–Medvedev bounded variation spaces, i.e., we shall present necessary and sufficient conditions for the Young functions φ

vol. 56, no. 2 (2016), 225–237

Characterization of inclusion among Riesz–Medvedev variation spaces

Wadie Aziz and Tomás Ereú

Summary. We present a characterization of inclusion among Riesz–

–Medvedev bounded variation spaces, i.e., we shall present necessary and sufficient conditions for the Young functions φ

and φ

so that RV

[a, b] ⊂ RV

[a, b] or RV

[a, b] ⊂ RV

[a, b].

Keywords Banach spaces;

φ-bounded variation

MSC 2010 93B05; 93C25 Received: 2016-03-10, Accepted: 2016-10-21

1. Introduction

J. Musielak and W. Orlicz [8] established necessary and sufficient conditions for Young functions φ ₁ and φ ₂ (cf. [2]) so that BV φ

[a, b] ⊂ BV ^φ

[a, b] where BV ^φ [a, b] is the class of all functions x ∶ [a, b] → R of bounded Riesz–Medvedev variation in the sense of Wiener.

More precisely, BV φ

[a, b] ⊂ BV ^φ

[a, b] if and only if there exist positive constants C and T such that φ ₂ (t) ⩽ Cφ 1 (t) for all 0 < t ⩽ T.

According to W. Orlicz’s classical result [3], it is known that BV φ

[a, b] ⊂ BV ^φ

[a, b]

if and only if ℓ φ

⊂ ℓ ^φ

where ℓ φ is the class of all sequences {a ⁿ } ⁿ ⩾1 such that

∞

∑

n =1

φ (∣a ⁿ ∣) < ∞.

H. Herda [4] found necessary and sufficient conditions for Young functions φ 1 and φ ₂ so that BV _φ ^∗

1

[a, b] ⊂ BV φ ^∗

[a, b], where BV φ ^∗ [a, b] is the space generated by the class

Wadie Aziz, Universidad de Los Andes, Departamento de Física y Matemáticas, Trujillo-Venezuela (e-mail: wadie@ula.ve)

Tomás Ereú, Universidad Nacional Abierta, Centro Local Lara, Lara-Venezuela (e-mail: tomasereu@gmail.com)

DOI 10.14708/cm.v56i2.1133 © 2016 Polish Mathematical Society

BV φ [a, b]. More precisely, BV ^{φ ∗}

[a, b] ⊂ BV ^{φ ∗}

[a, b] if and only if there exist positive con- stants C and T such that φ ₂ (t) ⩽ φ 1 (Ct) for all 0 < t ⩽ T. By a classical result due to Orlicz [3], it follows that ℓ ^∗ _φ

1

⊂ ℓ ^∗ φ

if and only if BV _φ ^∗

1

[a, b] ⊂ BV φ ^∗

[a, b], where ℓ ^∗ φ is the space generated by the class ℓ φ .

In the present work, we shall establish a similar characterization for the class RV φ [a, b]

of all functions x ∶ [a, b] → R of bounded φ-variation in the sense of Riesz. Moreover, we shall show that RV _φ ^∗

1

[a, b] ⊂ RV φ ^∗

[a, b] if and only if there exists positive constant C and T such that φ 2 (t) ⩽ φ ¹ (Ct) for all t ⩾ T. From W. Orlicz’s classical result [ 5, p. 63] we conc- lude that BV _φ ^∗

1

[a, b] ⊂ BV φ ^∗

[a, b] if and only if L ^φ

[a, b] ⊂ L ^φ

[a, b], where L ^φ [a, b] is the so-called Orlicz class. Concerning the space RV _φ ^∗ [a, b] generated by the class RV ^φ [a, b], we establish the fact that RV _φ ^∗

1

[a, b] ⊂ RV φ ^∗

[a, b] if and only if there exist positive con- stants C and T such that φ 2 (t) ⩽ φ ¹ (Ct) for all t ⩾ T. As a corollary one obtains that RV _φ ^∗

1

[a, b] ⊂ RV ^{φ ∗}

[a, b] if and only if L ^{∗ φ}

[a, b] ⊂ L ^{∗ φ}

[a, b], where L ^{∗ φ} [a, b] is the Orlicz space generated by the class L φ [a, b].

2. Preliminary

In this section we introduce some definitions and notation. A function φ ∶ [0, ∞) → [0, ∞) is called a Young function if it satisfies the following conditions: φ (t) = 0 if and only if t = 0, φ is continuous and non-decreasing on [0, ∞), and φ(t) → ∞ as t → ∞. In our considerations the condition ( ∞ ¹ ) is given by:

lim sup

t →∞

φ (t) t = ∞.

The Young function φ satisfies the condition ∆ ₂ for large t when for some constants k > 0, t 0 > 0 we have φ(2t) ⩽ kφ(t) for t ⩾ t 0 . If this inequality holds for t ⩾ 0 we say the condition ∆ ₂ is satisfied for all t. Throughout this paper F will denote the vector space of real-valued functions x defined in a finite interval [a, b]. Let x ∈ F. For a Young function φ, we can define the Riesz–Medvedev variation as the number

v _φ ^R (x; [a, b]) ∶= sup

π n

∑

i =1

φ ( ∣x(t ⁱ ) − x(t ⁱ −1 )∣

∣t ⁱ − t ⁱ −1 ∣ )∣t ⁱ − t ⁱ −1 ∣,

where supremun is taken over all partitions π ∶ a = t ⁰ < ⋯ < t ⁿ = b of the interval [a, b].

In the literature well known is also the so-called Wiener–Young variation (cf. [2]) v φ (x; [a, b]) ∶= sup

π n

∑

i =1

φ (∣x(t ⁱ ) − x(t ⁱ −1 )∣),

where supremun is taken over all partitions π of [a, b].

The function x ∈ F is said to have bounded (or finite) Riesz–Medvedev variation if v ^R _φ (x; [a, b]) < ∞. By RV ^φ [a, b] we denote the class of all functions x ∈ F such that v ^R _φ (x; [a, b]) < ∞. By BV ^φ [a, b] we denote the class of all functions x ∈ F such that v φ (x; [a, b]) < ∞. The space RV φ ^∗ [a, b] of all functions x ∈ F such that v ^R φ (λx; [a, b]) <

∞ for some λ > 0 is the space generated by the class RV ^φ [a, b]. Similarly, the space BV _φ ^∗ [a, b] of all functions x ∈ F such that v ^φ (λx; [a, b]) < ∞ for some λ > 0 is the space generated by the class BV φ [a, b].

The spaces RV _φ ^∗ [a, b] and BV φ ^∗ [a, b] appeared for the first time in papers [ 1] and [8], respectively.

When φ (t) = t ^p for some p ⩾ 1, then we have the classical space RV ^p [a, b] of func- tions of bounded Riesz p-variation.

Note that the assumption ( ∞ ¹ ) in the case of a convex function φ is just

t lim →∞

φ (t) t = ∞.

Moreover, as was observed in [6], if φ is a convex Young function and condition ( ∞ ¹ ) is not satisfied, i.e.

t lim →∞

φ (t)

t = r < ∞,

then RV _φ ^∗ [a, b] = BV[a, b] where BV[a, b] denotes the usual space of functions of boun- ded variation.

If φ is a convex Young function, then the space RV _φ ^∗ [a, b] is a Banach space with the norm

∥x∥ ^R φ ∶= ∣x(a)∣ + ∥x∥ ^{0 φ} , where

∥x∥ ⁰ φ ∶= inf{ε > 0 ∶ v φ ^R ( x ε

; [a, b]) ⩽ 1}.

Similarly, we can obtain a Banach space BV _φ ^∗ [a, b] with the following norm:

∥x∥ ^φ ∶= ∣x(a)∣ + ∥x∥ ^{0 φ} , where

∥x∥ ^{0 φ} ∶= inf{ε > 0 ∶ v ^φ ( x ε

; [a, b]) ⩽ 1}.

For a convex Young function φ which satisfies ( ∞ 1 ), some useful properties of Riesz–

–Medvedev variation are stated in the following lemma.

2.1. Lemma. Let φ be a convex Young function.

(i) (Musielak–Orlicz [8]) If x ∈ RV φ ^∗ [a, b] and ∥x∥ ⁰ φ > 0, then v _φ ^R ( x

∥x∥ ⁰ φ

; [a, b]) ⩽ 1.

(ii) (Maligranda–Orlicz [6]) If x ∈ RV ^{φ ∗} [a, b], then x is bounded on [a, b] and sup

t ∈[a,b] ∣x(t)∣ ⩽ C ^φ (h)∥x∥ ⁰ φ , where

C φ (h) ∶= max{min{ 1 φ (1) ,

1

hφ (1/h) }, h

φ ⁻¹ (1/h) }, h = b − a.

Moreover, if additionally φ satisfies the condition

t lim →∞

φ (t) t = ∞, then

∥x∥ ^φ = sup

t ∈[a,b] ∣x(t)∣ + ∥x∥ ^{R φ} or ∥x∥ ^φ = 2C ^φ (h)∥x∥ ^R φ

is a normed Banach algebra.

(iii) (Medvedev [7]) v ^R _φ (x; [a, b]) < ∞ if and only if x is absolutely continuous on [a, b]

and

∫

b a

φ (∣x ^′ (t)∣)dt < ∞.

In this case we also have the equality

v ^R _φ (x; [a, b]) = ∫ _a ^b φ (∣x ^′ (t)∣)dt.

The purpose of this paper is to solve the inclusion problem for the class RV φ [a, b]

and the space RV _φ ^∗ [a, b]. Before presenting our main results in Theorems 3.1–3.5 below, we briefly review what is known in the literature of the class BV φ [a, b] and the space BV _φ ^∗ [a, b].

(i) Musielak–Orlicz [8] proved that BV φ

[a, b] ⊂ BV ^φ

[a, b] if and only if there exist positive constants C and T such that φ 2 (t) ⩽ Cφ ¹ (t) for all 0 < t ⩽ T.

(ii) Musielak–Orlicz [8] proved that BV φ [a, b] is a vector space if and only if the function φ satisfies the condition ∆ 2 for small t, i.e. if there exist positive constants k and t 0

such that φ (2t) ⩽ kφ(t) for all 0 < t ⩽ t ⁰ . (iii) Herda [4] proved that BV _φ ^∗

1

[a, b] ⊂ BV φ ^∗

[a, b] if and only if there exist positive con- stants C and T such that φ 2 (t) ⩽ φ ¹ (Ct) for all 0 < t ⩽ T.

(iv) According to W. Orlicz’s classical results [3], it is known that BV φ

[a, b] ⊂ BV ^φ

[a, b]

if and only if ℓ φ

[a, b] ⊂ ℓ ^φ

[a, b], where ℓ ^φ [a, b] is the class of all sequences {a ⁿ } ⁿ ⩾1

such that

∞

∑ n =1

φ (∣a ⁿ ∣) < ∞.

(v) BV _φ ^∗

1

[a, b] ⊂ BV ^{φ ∗}

[a, b] if and only if ℓ ^{∗ φ}

[a, b] ⊂ ℓ ^{∗ φ}

[a, b],where ℓ ^{∗ φ} [a, b] is the space generated by the class ℓ φ [a, b].

3. Main results

In this section we shall present necessary and sufficient conditions for the Young functions φ ₁ and φ ₂ so that RV φ

[a, b] ⊂ RV ^φ

[a, b] or RV φ ^∗

[a, b] ⊂ RV φ ^∗

[a, b].

3.1. Theorem. Let φ 1 , φ 2 be convex Young functions which satisfy the condition ( ∞ ¹ ). Then RV φ

[a, b] ⊂ RV ^φ

[a, b] if and only if there exist positive constants C and T such that

φ ₂ (t) ⩽ Cφ ¹ (t) for all t ⩾ T. (1)

Proof. Suppose that there exist positive constants C and T such that inequality (1) holds.

Let π ∶ a = t ⁰ < ⋯ < t ⁿ = b be a partition of [a, b]. Let x ∈ RV ^φ

[a, b] and define the set e T by

e T ∶= {i = 0, 1, . . . , n ∶ ∣x(t ⁱ ) − x(t ⁱ −1 )∣

∣t ⁱ − t ⁱ −1 ∣ ⩾ T}.

Then the following estimate can be obtained

n

∑

i =1

φ 2 ( ∣x(t ⁱ ) − x(t ⁱ −1 )∣

∣t ⁱ − t ⁱ −1 ∣ )∣t ⁱ − t ⁱ −1 ∣ ⩽ C

n

∑

i =1

φ 1 ( ∣x(t ⁱ ) − x(t ⁱ −1 )∣

∣t ⁱ − t ⁱ −1 ∣ ) + φ ² (T)

n

∑

i =1 (t ⁱ − t ⁱ −1 )

⩽ Cv φ ^R

(x; [a, b]) + φ ² (T)(b − a).

Thus

v ^R _φ

2

(x; [a, b]) ⩽ Cv ^{φ R}

(x; [a, b]) + φ ² (T)(b − a), consequently RV φ

[a, b] ⊂ RV ^φ

[a, b].

Suppose now that RV φ

[a, b] ⊂ RV ^φ

[a, b] and that φ ¹ and φ 2 does not satisfy the inequality (1), that is, there exists a sequence {t ⁿ } ⁿ ⩾1 such that t n ↑ ∞ as n → ∞ and

φ ₂ (t ⁿ ) > 2 ⁿ φ ₁ (t ⁿ ), n ∈ N. (2) Without loss of generality, we can assume that [a, b] = [0, 1]. We shall prove that there exists x ∈ RV ^φ

[0, 1] such that x /∈ RV ^φ

[0, 1]. By considering subsequences, if necessary, we may define the sequence {a ⁿ } ⁿ ⩾1 on [0, 1] in the following way:

a n = 1

2 ⁿ φ 1 (t ⁿ +1 ) , n ∈ N.

Then a n ↓ 0 as n → ∞, and the series

∞

∑ n =1

t n +1 − t ⁿ

2 ⁿ φ ₁ (t ⁿ +1 ) (3)

is convergent. Indeed, since the function φ ₁ is convex and the function t ↦ t/φ ¹ (t) is non-increasing, we have

∣t ⁿ +1 − t ⁿ ∣ 2 ⁿ φ ₁ (t ⁿ +1 ) ⩽ 1

2 ⁿ ( t n +1

φ ₁ (t ⁿ +1 ) + t n

φ ₁ (t ⁿ +1 ) ) ⩽ 1 2 ⁿ

t n +1

φ ₁ (t ⁿ +1 ) ⩽ 1 2 ⁿ

t 2

φ ₁ (t ² ) , for all n ∈ N. Consequently the series ( 3) is convergent.

Consider the set sequence {I ⁿ } ⁿ ⩾1 given by

I 1 = [a ¹ , 1 ] and I ⁿ = [a ⁿ , a n −1 ), n = 2, 3, . . . Then I m ∩ I ⁿ = ∅ (n, m ∈ N, n ≠ m) and

∞

⋃ n =1

I n = (0, 1].

Define the function x ∶ [0, 1] → R in the following way:

x (τ) = ⎧⎪⎪⎪ ⎪⎨

⎪⎪⎪⎪ ⎩

t n τ − ∑ ^∞

i =n

t i +1 − t ⁱ

2 ⁱ φ 1 (t ⁱ +1 ) , τ ∈ I ⁿ ,

0, τ = 0.

We claim that x ∈ RV ^φ

[0, 1], but x /∈ RV ^φ

[0, 1]. Indeed, the function x is continuous and differentiable in the interior of I n , n ⩾ 1, and since

τ lim ↑a

x (τ) = lim _τ

↑a

(t ⁿ +1 τ − ∑ ^∞

i =n+1

t i +1 − t ⁱ 2 ⁱ φ ₁ (t ⁱ +1 ) )

= t n +1

2 ⁿ φ 1 (t ⁿ +1 ) − ∑ ^∞

i =n+1

t i +1 − t ⁱ 2 ⁱ φ 1 (t ⁱ +1 )

= t n

2 ⁿ φ ₁ (t ⁿ +1 ) − ∑ ^∞

i =n

t i − t ⁱ −1

2 ^{i −1} φ ₁ (t ⁱ )

= lim _{τ ↓a}

n

x (τ),

the function x is continuous on (0, 1]. Next we shall prove that the function x is continuous at τ = 0.

Let {s ⁿ } ⁿ ⩾1 be a sequence in (0, 1] such that s ⁿ ↓ 0 as n → ∞. Since a ⁿ ↓ 0 as n → ∞, for all n ∈ N there exists a positive number m ⁿ such that s n ∈ I ^m

, and consequently

0 ⩽ lim _{n →∞} x (s ⁿ ) = lim _{n →∞} (t ^m

a m

−1 − ∑ ^∞

i =m

t i +1 − t ⁱ 2 ⁱ φ 1 (t ⁱ +1 ) )

= lim _{n →∞} t m

2 ^m

φ 1 (t ^m

) − ∑ ^∞

i =m

t i +1 − t ⁱ

2 ⁱ φ 1 (t ⁱ +1 )

= lim _{n →∞} t m

2 ^m

φ ₁ (t ^m

) − lim _{n →∞} ∑ ^∞

i =m

t i +1 − t ⁱ 2 ⁱ φ ₁ (t ⁱ +1 ) = 0.

Thus x (s ⁿ ) → 0 as n → ∞ for every sequence {s ⁿ } ⁿ ⩾1 in (0, 1] such that s ⁿ ↓ 0. Consequ- ently x is continuous at τ = 0.

Since x is continuous on [0, 1] and differentiable in the interior of I ⁿ , n ⩾ 1, x is absolutely continuous on [0, 1].

Next we shall prove that x ∈ RV ^φ

[0, 1]. By the definition of the function x and Lem- ma 2.1, we get the following estimate:

v ^R _φ

1

(x; [0, 1]) = ∑ ^∞

n =1

φ 1 (t ⁿ )(a ⁿ +1 − a ⁿ )

= ∑ ^∞

n =1

φ ₁ (t ⁿ )( 1

2 ^{n −1} φ 1 (t ⁿ ) − 1 2 ⁿ φ 1 (t ⁿ +1 ) )

⩽ ∑ ^∞

n =1

1 2 ^{n −1} + ∑ ^∞

n =1

φ ₁ (t ⁿ ) 2 ⁿ φ 1 (t ⁿ +1 ) ⩽ 3.

Consequently x ∈ RV ^φ

[0, 1].

As a result of the definition of the function x , inequality (2) and Lemma 2.1, we get the following estimate:

v ^R _φ

2

(x; [0, 1]) = ∑ ^∞

n =1

φ 2 (t ⁿ )(a ⁿ +1 − a ⁿ )

= ∑ ^∞

n =1

φ ₂ (t ⁿ ) 2 ^{n −1} ( 1

φ 1 (t ⁿ ) − 1 2φ 1 (t ⁿ +1 ) )

⩾ ∑ ^∞

n =1

2φ ₁ (t ⁿ )( 1

φ ₁ (t ⁿ ) − 1 2φ ₁ (t ⁿ +1 ) )

⩾ ∑ ^∞

n =1 (2 − φ ₁ (t ⁿ )

φ 1 (t ⁿ +1 ) ) = ∞.

Thus x ∈ RV ^φ

[0, 1] and x /∈ RV ^φ

[0, 1], which is a contradiction.

From W. Orlicz’s classical result [5, p. 63], we obtain the following corollary.

3.2. Corollary. Let φ ₁ , φ ₂ be convex Young functions which satisfy the condition ( ∞ 1 ). Then RV φ

[a, b] ⊂ RV ^φ

[a, b] if and only if L ^φ

[a, b] ⊂ L ^φ

[a, b] where L ^φ [a, b] is the so-called Orlicz class.

In the following theorem we shall establish a necessary and sufficient condition for Young functions φ ₁ and φ ₂ so that

RV _φ ^∗

1

[a, b] ⊂ RV ^{φ ∗}

[a, b].

3.3. Theorem. Let φ ₁ , φ ₂ be convex Young functions which satisfy the condition ( ∞ ¹ ). Then RV _φ ^∗

1

[a, b] ⊂ RV φ ^∗

[a, b] if and only if there exist positive constants C and T such that

φ ₂ (t) ⩽ φ 1 (Ct), t ⩾ T. (4)

Proof. Suppose that there exist positive constants C and T such that inequality (4) holds.

Let x ∈ RV φ ^∗

[a, b]. Then by Lemma 2.1 and the definition of the space RV _φ ^∗

1

[a, b] we have that the function x is absolutely continuous on [a, b] and there exists λ > 0 such that

v ^R _φ

1

(λx; [a, b]) = ∫ _a ^b φ ₁ (λ∣x ^′ (t)∣)dt < ∞.

Define the set e T , λ by

e

∶= {t ∈ [a, b] ∶ λ∣x ^′ (t)∣ > CT}.

The following estimate can be obtained

∫

b a

φ ₂ ( λ

C ∣x ^′ (t)∣)dt = ∫ _e

T , λ

φ ₂ ( λ

C ∣x ^′ (t)∣)dt + ∫ _[a,b]/e

T , λ

φ ₂ ( λ

C ∣x ^′ (t)∣)dt

⩽ ∫ _e

T , λ

φ ₁ (λ∣x ^′ (t)∣)dt + φ ² (T)(b − a) < ∞, consequently x ∈ RV φ ^∗

[a, b].

Suppose now that RV _φ ^∗

1

[a, b] ⊂ RV φ ^∗

[a, b] and that φ ¹ and φ 2 do not satisfy inequali- ty (4), hence there exists a sequence {t ⁿ } ⁿ ⩾1 such that t n ↑ ∞ as n → ∞, and

φ ₂ (t ⁿ ) > 2 ⁿ φ ₁ (t ⁿ ) n ∈ N. (5) Without loss of generality we can assume that [a, b] = [0, 1]. We shall prove that there exists x ∈ RV ^{φ ∗}

[a, b] such that x /∈ RV ^{φ ∗}

[a, b]. Define a sequence {a ⁿ } ⁿ ⩾1 in [0, 1] by

a n = 1

2 ⁿ φ 1 ((n + 1) ² t n +1 ) n ∈ N.

Then a n ↓ 0 as n → ∞, and the series

∞

∑

n =1

a n ((n + 1)t ⁿ +1 − nt ⁿ )

is convergent. Indeed, since φ ₁ is a convex function and the function t ↦ t/φ 1 (t) is non- -increasing, the following estimate can be obtained:

a n ((n + 1)t ⁿ +1 − nt ⁿ ) ⩽ 2(n + 1)t ⁿ +1 a n

= (n + 1)t ⁿ +1

2 ^{n −1} φ 1 ((n + 1) ² t n +1 )

⩽ (n + 1)t ⁿ +1

2 ^{n −1} (n + 1) ² φ 1 (t ⁿ +1 )

⩽ t n +1

2 ^{n −1} φ 1 (t ⁿ +1 ) ⩽ 1 2 ^{n −1}

t ₂ φ 1 (t ² ) . Thus the series

∞

∑ n =1

a n ((n + 1)t ⁿ +1 − nt ⁿ ) is convergent.

Consider the set sequence I n , n ⩾ 1, given by

I 1 = [a ¹ , 1 ] and I ⁿ = [a ⁿ , a n −1 ), n = 2, 3, . . . Then I m ∩ I ⁿ = ∅ (n, m ∈ N, n ≠ m) and

∞

⋃ n =1

I n = (0, 1].

Define the function x ∶ [0, 1] → R in the following way:

x (τ) = ⎧⎪⎪⎪ ⎪⎨

⎪⎪⎪⎪ ⎩

nt n τ − ∑ ^∞

i =n

a i ((i + 1)t ⁱ +1 − it ⁱ ), τ ∈ I ⁿ

0, τ = 0.

We claim that x ∈ RV φ ^∗

[0, 1] but x /∈ RV φ ^∗

[0, 1]. Indeed, the function x is continuous and differentiable in the interior of I n , n ⩾ 1, and since

τ lim ↑a

x (τ) = lim _τ

↑a

((n + 1)t ⁿ +1 τ − ∑ ^∞

i =n+1

a i ((i + 1)t ⁱ +1 − it ⁱ ))

= nt ⁿ +1 a n − ∑ ^∞

i =n

a i −1 (it ⁱ − (i − 1)t ⁱ −1 )

= lim _τ

↓a

x (τ),

the function x is continuous on (0, 1]. Next we shall prove that the function x is continuous at τ = 0.

Let {s ⁿ } ⁿ ⩾1 be a sequence in (0, 1] such that s ⁿ ↓ 0 as n → ∞. Since a ⁿ ↓ 0 as n → ∞, for all n ∈ N there exists a positive number m ⁿ such that s n ∈ I ^m

and consequently

0 ⩽ lim _{n →∞} x (s ⁿ ) ⩽ lim _{n →∞} (m ⁿ t m

a m

−1 − ∑ ^∞

i =m

a i ((i + 1)t ⁱ +1 − it ⁱ ))

= lim _{n →∞} m n t m

2 ^m

⁻¹ φ ₁ (m ⁿ t m

−1 ) = 0.

Thus x (s ⁿ ) → 0 as n → ∞ for all sequences {s ⁿ } ⁿ ⩾1 in (0, 1] such that s ⁿ ↓ 0. Consequently, x is continuous at τ = 0.

Since x is continuous on [0, 1] and differentiable in the interior of I ⁿ , n ⩾ 1, x is absolutely continuous on [0, 1].

Next we shall prove that x ∈ RV φ ^∗

[0, 1]. By the definition of the function x and Lem- ma 2.1, we get the following estimate:

v _φ ^R

1

(x; [0, 1]) = ∑ ^∞

n =1

φ ₁ (nt ⁿ )(a ⁿ −1 − a ⁿ )

= ∑ ^∞

n =1

φ 1 (nt ⁿ )( 1

2 ^{n −1} φ ₁ (n ² t n ) − 1

2 ⁿ φ ₁ ((n + 1) ² t n +1 ) )

⩽ ∑ ^∞

n =1

1 2 ^{n −1} + ∑ ^∞

n =1

1 2 ⁿ = 3.

Consequently x ∈ RV φ ^∗

[0, 1]. By the definition of the function x, inequality ( 5) and Lem- ma 2.1, for all λ > 0 we get the following estimate:

v _φ ^R

2

(λx; [0, 1]) = ∑ ^∞

n =1

φ ₂ (λnt ⁿ )(a ⁿ −1 − a ⁿ ) ⩾ ∑ ^∞

n ⩾

φ ₂ (t ⁿ )(a ⁿ −1 − a ⁿ )

= ∑ ^∞

n ⩾

φ ₂ (t ⁿ ) 2 ^{n −1} ( 1

φ ₁ (n ² t n ) − 1

2φ ₁ ((n + 1) ² t n +1 ) )

⩾ ∑ ^∞

n ⩾

φ 1 (n ² 2 ⁿ t n ) 2 ^{n −1} ( 1

φ 1 (n ² t n ) − 1

2φ 1 ((n + 1) ² t n +1 ) )

⩾ ∑ ^∞

n ⩾

2 ⁿ φ ₁ (n ² t n ) 2 ^{n −1} ( 1

φ ₁ (n ² t n ) − 1

2φ ₁ ((n + 1) ² t n +1 ) )

⩾ ∑ ^∞

n ⩾

(2 − φ 1 (n ² t n )

φ 1 ((n + 1) ² t n +1 ) ) = ∞.

Thus x ∈ RV φ ^∗

[0, 1] and x /∈ RV φ ^∗

[0, 1], which is a contradiction.

From W. Orlicz’s classical result [5, p. 110], we obtain the following corollary.

3.4. Corollary. Let φ 1 , φ 2 be convex Young functions which satisfy condition (∞ ¹ ). Then RV _φ ^∗

1

[a, b] ⊂ RV φ ^∗

[a, b] if and only if L ^∗ φ

[a, b] ⊂ L ^∗ φ

[a, b] where L ^∗ φ [a, b] is the Orlicz space generated by the class L φ [a, b].

Now we present equivalent conditions for inequality (4).

3.5. Theorem. Let φ ₁ , φ 2 be convex Young functions which satisfy the condition (∞ ¹ ). Then

the following conditions are equivalent:

(i) there exist positive constants C and T such that

φ ₁ (t) ⩽ φ 2 (Ct), t ⩾ T, (ii) there exist positive constants C and T such that

φ 1 (Ct) ⩽ φ ² (t), t ⩾ T, (iii) there exist positive constants C 1 , C 2 and T such that

φ 1 (t) ⩽ C ¹ φ 2 (C ² t ), t ⩾ T, (iv) there exist positive constants n ∈ N and T such that

φ ₁ (t) ⩽ φ ² (nt), t ⩾ T, (v) there exist positive constants n, m ∈ N and T such that

φ ₁ (t) ⩽ mφ 2 (nt), t ⩾ T, (vi) there exist positive constants C and r such that

lim sup

t →∞

φ ₁ (t)

φ ₂ (Ct) = r < ∞, (vii) there exist positive constants C and r such that

lim inf

t →∞

φ 2 (Ct)

φ ₁ (t) = r > 0, (viii) there exist a constant n ∈ N such that

nRV _φ ^∗

2

[a, b] ⊂ RV φ ^∗

[a, b].

From this theorem we obtain the following corollary.

3.6. Corollary.

(i) If there exists a positive constant C such that

t lim →∞

φ 2 (Ct)

φ ₁ (t) > 0, then RV φ ^∗

[a, b] ⊂ RV φ ^∗

[a, b], (ii) If there exists a positive constant C such that

0 < lim _{t →∞} φ ₂ (Ct)

φ 1 (t) < ∞, then RV φ ^∗

[a, b] = RV φ ^∗

[a, b],

(iii) RV φ

[a, b] = RV ^φ

[a, b] if and only if there exists a positive constant C such that

0 < lim _{t →∞} φ ₁ (t)

t ^p < ∞ if and only if RV φ ^∗

[a, b] = RV ^p [a, b].

Now we present a necessary and sufficient condition on the Young function φ so that the class RV φ [a, b] is a vector space.

3.7. Theorem. Let φ be a convex Young function which satisfies the condition (∞ ¹ ). The class RV φ [a, b] is a vector space if and only if the function φ satisfies the condition ∆ ² for large t, i.e. if there exist positive constants k and t 0 such that

φ (2t) ⩽ kφ(t) for all t ⩾ t 0 . Proof. Similar to the proof of Theorem 1.12, p. 14 in [8].

3.8. Remark.

(i) Let φ ₁ , φ ₂ be convex Young functions which satisfy the condition (∞ 1 ). If the inequ- ality (1) holds for all t ⩾ 0, then

– Lip [a, b] ⊂ RV ^φ

[a, b] ⊂ RV ^φ

[a, b],

– BV φ

[a, b] ⊂ BV ^φ

[a, b]. Moreover, if additionally φ ¹ satisfies the following con- dition: there exists a positive constant C ₁ such that φ ₁ (t) ⩽ C ¹ t for all t ⩾ 0, then

– Lip [a, b] ⊂ RV ^φ

[a, b] ⊂ RV ^φ

[a, b] ⊂ BV[a, b] ⊂ BV ^φ

[a, b] ⊂ BV ^φ

[a, b].

(ii) Let φ 1 , φ 2 be convex Young functions which satisfy the condition (∞ ¹ ). If the inequ- ality (4) holds for all t ⩾ 0, then

– Lip [a, b] ⊂ RV φ ^∗

[a, b] ⊂ RV φ ^∗

[a, b], – BV _φ ^∗

1

[a, b] ⊂ BV φ ^∗

[a, b]. Moreover, if additionally φ ¹ satisfies the following con- dition: there exists a positive constant C 1 such that φ 1 (t) ⩽ C ¹ t for all t ⩾ 0, then

– Lip [a, b] ⊂ RV φ ^∗

[a, b] ⊂ RV φ ^∗

[a, b] ⊂ BV[a, b] ⊂ BV φ ^∗

[a, b] ⊂ BV φ ^∗

[a, b].

(iii) If the φ-function φ satisfies the condition ∆ ₂ for all t ⩾ 0, then the classes RV ^φ [a, b]

and BV φ [a, b] are vector spaces.

Acknowledgements. The authors would like to thank the anonymous referees and the

editors for their valuable comments and suggestions.

References

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