On Functions of Bounded (p,k)-Variation

Abstract

We introduce and study the concept of (𝑝,𝑘)-variation (1<𝑝<∞, 𝑘∈ℕ) of a real function on a compact interval. In particular, we prove that a function 𝑢∶[𝑎,𝑏]→ℝ has bounded (𝑝,𝑘)-variation if and only if 𝑢^(𝑘−1) is absolutely continuous on [𝑎,𝑏] and 𝑢^(𝑘) belongs to 𝐿_𝑝[𝑎,𝑏]. Moreover, an explicit connection between the (𝑝,𝑘)-variation of 𝑢 and the 𝐿_𝑝-norm of 𝑢^(𝑘) is given which is parallel to the classical Riesz formula characterizing functions in the spaces 𝑅𝑉_𝑝[𝑎,𝑏] and 𝐴_𝑝[𝑎,𝑏]. This may also be considered as an alternative characterization of the one variable Sobolev space 𝑊^𝑘𝑝[𝑎,𝑏].

1. Introduction

About 120 years ago, Jordan [1] introduced the notion of a function of bounded variation and the corresponding function space BV[𝑎,𝑏]. He also proved the important result that 𝑢∈BV[𝑎,𝑏] if and only if 𝑢=𝑓−𝑔 with both 𝑓 and 𝑔 being monotonically increasing. Later the concept of bounded variation was generalized in various directions. In 1908, De la Vallée Poussin [2] introduced the space BV²[𝑎,𝑏] of functions with bounded second variation. It is known that 𝑢∈BV²[𝑎,𝑏] if and only if 𝑢 can be represented in the form 𝑢=𝑓−𝑔, where both 𝑓 and 𝑔 are convex.

This was further generalized by Popoviciu [3] who introduced, for any 𝑘∈ℕ, the notion of 𝑘th variation and defined the corresponding space BV^𝑘[𝑎,𝑏] of functions 𝑢∶[𝑎,𝑏]→ℝ of bounded 𝑘th variation on [𝑎,𝑏]. It is known that, for any 𝑢∈BV^𝑘[𝑎,𝑏], the derivative 𝑢^(𝑘−2) belongs to BV²[𝑎,𝑏]; therefore, there exist the right- and left-hand derivatives 𝑢^+(𝑘−1) and 𝑢^{−(𝑘−1)} on [𝑎,𝑏]. Moreover, the set 𝐸 of points 𝑥 where 𝑢^(𝑘−1)(𝑥) does not exist is at most countable, the derivative 𝑢^(𝑘−1) is continuous on [𝑎,𝑏]⧵𝐸, and the unilateral right derivative 𝑢^+(𝑘−1) and the unilateral left derivative 𝑢^{−(𝑘−1)} are right continuous and left continuous, respectively.

In [4], the first author defined and studied the notion of the so-called bounded (𝑝,2)–variation (1<𝑝<∞) and proved a generalization of the well known Riesz lemma. More precisely, a function 𝑢∶[𝑎,𝑏]→ℝ has bounded (𝑝,2)-variation if and only if 𝑢^∈AC[𝑎,𝑏], where AC[𝑎,𝑏] denotes the space of all absolutely continuous functions on [𝑎,𝑏], and 𝑢^∈𝐿_𝑝[𝑎,𝑏]. Moreover, the (𝑝,2)-variation of 𝑢 on [𝑎,𝑏] is then given by the formula 𝑉_(𝑝,2)(𝑢;[𝑎,𝑏])=‖‖𝑢^‖‖^𝑝𝐿_𝑝.(1.1)

In this paper we will prove the following parallel result: given 𝑝∈(1,∞) and 𝑘∈ℕ, a function 𝑢∶[𝑎,𝑏]→ℝ has bounded (𝑝,𝑘)-variation on [𝑎,𝑏] if and only if 𝑢^(𝑘−1)∈AC[𝑎,𝑏] and 𝑢^(𝑘)∈𝐿_𝑝[𝑎,𝑏]. Moreover, the (𝑝,𝑘)-variation of 𝑢 on [𝑎,𝑏] is then given by 𝑉_(p,𝑘)(𝑢;[𝑎,𝑏])=‖‖𝑢^(𝑘)‖‖^𝑝𝐿_𝑝((𝑘−1)!)^𝑝.(1.2)

This characterization can be considered as a natural generalization of the classical Riesz lemma [5] for the class 𝐴_𝑝[𝑎,𝑏] of all functions 𝑢∶[𝑎,𝑏]→ℝ such that 𝑢∈AC[𝑎,𝑏] and 𝑢^∈𝐿_𝑝[𝑎,𝑏]. We point out that the Riesz lemma provides a criterion for functions to belong to the Sobolev space 𝑊^1𝑝[𝑎,𝑏] more than 20 years before this space has been introduced by Sobolev [6] in 1934.

2. Preliminaries

In this section, we recall some definitions and known results concerning the Riesz 𝑝-variation, the De la Vallée Poussin second variation, and the Popoviciu 𝑘th variation.

Given a function 𝑢∶[𝑎,𝑏]→ℝ and a partition 𝜋∶={𝑎=𝑡₀<𝑡₁<⋯<𝑡_𝑚=𝑏} of [𝑎,𝑏], consider the expression 𝜎_𝑝(𝑢;𝜋)=^𝑚_𝑗=1||𝑢𝑡_𝑗−𝑢𝑡_𝑗−1||^𝑝||𝑡_𝑗−𝑡_𝑗−1||^𝑝−1.(2.1)

The number 𝑉_𝑝(𝑢;[𝑎,𝑏])∶=sup_𝜋𝜎_𝑝(𝑢;𝜋), where the supremum is taken over all partitions 𝜋 of [𝑎,𝑏], is called the Riesz 𝑝-variation of 𝑢 on [𝑎,𝑏]. In case 𝑉_𝑝(𝑢;[𝑎,𝑏])<∞, the function 𝑢 is said to have bounded 𝑝-variation. In what follows, by 𝑅𝑉_𝑝[𝑎,𝑏] we will denote the class of all functions 𝑢∶[𝑎,𝑏]→ℝ of bounded 𝑝-variation on [𝑎,𝑏] (in the Riesz sense).

The class 𝑅𝑉_𝑝[𝑎,𝑏] is a Banach space with respect to the norm ‖𝑢‖_𝑝∶=||𝑢(𝑎)||+𝑉_𝑝(𝑢;[𝑎,𝑏])^1/𝑝.(2.2)

The following characterization of functions 𝑢∈𝑅𝑉_𝑝[𝑎,𝑏] is known in the literature as the Riesz lemma [5].

Proposition 2.1. A function 𝑢 belongs to 𝑅𝑉_𝑝[𝑎,𝑏](1<𝑝<∞) if and only if 𝑢∈𝐴_𝑝[𝑎,𝑏], that is, 𝑢∈AC[𝑎,𝑏] and 𝑢^∈𝐿_𝑝[𝑎,𝑏]. Moreover, the equality 𝑉_𝑝(𝑢;[𝑎,𝑏])=‖𝑢′‖^𝑝𝐿_𝑝(2.3) holds in this case.

In 1908, De la Vallée Poussin [2] introduced the class of functions of bounded second variation as follows. Given a function 𝑢∶[𝑎,𝑏]→ℝ and a partition 𝜋 of [𝑎,𝑏] of the form 𝑎=𝑎₁<𝑐₁≤𝑑₁<𝑏₁≤𝑎₂<⋯<𝑏_𝑚−1≤𝑎_𝑚<𝑐_𝑚≤𝑑_𝑚<𝑏_𝑚=𝑏,(2.4) consider the expression 𝜎²(𝑢;𝜋)∶=^𝑚_𝑗=1||||𝑢𝑏_𝑗−𝑢𝑑_𝑗𝑏_𝑗−𝑑_j−𝑢𝑐^𝑗−𝑢𝑎_𝑗𝑐_𝑗−𝑎_𝑗||||(2.5) and let 𝑉²(𝑢;[𝑎,𝑏])=sup_𝜋𝜎²(𝑢;𝜋),(2.6) where the supremum is taken over all partitions 𝜋 of the form (2.4). The number 𝑉²(𝑢;[𝑎,𝑏]) is called the De La Vallée Poussin second variation of 𝑢 on [𝑎,𝑏]. In case 𝑉²(𝑢;[𝑎,𝑏])<∞, the function 𝑢 is said to have bounded second variation. In what follows, by BV²[𝑎,𝑏] we will denote the class of all functions 𝑢∶[𝑎,𝑏]→ℝ of bounded second variation on [𝑎,𝑏].

The following result has been proved in [7], see also [8].

Proposition 2.2. Every function 𝑢∈BV²[𝑎,𝑏] is Lipschitz continuous on [𝑎,𝑏] and can be expressed as a difference of two convex functions.

If 𝑢∈BV²[𝑎,𝑏], then from standard properties of convex functions it follows that the unilateral right derivative 𝑢′₊ and the unilateral left derivative 𝑢′₋ exist on [𝑎,𝑏]. Moreover, the set 𝐸 of points 𝑥 where 𝑢^′(𝑥) fails to exist is countable, 𝑢^ is continuous on [𝑎,𝑏]⧵𝐸, the right unilateral derivative 𝑢^+ is right continuous, and the left unilateral derivative 𝑢^− is left continuous on [𝑎,𝑏].

Finally, let us recall Popoviciu’s more general definition of bounded 𝑘th variation [3]. To this end, we need the concept of the 𝑘th divided difference of a function 𝑢∶[𝑎,𝑏]→ℝ with respect to distinct points 𝑡₀,𝑡₁,…,𝑡_𝑘∈[𝑎,𝑏] (not necessarily in increasing order) defined by 𝑢𝑡₀,𝑡₁,…,𝑡_𝑘∶=^𝑘_𝑗=0𝑢𝑡_𝑗𝑡_𝑗−𝑡₀⋯𝑡_𝑗−𝑡_𝑗−1𝑡_𝑗−𝑡_𝑗+1⋯𝑡_𝑗−𝑡_𝑘.(2.7)

By definition, the 𝑘th divided difference (2.7) is independent of the order in which the points 𝑡₀,𝑡₁,…,𝑡_𝑘 appear. The following two results are well known (see, e.g., [3, 9]).

Proposition 2.3. Suppose that 𝑢∶[𝑎,𝑏]→ℝ belongs to 𝐶^𝑘[𝑎,𝑏] for some 𝑘≥1. Then, 𝑢𝑡₀,𝑡₁,…,𝑡_𝑘∶=𝑢^(𝑘)(𝜉)𝑘!(2.8) for some 𝜉∈conv{𝑡₀,𝑡₁,…,𝑡_𝑘}, where conv𝑀 denotes the smallest interval containing 𝑀.

Proposition 2.4. Suppose that 𝑢∶[𝑎,𝑏]→ℝ is such that 𝑢^(𝑘) is Riemann integrable on [𝑎,𝑏] for some 𝑘≥1. Then, 𝑢𝑡₀,𝑡₁,…,𝑡_𝑘=¹⁰^𝑥1₀⋯^𝑥𝑘−1₀𝑢^(𝑘)𝑥_𝑘𝑡_𝑘−𝑡_𝑘−1+⋯+𝑥₁𝑡₁−𝑡₀+𝑡₀𝑑𝑥_𝑘…𝑑𝑥₁.(2.9)

If the function 𝑢 has a Riemann integrable derivative of order 𝑘≥1, then the last result allows us to generalize the concept of the 𝑘th divided difference for points 𝑡₀,𝑡₁,…,𝑡_𝑘 which are not necessarily distinct. Moreover, if the unilateral derivatives 𝑢^+(𝑘−1) and 𝑢^{−(𝑘−1)} are right and left continuous on [𝑎,𝑏], respectively, then the function of 𝑘+1 variables (𝑡₀,𝑡₁,…,𝑡_𝑘)↦𝑢[𝑡₀,𝑡₁,…,𝑡_𝑘] is continuous in each variable separately.

As a consequence of the source Propositions 2.3 and 2.4 we have that, if 𝑢∈𝐶^𝑘[𝑎,𝑏] for some 𝑘≥1, then lim_ℎ→0+𝑢𝑡₀,…,𝑡₀,𝑡₀+ℎ=𝑢^(𝑘)𝑡₀𝑘!=lim^ℎ→0+𝑢𝑡₀−ℎ,𝑡₀,…,𝑡₀.(2.10)

However, as a corollary to Proposition 2.4 we can obtain a weaker version of the preceding results.

Proposition 2.5. Suppose that 𝑢∶[𝑎,𝑏]→ℝ is such that 𝑢^(𝑘) is Riemann integrable on [𝑎,𝑏] for some 𝑘≥1. Assume, in addition, that the unilateral right derivative 𝑢^+(𝑘−1) is right continuous and the unilateral left derivative 𝑢^{−(𝑘−1)} is left continuous on [𝑎,𝑏]. Then, lim_ℎ→0+𝑢𝑡₀,…,𝑡₀,𝑡₀+ℎ=𝑢^+(𝑘)𝑡₀𝑘!𝑎<𝑡⁰lim_ℎ→0+<𝑏,𝑢𝑡₀−ℎ,𝑡₀,…,𝑡₀=𝑢^−(𝑘)𝑡₀𝑘!𝑎<𝑡⁰<𝑏.(2.11)

Let 𝑢∶[𝑎,𝑏]→ℝ be a function and 𝜋 a partition of the form 𝑎=𝑡_1,1<𝑡_1,2<⋯<𝑡_1,𝑘≤𝑡_1,𝑘+1<⋯<𝑡_1,2𝑘≤⋯<𝑡_2,𝑘+1<⋯<𝑡_2,2𝑘≤⋯<𝑡_3,1<⋯<𝑡_𝑗,1<⋯<⋯<𝑡_𝑗,𝑘≤𝑡_𝑗,𝑘+1<⋯<𝑡_𝑗,2𝑘≤⋯<𝑡_{𝑚−1,2𝑘}≤𝑡_𝑚−1<⋯<𝑡_𝑚,𝑘≤𝑡_𝑚,𝑘+1<⋯<𝑡_𝑚,2𝑘=𝑏.(2.12) Moreover, let 𝜎^𝑘(𝑢;𝜋)∶=^𝑚_𝑗=1||𝑢𝑡_𝑗,𝑘+1,…,𝑡_𝑗,2𝑘−𝑢𝑡_𝑗,1,…,𝑡_𝑗,𝑘||,(2.13)𝑉^𝑘(𝑢;[𝑎,𝑏])∶=sup_𝜋𝜎^𝑘(𝑢;𝜋),(2.14) where the supremum is taken over all partitions 𝜋 of the form (2.12). The number 𝑉^𝑘(𝑢;[𝑎,𝑏]) is called the Popoviciu 𝑘th variation of 𝑢 on [𝑎,𝑏]. In case 𝑉^𝑘(𝑢;[𝑎,𝑏])<∞, the function 𝑢 is said to have bounded 𝑘th variation on [𝑎,𝑏] and the set of such functions is denoted by BV^𝑘[𝑎,𝑏].

The following result on the regularity of higher derivatives is also well known [7].

Proposition 2.6. Suppose that 𝑢∶[𝑎,𝑏]→ℝ belongs to BV^𝑘[𝑎,𝑏] for some 𝑘≥2. Then, the derivative 𝑢^(𝑘−2) belongs to BV²[𝑎,𝑏]; therefore, the unilateral right derivative 𝑢^+(𝑘−1) and the unilateral left derivative 𝑢^{−(𝑘−1)} exist on [𝑎,𝑏]. Moreover, the set 𝐸 of points 𝑥 where the derivative 𝑢^(𝑘−1)(𝑥) fails to exist is countable, 𝑢^(𝑘−1) is continuous on [𝑎,𝑏]⧵𝐸, the unilateral right derivative 𝑢^+(𝑘−1) is right continuous, and the unilateral left derivative 𝑢^{−(𝑘−1)} is left continuous on [𝑎,𝑏].

3. Main Result

In what follows, for 1≤𝑝<∞ and 𝑘∈ℕ we denote by 𝐴_(𝑝,𝑘)[𝑎,𝑏] the class of all functions 𝑢∶[𝑎,𝑏]→ℝ such that 𝑢^(𝑘−1)∈AC[𝑎,𝑏] and 𝑢^(𝑘)∈𝐿_𝑝[𝑎,𝑏]. Thus, 𝐴_(𝑝,1)[𝑎,𝑏]=𝐴_𝑝[𝑎,𝑏].

In this section we introduce the notion of (𝑝,𝑘)-variation, where 𝑝 always denotes a real number in [1,∞) and 𝑘≥1 denotes a natural number. For 𝑝>1 we will prove that a function 𝑢∶[𝑎,𝑏]→ℝ has bounded (𝑝,𝑘)-variation (for the definition see below) if and only if 𝑢∈𝐴_(𝑝,𝑘)[𝑎,𝑏]. Moreover, we will show that equality (1.2) holds, as one should expect.

So given 𝑢∶[𝑎,𝑏]→ℝ, consider a partition 𝜋 of the form 𝑎=𝑡_1,1<𝑡_1,2<⋯<𝑡_1,𝑘≤𝑡_1,𝑘+1<⋯<𝑡_1,2𝑘≤⋯<𝑡_2,𝑘+1<⋯<𝑡_2,2𝑘≤⋯<𝑡_3,1<⋯<𝑡_𝑗,1<⋯<𝑡_𝑗,𝑘≤𝑡_𝑗,𝑘+1<⋯<𝑡_𝑗,2𝑘≤⋯<𝑡_{𝑚−1,2𝑘}≤𝑡_𝑚−1<⋯<𝑡_𝑚,𝑘≤𝑡_𝑚,𝑘+1<⋯<𝑡_𝑚,2𝑘=𝑏.(3.1)

Moreover, define 𝜎_(𝑝,𝑘)(𝑢;𝜋)∶=∑^𝑚_𝑗=1||𝑢𝑡_𝑗,𝑘+1,…,𝑡_𝑗,2𝑘−𝑢𝑡_𝑗,1,…,𝑡_𝑗,𝑘||^𝑝||𝑡_𝑗,2𝑘−𝑡_𝑗,1||^𝑝−1𝑉_(𝑝,𝑘),(𝑢;[𝑎,𝑏])∶=sup_𝜋𝜎_(𝑝,𝑘)(𝑢;𝜋),(3.2) where the supremum is taken over all partitions 𝜋 of the form (3.1). We call the number 𝑉_(𝑝,𝑘)(𝑢;[𝑎,𝑏]) the (𝑝,𝑘)-variation of 𝑢 on [𝑎,𝑏]. In case 𝑉_(𝑝,𝑘)(𝑢;[𝑎,𝑏])<∞, we say that the function 𝑢 has bounded (𝑝,𝑘)-variation, and we denote the space of such functions by 𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏].

Proposition 3.1. For 𝑘≥1 and 𝑝∈(1,∞), the inclusion 𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏]⊆BV^𝑘[𝑎,𝑏](3.3) and the inequality 𝑉^𝑘(𝑢;[𝑎,𝑏])≤(𝑏−𝑎)^1−1/𝑝𝑉_(𝑝,𝑘)(𝑢;[𝑎,𝑏])^1/𝑝(3.4) hold true, where 𝑉^𝑘(𝑢;[𝑎,𝑏]) is given by (2.14).

Proof. Let 𝜋 be a partition of [𝑎,𝑏] of the form (3.1). By Hölder’s inequality, we obtain ^𝑚_𝑗=1||𝑢𝑡_𝑗,𝑘+1,…,𝑡_𝑗,2𝑘−𝑢𝑡_𝑗,1,…,𝑡_𝑗,𝑘||||𝑡_𝑗,2𝑘−𝑡_𝑗,1||||𝑡^𝑗,2𝑘≤_𝑚−𝑡_𝑗,1||_𝑗=1||𝑢𝑡_𝑗,𝑘+1,…,𝑡_𝑗,2𝑘−𝑢𝑡_𝑗,1,…,𝑡_𝑗,𝑘||||𝑡_𝑗,2𝑘−𝑡_𝑗,1||^𝑝−1^1/𝑝^𝑚_𝑗=1||𝑡_𝑗,2𝑘−𝑡_𝑗,1||^1−1/𝑝.(3.5) Passing on both sides of to the supremum over all partitions 𝜋 of the form (3.1), we conclude that (3.4) is true, and so also (3.3), by definition of BV^𝑘[𝑎,𝑏].

From Proposition 2.6 and Proposition 3.1 we immediately deduce the following.

Corollary 3.2. Let 1<𝑝<∞, 𝑘≥2, and 𝑢∈𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏]. Then, the derivative 𝑢^(𝑘−2) belongs to 𝐵𝑉²[𝑎,𝑏]; therefore, the unilateral right and the unilateral left derivatives 𝑢^+(𝑘−1) and 𝑢^{−(𝑘−1)} exist on [𝑎,𝑏] and are right and left continuous, respectively.

Now we prove a statement on the existence of the derivative 𝑢^(𝑘−1)(𝑡₀) for all 𝑡₀∈(𝑎,𝑏) in case 𝑢∈𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏].

Proposition 3.3. Let 1<𝑝<∞, 𝑘≥2, and 𝑢∈𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏]. Then, the derivative 𝑢^(𝑘−1)(𝑡) exists for all 𝑡∈(𝑎,𝑏).

Proof. Suppose that there exists 𝑥₀∈(𝑎,𝑏) such that 𝛼_𝑥0∶=|𝑢^+(𝑘−1)(𝑥₀)−𝑢^{−(𝑘−1)}(𝑥₀)|>0. Let 𝜋 be a partition of [𝑎,𝑏] of the form 𝑎=<𝑡𝑡_1,1𝑗,2<𝑡_1,2<⋯<𝑡_1,𝑘≤𝑡_1,𝑘+1<⋯<𝑡_1,2𝑘≤⋯<𝑡_𝑗,1=𝑥₀−ℎ⋯<⋯<𝑡_𝑗,𝑘<⋯<𝑡_{𝑗,2𝑘−1}=𝑥₀<𝑡_𝑗,2𝑘=𝑥₀+ℎ<⋯<𝑡_𝑚,1≤𝑡_𝑚,2<⋯𝑡_𝑚,𝑘≤𝑡_𝑚,𝑘+1<⋯𝑡_𝑚,2𝑘=𝑏,(3.6) where ℎ satisfies 0<ℎ<𝑚𝑖𝑛𝑡^𝑗,2𝑘−𝑡_𝑗,12∶𝑗=1,2,…,𝑚.(3.7)
By definition of the (𝑝,𝑘)-variation of 𝑢, we have then 𝑉_(𝑝,𝑘)(𝑢;[𝑎,𝑏])≥||𝑢𝑡^𝑗,𝑘+1,𝑡_𝑗,𝑘+2,…,𝑡_{𝑗,2𝑘−2},𝑥₀,𝑥₀+ℎ−𝑢𝑥₀−ℎ,𝑡_𝑗,2,…,𝑡_𝑗,𝑘||^𝑝||2ℎ||^𝑝−1.(3.8)
From Corollary 3.2 we know that the unilateral right derivative 𝑢^+(𝑘−1) and the unilateral left derivative 𝑢^{−(𝑘−1)} are right and left continuous, respectively, on [𝑎,𝑏]. Now, by Corollary 3.2, we further have 𝑉_(𝑝,𝑘)(𝑢;[𝑎,𝑏])≥lim_ℎ→0+lim_{𝑡𝑗,2→𝑥0}||𝑢𝑡_𝑗,𝑘+1,𝑡_𝑗,𝑘+2,…,𝑡_{𝑗,2𝑘−2},𝑥₀,𝑥₀+ℎ−𝑢𝑥₀−ℎ,𝑡_𝑗,2,…,𝑡_𝑗,𝑘||^𝑝||2ℎ||^𝑝−1=𝛼^𝑝𝑥02^𝑝−1((𝑘−1)!)^𝑝lim_ℎ→0+1ℎ^𝑝−1=∞,(3.9) contradicting our assumption 𝑢∈𝑅𝑉_(𝑝,𝑘). Consequently, the function 𝑢 has a derivative 𝑢^(𝑘−1) everywhere on (𝑎,𝑏).

We are now in a position to formulate and prove our main result.

Theorem 3.4. Let 1<𝑝<∞ and 𝑘≥2. Then, 𝑢∈𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏] if and only if 𝑢∈𝐴_(𝑝,𝑘)[𝑎,𝑏]. Moreover, in this case the (𝑝,𝑘)-variation of 𝑢 on [𝑎,𝑏] is given by (1.2).

Proof. Suppose first that 𝑢∶[𝑎,𝑏]→ℝ belongs to 𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏]. By Proposition 3.3, we know then that the derivative 𝑢^(𝑘−1) exists on [𝑎,𝑏].
Let 𝜋={𝑎=𝑡₀<𝑡₁<⋯<𝑡_𝑚=𝑏} be a partition of [𝑎,𝑏]. For any subinterval [𝑡_𝑗,𝑡_𝑗+1], 𝑗=0,1,…, 𝑚−1, we define a partition 𝜋_𝑗 in the form 𝑡_𝑗=𝑡_𝑗,1<𝑡_𝑗,2<⋯<𝑡_{𝑗,𝑘−1}<𝑡_𝑗,𝑘=𝑡_𝑗+ℎ<⋯<𝑡_𝑗,𝑘+1=𝑡_𝑗+1−ℎ<𝑡_𝑗,𝑘+2<⋯<𝑡_𝑗,2𝑘=𝑡_𝑗+1,(3.10) where ℎ∶=min𝑡^𝑗+1−𝑡_𝑗2∶𝑗=1,2,…,𝑚−1.(3.11)
Using the shortcut 𝑈_𝑗,𝑘𝜋_𝑗;ℎ∶=||𝑢𝑡^𝑗+1−ℎ,𝑡_𝑗,𝑘+2,…,𝑡_{𝑗,2𝑘−1}−𝑢𝑡_𝑗,𝑡_𝑗,2,…,𝑡_{𝑗,𝑘−1},𝑡_𝑗+ℎ||^𝑝||𝑡_𝑗+1−𝑡_𝑗||^𝑝−1,(3.12) by Corollary 3.2, we have then ||𝑢^(𝑘−1)𝑡_𝑗+1−𝑢^(𝑘−1)𝑡_𝑗||^𝑝||𝑡_𝑗+1−𝑡_𝑗||^𝑝−1=lim_ℎ→0+lim_{𝑡𝑗,𝑘−1→𝑡𝑗}lim_{𝑡𝑗,𝑘+2→𝑡𝑗+1}𝑈_𝑗,𝑘𝜋_𝑗;ℎ.(3.13)
Consequently, 1((𝑘−1)!)^𝑝𝑚_𝑗=1||𝑢^(𝑘−1)𝑡_𝑗+1−𝑢^(𝑘−1)𝑡_𝑗||^𝑝||𝑡_𝑗+1−𝑡_𝑗||^𝑝−1=lim_ℎ→0+lim_{𝑡𝑗,𝑘−1→𝑡𝑗}lim_{𝑡𝑗,𝑘+2→𝑡𝑗+1}𝑈_𝑗,𝑘𝜋_𝑗;ℎ≤𝑉_(𝑝,𝑘)(𝑢;[𝑎,𝑏]).(3.14) Hence, by the Riesz lemma, we see that 𝑢^(𝑘−1)∈AC[𝑎,𝑏] and 𝑢^(𝑘)∈𝐿_𝑝[𝑎,𝑏], that is, 𝑢∈𝐴_(𝑝,𝑘)[𝑎,𝑏]. Moreover, ^𝑏𝑎||𝑢^(𝑘)(𝑡)||^𝑝((𝑘−1)!)^𝑝𝑑𝑡≤𝑉_(𝑝,𝑘)(𝑢;[𝑎,𝑏]).(3.15)
Conversely, suppose now that 𝑢∶[𝑎,𝑏]→ℝ belongs to 𝐴_(𝑝,𝑘)[𝑎,𝑏], and let 𝜋 be a partition of [𝑎,𝑏] of the form (3.10). Since 𝑢^(𝑘−1)∈AC[𝑎,𝑏] and 𝑢^(𝑘)∈𝐿_𝑝[𝑎,𝑏], by Proposition 2.4, we have ^𝑚_𝑗=1||𝑢𝑡_𝑗,𝑘+1,…,𝑡_𝑗,2𝑘−𝑢𝑡_𝑗,1,…,𝑡_𝑗,𝑘||^𝑝||𝑡_𝑗,2𝑘−𝑡_𝑗,1||^𝑝−1=^𝑚_𝑗=1||𝑢^(𝑘−1)𝜉_𝑗,2𝑘−𝑢^(𝑘−1)𝜉_{𝑗−1,𝑘}||^𝑝((𝑘−1)!)^𝑝||𝑡_𝑗,2𝑘−𝑡_𝑗,1||^𝑝−1=^𝑚_𝑗=1||||^𝜉_𝑗^𝜉,2𝑘_{𝑗−1,𝑘}𝑢^(𝑘)(𝜎)𝑑𝜎||||^𝑝1((𝑘−1)!)^𝑝||𝑡_𝑗,2𝑘−𝑡_𝑗,1||^𝑝−1,(3.16) where 𝜉_𝑗,2𝑘∈conv{𝑡_𝑗,𝑘+1,…,𝑡_𝑗,2𝑘} and 𝜉_{𝑗−1,𝑘}∈conv{𝑡_𝑗,1,…,𝑡_𝑗,𝑘}. By Hölder’s inequality, we get then the estimates ^𝑚_𝑗=1||||^𝜉_{𝑗,2𝑘𝜉𝑗−1,𝑘}𝑢^(𝑘)(𝑡)𝑑𝑡||||^𝑝1((𝑘−1)!)^𝑝||𝑡_𝑗,2𝑘−𝑡_𝑗,1||^𝑝−1≤^𝑚_𝑗=1^{𝜉𝑗,2𝑘}_{𝜉𝑗−1,𝑘}||𝑢^(𝑘)(𝑡)||^𝑝𝑑𝑡||𝜉^𝑗,2𝑘−𝜉_{𝑗−1,𝑘}||^𝑝−1((𝑘−1)!)^𝑝||𝑡_𝑗,2𝑘−𝑡_𝑗,1||^𝑝−1≤^𝑚_𝑗=1^{𝜉𝑗,2𝑘}_{𝜉𝑗−1,𝑘}||𝑢^(𝑘)(𝑡)||^𝑝((𝑘−1)!)^𝑝𝑑𝑡≤^𝑚_𝑗=1^{𝑡𝑗,2𝑘}_𝑡𝑗,1||𝑢^(𝑘)(𝑡)||^𝑝((𝑘−1)!)^𝑝𝑑𝑡=^𝑏𝑎||𝑢^(𝑘)(𝑡)||^𝑝((𝑘−1)!)^𝑝𝑑𝑡.(3.17)
This implies that 𝑢∈𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏] and ^𝑏𝑎||𝑢^(𝑘)(𝑡)||^𝑝((𝑘−1)!)^𝑝𝑑𝑡≥𝑉_(𝑝,𝑘)(𝑢;[𝑎,𝑏]).(3.18)
Combining estimates (3.15) and (3.18), we conclude that (1.2) is true, and so the proof is complete.

The characterization of functions in 𝑅𝑉_(𝑝,𝑘)[𝑎,𝑏] included in Theorem 3.4 can be considered as a natural generalization of that given by Riesz [5] for the class 𝐴_𝑝[𝑎,𝑏](1<𝑝<∞), and by Merentes [4] for the class 𝐴_(𝑝,2)[𝑎,𝑏].

Acknowledgments

This paper has been partly supported by the Central Bank of Venezuela which is gratefully acknowledged. the authors also express their gratitude to the staff of the math library for compiling the references.