → [0, ∞) be a continuous, strictly increasing function with φ(0) = 0 and limt→∞ φ(t) = ∞. 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) < ∞, 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 | V(φ,α)(f) < ∞ .

p Note, if we set φ(t) = t (1 6 p < ∞) 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 | ∃λ > 0 such that λf ∈ BV(φ,α)[a, b] =  R f :[a, b] → R | ∃λ > 0 such that V(φ,α)(λf) < +∞ 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 | f ∈ 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 |·|(φ,α) : RBV(φ,α)[a, b] → R by the formula R  R |f|(φ,α) = inf ε > 0 | V(φ,α)(f/ε) 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→ |f(a)| + |f − f(a)|R = |f(a)| + inf ε > 0 VR 1 . (φ,α) (φ,α) ε 6 The following results were shown in [3] 0 Lemma 2.9. Let φ be a convex φ-function. Let f ∈ RBV(φ,α)[a, b]. Then

R R  R  i) |f|(φ,α) 6= 0 =⇒ V(φ,α) f/|f|(φ,α) 6 1;

R R
ii) |f|(φ,α) < k ⇐⇒ V(φ,α) f/k 6 1, k > 0; R R R iii) 0 6 |f|(φ,α) 6 1 =⇒ V(φ,α)(f) 6 |f|(φ,α);

Lemma 2.10. Let φ be a φ-function, then f ∈ RBV(φ,α)[a, b] if and only if 0 f − f(a) ∈ RBV(φ,α)[a, b].

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3. Embedding Results φ(x) Definition 3.1. Let φ be a convex φ-function. If limn→∞ x = +∞, then we say that φ satisfies the ∞1-condition.

Using the ∞1-condition we obtain several relations among some spaces.

Theorem 3.2. Let φ be a convex φ-function which satisfy the ∞1-condition. Let f ∈ 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 ∈ RBV(φ,α)[a, b]. Given ε > 0, let us consider (aj, bj), j = 1, 2, . . . , n a finite collection of disjoint subintervals contained in [a, b]. Let R mε m > 0 such that V(φ,α)(f) < 2 . Since φ satisfy the ∞1-condition, there exists x0 ∈ (0, ∞) such that φ(x) > mx for x > x0. Next, let us consider the following set ( ) f(b ) − f(a ) j j E = j ∈ {1, 2, . . . , n} > x0 . α(bj) − α(aj) If j ∈ E, then

f(bj) − f(aj) x0 6 . α(bj) − α(aj)

Since φ satisfy the ∞1-condition we have ! f(bj) − f(aj) f(bj) − f(aj) m 6 φ α(bj) − α(aj) α(bj) − α(aj) and thus ! 1 f(bj) − f(aj)
f(bj) − f(aj) 6 φ α(bj) − α(aj) . m α(bj) − α(aj) From this last inequality we obtain m X X X f(bj) − f(aj) = f(bj) − f(aj) + f(bj) − f(aj) j=1 j∈E j∈ /E ! 1 X f(bj) − f(aj)
X
6 φ α(bj) − α(aj) + x0 α(bj) − α(aj) m α(bj) − α(aj) j∈E j∈ /E n ! n 1 X f(bj) − f(aj) X φ α(b ) − α(a )
+ x α(b ) − α(a )
6m α(b ) − α(a ) j j 0 j j j=1 j j j=1 n 1 X < VR (f) + x α(b ) − α(a )
. m (φ,α) 0 j j j=1

Revista Colombiana de Matem´aticas 102 RENE´ ERLIN CASTILLO, HUMBERTO RAFEIRO & EDUARD TROUSSELOT Pn
Now, choose 0 < δ < ε/(2x0). Thus, if j=1 α(bj) − α(aj) < δ, then

n X ε f(bj) − f(aj) < + x0δ < ε. 2 j=1 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 (aj, bj) | j = Pn
Pn 1, 2, . . . , n} of [a, b] such that j=1 α(bj) − α(aj) < δ, then j=1 f(bj) − f(aj) < ε, which means that f ∈ α-AC[a, b]. X

Theorem 3.3. Let φ be a convex φ-function which satisfy the ∞1-condition. 0 0 Let f ∈ α-Lip [a, b], then f ∈ RBV(φ,α)[a, b] and there exists m > 0 such that R |f|(φ,α) 6 m|f|α-Lip[a,b] with

1 m >   −1 1 φ α(b)−α(a) and thus 0 0 α-Lip [a, b] ,→ RBV(φ,α)[a, b].

Proof. Let f ∈ α-Lip0[a, b], then

f(x) − f(y) |f| α(x) − α(y) 6 α-Lip[a,b] for all x, y ∈ [a, b] and x 6= y.  Let Π = a = x0 < x1 < ··· < xn = b be a partition of [a, b]. Since limx→0 φ(x) = 0, there exists m > 0 such that

 1  1 φ , m 6 α(b) − α(a)

−1 1  that is, m · φ α(b)−α(a) > 1. Now let us consider

n ! X f(xj) − f(xj−1) φ α(x ) − α(x )
=: I. m|f| α(x ) − α(x )
j j−1 j=1 α-Lip[a,b] j j−1

Since φ is an increasing function, we have

n X  1   1  I φ α(x ) − α(x )
φ α(b) − α(a)
1 6 m j j−1 6 m 6 j=1

Volumen 48, N´umero1, A˜no2014 EMBEDDINGS ON SPACES OF GENERALIZED BOUNDED VARIATION 103   for all partitions Π of [a, b]. Then VR f 1. By Lemma 2.9 ii) (φ,α) m|f|α-Lip[a,b] 6 we obtain

R |f|(φ,α) 6 m|f|α-Lip[a,b]. (1)

X

Corollary 3.4. Let φ be a convex φ-function which satisfy the ∞1-condition, then α-Lip[a, b] ,→ RBV(φ,α)[a, b] and there exists m > 0 such that

R kfk(φ,α) 6 max{1, m}kfkα-Lip[a,b]

−1 1  with m · φ α(b)−α(a) > 1.

Proof. Let f ∈ α-Lip[a, b]. Then f − f(a) ∈ α-Lip0[a, b] (see Proposition 2.3) 0 and by Lemma 3.3 we have that f − f(a) ∈ RBV(φ,α)[a, b]. Moreover, by (1) there exists m > 0 such that R f − f(a) (φ,α) 6 m f − f(a) α-Lip[a,b].

Observe that
f − f(a) α-Lip[a,b] = Lipα f − f(a)

f − f(a) (x) − f − f(a) (y) = sup 0 x6=y α(x) − α(y)

f(x) − f(y) = sup x6=y α(x) − α(y)

= Lipα(f). Then there exists m > 0 such that

R R f (φ,α) = f(a) + f − f(a) (φ,α) 6 f(a) + m f − f(a) α-Lip[a,b]

= f(a) + mLipα(f) 6 max{1, m} f α-Lip[a,b]. X

Theorem 3.5. Let φ be a convex φ-function which satisfy the ∞1-condition. 0 0 Let f ∈ RBV(φ,α)[a, b]. Then f ∈ BV [a, b] and !
1 R V f;[a, b] α(b) − α(a) + f . 6 φ(1) (φ,α)

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Therefore 0 0 RBV(φ,α)[a, b] ,→ BV [a, b]. where BV0[a, b] is the BV[a, b] space of vanishing functions at the point a.

R Proof. From Theorem 3.2 we know that f is of bounded variation. If f (φ,α) = 0 then f = 0 and thus V(f) = 0; then the inequality holds trivially. R  If f (φ,α) 6= 0, let Π = a = x0 < x1 < ··· < xn = b be a partition R of [a, b]. Let us consider in the proof of Theorem 3.2, f/ f (φ,α) in place of f. Then we obtain

n ! X f(xj) − f(xj−1) 1 f α(b) − α(a)
+ VR R 6 φ(1) (φ,α) |f|R j=1 f (φ,α) (φ,α) 1 α(b) − α(a)
+ 6 φ(1) where the last inequality follows from Lemma 2.9 i). Thus n   X 1 R f(xj) − f(xj−1) α(b) − α(a) + f . 6 φ(1) (φ,α) j=1

Since this last inequality holds for any partition Π of [a, b], then  
1 R V f;[a, b] α(b) − α(a) + f . X 6 φ(1) (φ,α) 

Corollary 3.6. Let φ be a convex φ-function which satisfy the ∞1-condition. Let f ∈ RBV(φ,α)[a, b], then f ∈ BV[a, b] and   1 R f max 1, α(b) − α(a) + f . BV[a,b] 6 φ(1) (φ,α)

Then RBV(φ,α)[a, b] ,→ BV[a, b].

Proof. Taking f ∈ RBV(φ,α)[a, b], then, from Lemma 2.10, we have that the 0 function f − f(a) ∈ RBV(φ,α)[a, b] and by Theorem 3.5 that f − f(a) ∈ 0 BV(φ,α)[a, b]. Moreover  
1 R V f − f(a); [a, b] α(b) − α(a) + f − f(a) . 6 φ(1) (φ,α)

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Since Vf − f(a); [a, b]
= Vf;[a, b]
, then

f BV[a,b] = f(a) + V f;[a, b]   1 R f(a) + α(b) − α(a) + f − f(a) 6 φ(1) (φ,α)   1 R max 1, α(b) − α(a) + f . X 6 φ(1) (φ,α) 

Theorem 3.7. Let φ1 and φ2 be two convex φ-functions such that there exist t0 > 0 and c > 0 such that φ2(t) 6 φ1(ct) for t > t0. Then R R f K f , f ∈ RBV0 [a, b] (φ2,α) 6 (φ1,α) (φ,α)

with K > c 1 + α(b) − α(a) φ2(t0) , that is, RBV0 [a, b] ,→ RBV0 [a, b]. (φ1,α) (φ2,α)

R Proof. Let f ∈ RBV0 [a, b]. If f = 0 then the result is trivial. Let (φ1,α) (φ1,α) R
 f (φ ,α) 6= 0. Let M > 1 + α(b) − α(a) φ2(t0) and Π = a = x0 < x1 < ··· < 1 xn = b be a partition of [a, b]. Let us define ( ) f(xj) − f(xj−1) E = j ∈ {1, . . . , n} t . R
> 0 c f α(xj) − α(xj−1) (φ1,α)

Consider   n X f(xj) − f(xj−1) φ α(x ) − α(x )
=: I. 2  R
 j j−1 j=1 Mc f α(xj) − α(xj−1) (φ1,α)

Since φ2 is convex, φ2(0) = 0 and 1/M < 1 we have   n 1 X f(xj) − f(xj−1) I φ α(x ) − α(x )
6 M 2  R
 j j−1 j=1 c f α(xj) − α(xj−1) (φ1,α)   1 X f(xj) − f(xj−1) φ α(x ) − α(x )
+ 6 M 2  R
 j j−1 c f α(xj) − α(xj−1) j∈E (φ1,α)   1 X f(xj) − f(xj−1) φ α(x ) − α(x )
M 2  R
 j j−1 c f α(xj) − α(xj−1) j∈ /E (φ1,α)

Revista Colombiana de Matem´aticas 106 RENE´ ERLIN CASTILLO, HUMBERTO RAFEIRO & EDUARD TROUSSELOT
R

if j ∈ E then f(xj) − f(xj−1) / c f α(xj) − α(xj−1) t0 and (φ1,α) >     f(xj) − f(xj−1) f(xj) − f(xj−1) φ φ . 2  R
 6 1  R
 c f α(xj) − α(xj−1) f α(xj) − α(xj−1) (φ1,α) (φ1,α)
R

If j∈ / E, then f(xj)−f(xj−1) / c f α(xj)−α(xj−1) < t0. Since (φ1,α) φ2 is increasing we obtain   f(xj) − f(xj−1) φ φ (t ) 2  R
 6 2 0 c f α(xj) − α(xj−1) (φ1,α) and thus    1 X f(xj) − f(xj−1) I φ α(x ) − α(x )
+ 6 M  1  R
 j j−1 f α(xj) − α(xj−1) j∈E (φ1,α)  X
φ2(t0) α(xj) − α(xj−1)  j∈ /E    n 1 X f(xj) − f(xj−1) φ α(x ) − α(x )
+ 6 M  1  R
 j j−1 j=1 f α(xj) − α(xj−1) (φ1,α) n  X
φ2(t0) α(xj) − α(xj−1)  j=1     1 f VR + φ (t )α(b) − α(a)
6  (φ1,α)  R  2 0  M f (φ1,α) 1 1 + φ (t )α(b) − α(a)
 6 M 2 0 < 1 where the last inequality comes from the definition of M and the penultimate   R f from Lemma 2.9 ii). Thus σ R 1 for all partitions Π of [a, b], (φ2,α) Mc|f| 6 (φ1,α)   R f and then we get V R 1 and, one more time, by Lemma 2.9 (φ2,α) Mc|f| 6 (φ1,α) ii) we have R R f Mc f . (φ2,α) 6 (φ1,α) Let K = Mc; hence R R f K f . X (φ2,α) 6 (φ1,α) 

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Corollary 3.8. Let φ1 and φ2 be two convex φ-functions which satisfy the ∞1-condition such that there exists t0 > 0 and c > 0 such that φ2(t) 6 φ1(ct) for t > t0. Then R R f max{1,K} f , (φ2,α) 6 (φ1,α)
f ∈ RBV(φ1,α)[a, b] with K > c 1 + (α(b) − α(a) φ2(t0)) that is

RBV(φ1,α)[a, b] ,→ RBV(φ2,α)[a, b].

Proof. Let f ∈ RBV [a, b], then f − f(a) ∈ RBV0 [a, b] and f − f(a) ∈ (φ1,α) (φ1,α) RBV0 [a, b] by Theorem 3.7 (φ2,α) R R f − f(a) K f − f(a) (φ2,α) 6 (φ1,α) and thus R R f = f(a) + f − f(a) (φ2,α) (φ2,α) R R f(a) + K f − f(a) max{1,K} f X 6 (φ1,α) 6 (φ1,α) 

Corollary 3.9. Let φ be a convex φ-function which satisfy the ∞1-condition. p If there exists t0 > 0 such that t 6 φ(t) for t > t0, p > 1, then

RBV(φ,α)[a, b] ,→ RBV(p,α)[a, b]. p Remark 3.10. In the particular case φ(t) = t (1 6 p < ∞) we have R R f (φ,α) = f (p,α). Indeed         R f R f inf ε > 0 V 1 = inf ε > 0 V 1 (φ,α) ε 6 (p,α) ε 6   1 R = inf ε > 0 V (f) 1 εp (p,α) 6  1  h R i p = inf ε > 0 V (f) ε (p,α) 6 1 h R i p = V(p,α)(f) .

1 n   o h i p Thus f(a) + inf ε > 0 VR f 1 = f(a) + VR (f) . Therefore (φ,α) ε 6 (p,α) R R f (φ,α) = f (p,α). Corollary 3.11. Let 1 6 q 6 p, then

RBV(p,α)[a, b] ,→ RBV(q,α)[a, b].

p q Proof. Indeed, let us take φ1(t) = t and φ2(t) = t , next we may apply Theorem 3.7. X

Revista Colombiana de Matem´aticas 108 RENE´ ERLIN CASTILLO, HUMBERTO RAFEIRO & EDUARD TROUSSELOT

Acknowledgement. H. Rafeiro was partially supported by Pontificia Univer- sidad Javeriana under the research project “Study of variable bounded varia- tion spaces”, Id Pry: 5863.

References [1] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publications, Clarendon Press, 2000. [2] R. E. Castillo, N. Merentes, and H. Rafeiro, Bounded Variation Spaces with p-Variable, Mediterr. J. Math., Springer Basel (2013), 1–11, doi: 10.1007/s00009-013-0342-5. [3] R. E. Castillo, H. Rafeiro, and E. Trousselot, A Generalization for the Riesz p-Variation, Submitted, 2014. [4] R. E. Castillo and E. Trousselot, On Functions of (p, α)-Bounded Variation, Real Analysis Exchange 34 (2009), no. 1, 49–60. [5] C. Jordan, Sur la s´eriede Fourier, C. R. Acad. Paris 2 (1881), 228–230 (fr). [6] F. Riesz, Untersuchungen ¨uber Systeme Integrierbarer Funktionen, Mathe- matische Annalen 69 (1910), no. 4, 449–497 (de), 10.1007/BF01457637. [7] A. I. Vol’pert and S. I. Hudjaev, Analysis in Classes of Discontinuous Func- tions and Equations of Mathematical Physics, Mechanics: Analysis, vol. 8, Martinus Nijhoff Publishers, Dordrecht, Netherlands, 1985.

(Recibido en octubre de 2013. Aceptado en febrero de 2014)

Departamento de Matematicas´ Universidad Nacional de Colombia Facultad de Ciencias Carrera 30, calle 45 Bogota,´ Colombia e-mail: [email protected]

Departamento de Matematicas´ Pontificia Universidad Javeriana Facultad de Ciencias Cra 7a No 43-82 Bogota,´ Colombia e-mail: [email protected]

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