# On Functions of Bounded(p,k)-Variation

Authors: N. Merentes; S. Rivas; J. L. Sanchez
Journal: Journal of Function Spaces and Applications
 (2012)

---

## 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 Lp[a,b]. Moreover, an explicit connection between the (p,k)-variation of u and the Lp-norm of u(k) is given which is parallel to the classical Riesz formula characterizing functions in the spaces RVp[a,b] and Ap[a,b]. This may also be considered as an alternative characterization of the one variable Sobolev space Wpk[a,b].


---

## Body

## 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 BVk[a,b] of functions u:[a,b]→ℝ of bounded kth variation on [a,b]. It is known that, for any u∈BVk[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′′∈Lp[a,b]. Moreover, the (p,2)-variation of u on [a,b] is then given by the formula
(1.1)V(p,2)(u;[a,b])=‖u′′‖Lpp.In this paper we will prove the following parallel result: givenp∈(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)∈Lp[a,b]. Moreover, the (p,k)-variation of u on [a,b] is then given by
(1.2)V(p,k)(u;[a,b])=‖u(k)‖Lpp((k-1)!)p.This characterization can be considered as a natural generalization of the classical Riesz lemma [5] for the class Ap[a,b] of all functions u:[a,b]→ℝ such that u∈AC[a,b] and u′∈Lp[a,b]. We point out that the Riesz lemma provides a criterion for functions to belong to the Sobolev space Wp1[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 Rieszp-variation, the De la Vallée Poussin second variation, and the Popoviciu kth variation.Given a functionu:[a,b]→ℝ and a partition π:={a=t0<t1<⋯<tm=b} of [a,b], consider the expression
(2.1)σp(u;π)=∑j=1m|u(tj)-u(tj-1)|p|tj-tj-1|p-1.The numberVp(u;[a,b]):=supπσp(u;π), where the supremum is taken over all partitions π of [a,b], is called the Rieszp-variation of u on [a,b]. In case Vp(u;[a,b])<∞, the function u is said to have boundedp-variation. In what follows, by RVp[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 classRVp[a,b] is a Banach space with respect to the norm
(2.2)‖u‖p:=|u(a)|+(Vp(u;[a,b]))1/p.The following characterization of functionsu∈RVp[a,b] is known in the literature as the Riesz lemma [5].Proposition 2.1.
A functionu belongs to  RVp[a,b](1<p<∞) if and only if  u∈Ap[a,b], that is,  u∈AC[a,b] and  u′∈Lp[a,b]. Moreover, the equality
(2.3)Vp(u;[a,b])=‖u′‖Lpp
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
(2.4)a=a1<c1≤d1<b1≤a2<⋯<bm-1≤am<cm≤dm<bm=b,
consider the expression
(2.5)σ2(u;π):=∑j=1m|u(bj)-u(dj)bj-dj-u(cj)-u(aj)cj-aj|
and let
(2.6)V2(u;[a,b])=sup⁡πσ2(u;π),
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 functionu∈BV2[a,b] is Lipschitz continuous on  [a,b] and can be expressed as a difference of two convex functions.Ifu∈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 boundedkth 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,…,tk∈[a,b] (not necessarily in increasing order) defined by
(2.7)u[t0,t1,…,tk]:=∑j=0ku(tj)(tj-t0)⋯(tj-tj-1)(tj-tj+1)⋯(tj-tk).By definition, thekth divided difference (2.7) is independent of the order in which the points t0,t1,…,tk appear. The following two results are well known (see, e.g., [3, 9]).Proposition 2.3.
Suppose thatu:[a,b]→ℝ belongs to  Ck[a,b] for some  k≥1. Then,
(2.8)u[t0,t1,…,tk]:=u(k)(ξ)k!
for some ξ∈conv{t0,t1,…,tk}, where convM denotes the smallest interval containing M.Proposition 2.4.
Suppose thatu:[a,b]→ℝ is such that  u(k) is Riemann integrable on  [a,b] for some  k≥1. Then,
(2.9)u[t0,t1,…,tk]=∫01∫0x1⋯∫0xk-1u(k)[xk(tk-tk-1)+⋯+x1(t1-t0)+t0]dxk…dx1.If the functionu 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,…,tk 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,…,tk)↦u[t0,t1,…,tk] is continuous in each variable separately.As a consequence of the source Propositions2.3 and 2.4 we have that, if u∈Ck[a,b] for some k≥1, then
(2.10)lim⁡h→0+u[t0,…,t0,t0+h]=u(k)(t0)k!=lim⁡h→0+u[t0-h,t0,…,t0].However, as a corollary to Proposition2.4 we can obtain a weaker version of the preceding results.Proposition 2.5.
Suppose thatu:[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,
(2.11)lim⁡h→0+u[t0,…,t0,t0+h]=u+(k)(t0)k!(a<t0<b),lim⁡h→0+u[t0-h,t0,…,t0]=u-(k)(t0)k!(a<t0<b).Letu:[a,b]→ℝ be a function and π a partition of the form
(2.12)a=t1,1<t1,2<⋯<t1,k≤t1,k+1<⋯<t1,2k≤⋯<t2,k+1<⋯<t2,2k≤⋯<t3,1<⋯<tj,1<⋯<⋯<tj,k≤tj,k+1<⋯<tj,2k≤⋯<tm-1,2k≤tm-1<⋯<tm,k≤tm,k+1<⋯<tm,2k=b.
Moreover, let
(2.13)σk(u;π):=∑j=1m|u[tj,k+1,…,tj,2k]-u[tj,1,…,tj,k]|,(2.14)Vk(u;[a,b]):=sup⁡πσk(u;π),
where the supremum is taken over all partitions π of the form (2.12). The number Vk(u;[a,b]) is called the Popoviciukth variation of u on [a,b]. In case Vk(u;[a,b])<∞, the function u is said to have boundedkth variation on [a,b] and the set of such functions is denoted by BVk[a,b].The following result on the regularity of higher derivatives is also well known [7].Proposition 2.6.
Suppose thatu:[a,b]→ℝ belongs to  BVk[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, for1≤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)∈Lp[a,b]. Thus, A(p,1)[a,b]=Ap[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 givenu:[a,b]→ℝ, consider a partition π of the form
(3.1)a=t1,1<t1,2<⋯<t1,k≤t1,k+1<⋯<t1,2k≤⋯<t2,k+1<⋯<t2,2k≤⋯<t3,1<⋯<tj,1<⋯<tj,k≤tj,k+1<⋯<tj,2k≤⋯<tm-1,2k≤tm-1<⋯<tm,k≤tm,k+1<⋯<tm,2k=b.Moreover, define(3.2)σ(p,k)(u;π):=∑j=1m|u[tj,k+1,…,tj,2k]-u[tj,1,…,tj,k]|p|tj,2k-tj,1|p-1,V(p,k)(u;[a,b]):=sup⁡πσ(p,k)(u;π),
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.
Fork≥1 and  p∈(1,∞), the inclusion
(3.3)RV(p,k)[a,b]⊆BVk[a,b]
and the inequality
(3.4)Vk(u;[a,b])≤(b-a)1-1/pV(p,k)(u;[a,b])1/p
hold true, where Vk(u;[a,b]) is given by (2.14).Proof.
Letπ be a partition of [a,b] of the form (3.1). By Hölder’s inequality, we obtain
(3.5)∑j=1m|u[tj,k+1,…,tj,2k]-u[tj,1,…,tj,k]||tj,2k-tj,1||tj,2k-tj,1|≤(∑j=1m|u[tj,k+1,…,tj,2k]-u[tj,1,…,tj,k]||tj,2k-tj,1|p-1)1/p(∑j=1m|tj,2k-tj,1|)1-1/p.
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 BVk[a,b].From Proposition2.6 and Proposition 3.1 we immediately deduce the following.Corollary 3.2.
Let1<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 derivativeu(k-1)(t0) for all t0∈(a,b) in case u∈RV(p,k)[a,b].Proposition 3.3.
Let1<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 existsx0∈(a,b) such that αx0:=|u+(k-1)(x0)-u-(k-1)(x0)|>0. Let π be a partition of [a,b] of the form
(3.6)a=t1,1<t1,2<⋯<t1,k≤t1,k+1<⋯<t1,2k≤⋯<tj,1=x0-h⋯<tj,2<⋯<tj,k<⋯<tj,2k-1=x0<tj,2k=x0+h<⋯<tm,1≤tm,2<⋯tm,k≤tm,k+1<⋯tm,2k=b,
where h satisfies
(3.7)0<h<min{tj,2k-tj,12:j=1,2,…,m}.
By definition of the(p,k)-variation of u, we have then
(3.8)V(p,k)(u;[a,b])≥|u[tj,k+1,tj,k+2,…,tj,2k-2,x0,x0+h]-u[x0-h,tj,2,…,tj,k]|p|2h|p-1.
From Corollary3.2 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
(3.9)V(p,k)(u;[a,b])≥lim⁡h→0+lim⁡tj,2→x0|u[tj,k+1,tj,k+2,…,tj,2k-2,x0,x0+h]-u[x0-h,tj,2,…,tj,k]|p|2h|p-1=αx0p2p-1((k-1)!)plim⁡h→0+1hp-1=∞,
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.
Let1<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 thatu:[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].
Letπ={a=t0<t1<⋯<tm=b} be a partition of [a,b]. For any subinterval [tj,tj+1], j=0,1,…, m-1, we define a partition πj in the form
(3.10)tj=tj,1<tj,2<⋯<tj,k-1<tj,k=tj+h<⋯<tj,k+1=tj+1-h<tj,k+2<⋯<tj,2k=tj+1,
where
(3.11)h:=min⁡{tj+1-tj2:j=1,2,…,m-1}.
Using the shortcut(3.12)Uj,k(πj;h):=|u[tj+1-h,tj,k+2,…,tj,2k-1]-u[tj,tj,2,…,tj,k-1,tj+h]|p|tj+1-tj|p-1,
by Corollary 3.2, we have then
(3.13)|u(k-1)(tj+1)-u(k-1)(tj)|p|tj+1-tj|p-1=lim⁡h→0+lim⁡tj,k-1→tjlim⁡tj,k+2→tj+1Uj,k(πj;h).
Consequently,(3.14)1((k-1)!)p∑j=1m|u(k-1)(tj+1)-u(k-1)(tj)|p|tj+1-tj|p-1=lim⁡h→0+lim⁡tj,k-1→tjlim⁡tj,k+2→tj+1Uj,k(πj;h)≤V(p,k)(u;[a,b]).
Hence, by the Riesz lemma, we see that u(k-1)∈AC[a,b] and u(k)∈Lp[a,b], that is, u∈A(p,k)[a,b]. Moreover,
(3.15)∫ab|u(k)(t)|p((k-1)!)pdt≤V(p,k)(u;[a,b]).
Conversely, suppose now thatu:[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)∈Lp[a,b], by Proposition 2.4, we have
(3.16)∑j=1m|u[tj,k+1,…,tj,2k]-u[tj,1,…,tj,k]|p|tj,2k-tj,1|p-1=∑j=1m|u(k-1)(ξj,2k)-u(k-1)(ξj-1,k)|p((k-1)!)p|tj,2k-tj,1|p-1=∑j=1m|∫ξj-1,kξj,2ku(k)(σ)dσ|p1((k-1)!)p|tj,2k-tj,1|p-1,
where ξj,2k∈conv{tj,k+1,…,tj,2k} and ξj-1,k∈conv{tj,1,…,tj,k}. By Hölder’s inequality, we get then the estimates
(3.17)∑j=1m|∫ξj-1,kξj,2ku(k)(t)dt|p1((k-1)!)p|tj,2k-tj,1|p-1≤∑j=1m∫ξj-1,kξj,2k|u(k)(t)|pdt|ξj,2k-ξj-1,k|p-1((k-1)!)p|tj,2k-tj,1|p-1≤∑j=1m∫ξj-1,kξj,2k|u(k)(t)|p((k-1)!)pdt≤∑j=1m∫tj,1tj,2k|u(k)(t)|p((k-1)!)pdt=∫ab|u(k)(t)|p((k-1)!)pdt.
This implies thatu∈RV(p,k)[a,b] and
(3.18)∫ab|u(k)(t)|p((k-1)!)pdt≥V(p,k)(u;[a,b]).
Combining estimates (3.15) and (3.18), we conclude that (1.2) is true, and so the proof is complete.The characterization of functions inRV(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 Ap[a,b](1<p<∞), and by Merentes [4] for the class A(p,2)[a,b].



---

*Source: 202987-2012-02-08.xml*