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].
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
V(p,2)(u;[a,b])=‖u′′‖Lpp.
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)∈Lp[a,b]. Moreover, the (p,k)-variation of u on [a,b] is then given by
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 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<⋯<tm=b} of [a,b], consider the expression
σp(u;π)=∑j=1m|u(tj)-u(tj-1)|p|tj-tj-1|p-1.
The number Vp(u;[a,b]):=supπσp(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 Vp(u;[a,b])<∞, the function u is said to have bounded p-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 class RVp[a,b] is a Banach space with respect to the norm
‖u‖p:=|u(a)|+(Vp(u;[a,b]))1/p.
The following characterization of functions u∈RVp[a,b] is known in the literature as the Riesz lemma [5].
Proposition 2.1.
A function u 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
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
a=a1<c1≤d1<b1≤a2<⋯<bm-1≤am<cm≤dm<bm=b,
consider the expression
σ2(u;π):=∑j=1m|u(bj)-u(dj)bj-dj-u(cj)-u(aj)cj-aj|
and let
V2(u;[a,b])=supπσ2(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 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,…,tk∈[a,b] (not necessarily in increasing order) defined by
u[t0,t1,…,tk]:=∑j=0ku(tj)(tj-t0)⋯(tj-tj-1)(tj-tj+1)⋯(tj-tk).
By definition, the kth 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 that u:[a,b]→ℝ belongs to Ck[a,b] for some k≥1. Then,
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 that u:[a,b]→ℝ is such that u(k) is Riemann integrable on [a,b] for some k≥1. Then,
u[t0,t1,…,tk]=∫01∫0x1⋯∫0xk-1u(k)[xk(tk-tk-1)+⋯+x1(t1-t0)+t0]dxk…dx1.
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,…,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 Propositions 2.3 and 2.4 we have that, if u∈Ck[a,b] for some k≥1, then
limh→0+u[t0,…,t0,t0+h]=u(k)(t0)k!=limh→0+u[t0-h,t0,…,t0].
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,
limh→0+u[t0,…,t0,t0+h]=u+(k)(t0)k!(a<t0<b),limh→0+u[t0-h,t0,…,t0]=u-(k)(t0)k!(a<t0<b).
Let u:[a,b]→ℝ be a function and π a partition of the form
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
σk(u;π):=∑j=1m|u[tj,k+1,…,tj,2k]-u[tj,1,…,tj,k]|,Vk(u;[a,b]):=supπσk(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 Popoviciu kth variation of u on [a,b]. In case Vk(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 BVk[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 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, 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)∈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 given u:[a,b]→ℝ, consider a partition π of the form
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
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]):=supπσ(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.
For k≥1 and p∈(1,∞), the inclusion
RV(p,k)[a,b]⊆BVk[a,b]
and the inequality
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.
Let π be a partition of [a,b] of the form (3.1). By Hölder’s inequality, we obtain
∑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 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 αx0:=|u+(k-1)(x0)-u-(k-1)(x0)|>0. Let π be a partition of [a,b] of the form
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
0<h<min{tj,2k-tj,12:j=1,2,…,m}.
By definition of the (p,k)-variation of u, we have then
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 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])≥limh→0+limtj,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)!)plimh→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.
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<⋯<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
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
h:=min{tj+1-tj2:j=1,2,…,m-1}.
Using the shortcut
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
|u(k-1)(tj+1)-u(k-1)(tj)|p|tj+1-tj|p-1=limh→0+limtj,k-1→tjlimtj,k+2→tj+1Uj,k(πj;h).
Consequently,
1((k-1)!)p∑j=1m|u(k-1)(tj+1)-u(k-1)(tj)|p|tj+1-tj|p-1=limh→0+limtj,k-1→tjlimtj,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,
∫ab|u(k)(t)|p((k-1)!)pdt≤V(p,k)(u;[a,b]).
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)∈Lp[a,b], by Proposition 2.4, we have
∑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
∑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 that u∈RV(p,k)[a,b] and
∫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 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 Ap[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.
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