# Properties of Functions in the  Wiener ClassBVp[a,b] for 0<p<1

Authors: Yeli Niu; Heping Wang
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We will investigate properties of functions in the Wiener classBVp[a,b] with 0<p<1. We prove that any function in BVp[a,b](0<p<1) can be expressed as the difference of two increasing functions in BVp[a,b]. We also obtain the explicit form of functions in BVp[a,b] and show that their derivatives are equal to zero a.e. on [a,b].


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## Body

## 1. Introduction

Let0<p<∞. We say that a real valued function f on [a,b] is of bounded p-variation and is denoted by f∈BVp[a,b], if (1)Vpf=sup⁡T∑k=1n‍fxk-fxk-1p1/p<∞,where the supremum is taken over all partitions T:a=x0<x1<⋯<xn=b. When p=1, we get the well-known Jordan bounded variation BV[a,b]; and when 1<p<∞, we get Wiener’s definition of bounded p-variation. There are many other generalizations of BV, such as bounded Φ-variation in the sense of Young (see [1]) and Waterman’s Λ-bounded variation (see [2]). The class BVp and generalizations of BV have been studied mainly because of their applicability to the theory of Fourier series and some good approximative properties (see, e.g., [1–7]).However, it should be mentioned that results of most papers deal mostly with the casep≥1. This is because that in this case BVp[a,b] is a Banach space with the norm fBVp=|f(a)|+Vpf (see, e.g., [3]). In the case 0<p<1, BVp[a,b] is no longer a Banach space and has not been studied as far as we know. Nevertheless, functions in BVp[a,b]  (0<p<1) have many interesting properties; for example, their derivatives are equal to zero a.e. on [a,b].In this paper, we will investigate properties of functions in the classBVp[a,b] with 0<p<1. We will show that BVp[a,b] is a Frechet space with the quasinorm(2)qf=fap+Vpfp.We will get the Jordan type decomposition theorem which says that any function in BVp[a,b](0<p<1) can be expressed as the difference of two increasing functions in BVp[a,b]. We also get the representation theorem which gives the explicit form of functions in BVp[a,b](0<p<1).


## 2. Statement of Main Results

Clearly, for any fixedp∈(0,1), the Wiener class BVp[a,b] is a linear space. We define the functional q on BVp[a,b] by(3)qf=fap+Vpfp=fap+sup⁡T∑k=1n‍fxk-fxk-1p,f∈BVpa,b.From the inequality (a+b)p≤ap+bp  (a,b≥0,0<p<1), we get that q(f+g)≤q(f)+q(g). It then follows that q is a quasinorm on BVp[a,b].Our first result claims thatBVp[a,b]  (0<p<1) equipped with the quasinorm q is a Frechet space.Theorem 1.
The Wiener classBVp[a,b]  (0<p<1) equipped with the quasinorm q is a Frechet space.From the inequality(4)∑i=1∞‍aip21/p2≤∑i=1∞‍aip11/p1,ai≥0,0<p1≤p2<∞,we get that, for any f∈BVp1[a,b],(5)Vp2f≤Vp1f,which means that BVp1[a,b]⊆BVp2[a,b]. Specially, for 0<p<1, BVp[a,b]⊆BV1[a,b]≡BV[a,b]. This implies that BVp[a,b] functions are bounded, and the discontinuities of a BVp[a,b] function are simple and, therefore, at most denumerable (see [8, Theorem 13.7 and Lemma 13.2]). By the Jordan decomposition theorem, we know that every function f in BV[a,b] can be expressed as the difference of two increasing functions g and h defined on [a,b] (see [8, Corollary 13.6]). If f∈BVp[a,b]⊆BV[a,b], we can require that the above increasing functions g and h are still in BVp[a,b]. This is our next theorem.Theorem 2 (Jordan type decomposition theorem).
Any function inBVp[a,b]  (0<p<1) can be expressed as the difference of two increasing functions in BVp[a,b].Lett∈[a,b], d>0, and 0≤d′≤d. We set(6)ht,d,d′x=0,x<t,d′,x=t,d,x>t.Then ht,d,d′(x) is increasing on [a,b] with only one discontinuity point t. Also, (ht,d,d′x)′=0 for x≠t.Letf be an increasing function in BVp[a,b](0<p<1). Denote by A≡A(f) the set of points of discontinuity of f. Then A is at most countable (see [8, Theorem 2.17]). Since f is increasing, we get that, for any t∈A, the right and left limits f(t+0) and f(t-0) of the function f at t exist, f(t+0)-f(t-0)>0, and 0≤f(t)-f(t-0)≤f(t+0)-f(t-0). For t∈A, we define(7)ht~(x)≡ht,f~(x)=ht,f(t+0)-f(t-0),f(t)-f(t-0)(x).Our next theorem characterizes the form of an increasing function inBVp[a,b]. Any increasing function f in BVp[a,b] must be as follows:(8)fx=∑n=1N‍htn,dn,dn′x+c,where N≤∞, tn∈[a,b], dn>0, dn′∈[0,dn], and ∑n=1N‍dnp<∞.Theorem 3.
(1) Iff(x)=c+∑n=1N‍htn,dn,dn′(x), where N≤∞, tn∈[a,b],  dn>0, and dn′∈[0,dn], then f∈BVp[a,b]  (0<p<1) if and only if ∑n=1N‍dnp<∞. In this case,(9)∑n=1N‍dnp1/p≤Vpf≤2∑n=1N‍dnp1/p.
(2) Letf be an increasing function in BVp[a,b](0<p<1). Then f(x)=∑t∈A‍ht~(x)+c, where c is a constant, A is the set of points of discontinuity of f, and ht~(x) is defined by (7).Finally, for an increasing functionf in BVp[a,b]  (0<p<1), by Theorem 3 we have f(x)=∑t∈A‍ht~(x)+c, where A is the set of points of discontinuity of f and at most countable. Since (ht~x)′=0, a.e. x∈[a,b], by the Fubini term by term differentiation theorem (see [9, Proposition 4.6]), we get f′x=0, a.e. x∈[a,b]. By Theorem 2, any function f in BVp[a,b] can be expressed as the difference of two increasing functions g(x) and r(x) in BVp[a,b]. Applying Theorem 3, we get the representation theorem of functions in BVp[a,b]  (0<p<1) as follows.Corollary 4.
Letf∈BVp[a,b]  (0<p<1). Then f can be expressed in the following form:(10)fx=gx-rx=∑t∈A1‍ht,g~x-∑t∈A2‍ht,r~x+c,where c is a constant, g(x), r(x) are increasing functions in BVp[a,b], ht,g~(x) and ht,r~ are defined by (7), A1,A2⊆A, and A1, A2, A are the sets of points of discontinuity of g, r, and f, respectively. Furthermore, f′(x)=0, a.e. x∈[a,b].


## 3. Proofs of Theorems1–3

Proof of Theorem1.
It suffices to prove thatBVp[a,b] is complete. Let {fn} be a Cauchy sequence in BVp[a,b]; that is, q(fn-fm)=|fn(a)-fm(a)|p+(Vp(fn-fm))p→0 as n,m→∞. For any ξ∈[a,b], using the partition T:a≤ξ≤b and the definition of Vpf, we get that {fn(ξ)} is a Cauchy sequence in R and converges to a number denoted by f(ξ). For any ɛ>0, there exists an integer N such that q(fn-fm)≤ɛ for m,n>N. Let T:a=x0<x1<⋯<xk=b be an arbitrary partition of [a,b]. Then(11)fma-fnap+∑i=1k‍fm-fnxi-fm-fnxi-1p≤qfn-fm≤ɛ.Letting m→∞, we get that(12)fa-fnap+∑i=1k‍f-fnxi-f-fnxi-1p≤ɛ.Taking the supremum over all partitions T, we have q(f-fn)≤ɛ for n>N. This means that f=(f-fn)+fn∈BVp[a,b], and q(f-fn)→0 as n→∞. Hence, BVp[a,b]  (0<p<1) is complete. Theorem 1 is proved.Proof of Theorem2.
Suppose thatf∈BVp[a,b]  (0<p<1). Since f∈BVp[a,b]⊂BV[a,b], by the Jordan decomposition theorem (see [8, Corollary 13.6]), we have f(x)=g(x)-r(x), where g(x), r(x) are increasing functions on [a,b]. Indeed, we can choose g(x) to be Vax(f), the total variation function of f defined by(13)Vaxf=sup⁡T∑i=1n‍fxi-fxi-1,where the supremum is taken over all partitions T:a=x0<x1<⋯<xn=x of [a,x], r(x)=Vax(f)-f(x). It suffices to show that g(x)=Vax(f)∈BVp[a,b]. For any fixed partition T:a=x0<x1<⋯<xn=b, we note that(14)gxi-gxi-1p=Vxi-1xifp=sup⁡Ti∑j=1mi‍fξi,j-fξi,j-1p≤sup⁡Ti∑j=1mi‍fξi,j-fξi,j-1p,where the supremum is taken over all partitions Ti:xi-1=ξi,1<ξi,2<⋯<ξi,mi=xi of [xi-1,xi]. It follows that(15)∑i=1n‍gxi-gxi-1p≤∑i=1n‍sup⁡Ti∑j=1mi‍fξi,j-fξi,j-1p=sup⁡Ti,1≤i≤n∑i=1n‍∑j=1mi‍fξi,j-fξi,j-1p≤Vpfp,which implies g∈BVp[a,b]. This completes the proof of Theorem 2.To prove Theorem3, we introduce the next lemma.Lemma 5.
Iff∈BVp[a,b]⋂‍C[a,b]  (0<p<1), then f is a constant function.Proof.
It suffices to show that, for anyd∈[a,b], f(d)=f(a). Assume that there exists d∈(a,b] such that f(d)≠f(a). Without loss of generality, we assume that f(a)<f(d). Since f∈C[a,b], there exist n-1 points ξ1,ξ2,…,ξn-1 such that a=ξ0<ξ1<⋯<ξn-1<ξn=d and f(ξi)=f(a)+((f(d)-f(a))/n)i. Hence,(16)Vpfp≥∑i=1n‍fξi-fξi-1p=n1-pfd-fap⟶∞,as n→∞, which implies that f∉BVp[a,b]. This leads to a contradiction. Lemma 5 is proved.Proof of Theorem3.
(1) Without loss of generality, we may assume thatN=∞. Let T:a=y0<y1<⋯<ym=b be a partition of [a,b]. For j, 1≤j≤m, we note that (17)fyj-fyj-1p=∑n=1∞‍htn,dn,dn′yj-htn,dn,dn′yj-1p=∑nyj-1<tn<yjdn+∑ntn=yj-1dn-dn′+∑ntn=yjdn′p≤∑nyj-1≤tn≤yjdnp,where an empty sum denotes 0. It follows that(18)∑j=1m‍fyj-fyj-1p≤∑j=1m‍∑nyj-1≤tn≤yjdnp≤2∑n=1∞‍dnp.Taking the supremum over all partitions of [a,b], we obtain that(19)Vpfp≤2∑n=1∞‍dnp.
On the other hand, for any fixedm, by renumbering {tn}n=1m if necessary, we may assume that a≤t1<t2<⋯<tm≤b. We set yi=((ti+ti+1)/2)  (1≤i≤m-1). Then T:a=y0<y1<y2<⋯<ym-1<ym=b is a partition of [a,b]. It follows that(20)Vpfp≥∑j=1m‍fyj-fyj-1p≥∑j=1m‍∑nyj-1<tn<yjdnp≥∑j=1m‍djp.Letting m→∞, we get(21)Vpf≥∑n=1∞‍dnp1/p.Combining (19) with (21), we get (9). Hence, f∈BVp[a,b]  (0<p<1) if and only if ∑n=1∞‍dnp<∞.
(2) Letf be an increasing function in BVp[a,b]  (0<p<1) and A the set of points of discontinuity of f on [a,b]. We set hf(x)=∑t∈A‍ht~(x), where ht~(x) is defined by (7). Similar to the proof of (21), we have(22)∑t∈A‍ft+0-ft-0p≤Vpfp<∞.Applying the above proved result, we obtain that hf(x)∈BVp[a,b]. We set g(x)=f(x)-hf(x); then g∈BVp[a,b]. We will show that g(x) is continuous on [a,b].
Indeed, forx∈[a,b], we have(23)∑t∈A‍ht~x≤∑t∈A‍ft+0-ft-0≤∑t∈A‍ft+0-ft-0p1/p≤Vpf<∞.By Weierstrass M-test (see [10, Theorem 7.10]), we get that the series ∑t∈A‍ht~(x) converges uniformly on [a,b]. For x0∈[a,b]∖A, ht~(x)  (t∈A) is continuous at x0, so hf(x)=∑t∈A‍ht~(x) is also continuous at x0. It follows that g(x) is continuous at x0 for x0∈[a,b]∖A.
Forx0∈A, we set u(x)=∑t∈A∖{x0}‍ht~(x). Then u(x) is continuous at x0 and hf(x)=u(x)+hx0~(x). Hence,(24)hfx0+0=ux0+fx0+0-fx0-0,hf(x0-0)=u(x0),hfx0=ux0+fx0-fx0-0.Thus,(25)gx0+0=gx0=gx0-0=fx0-0-ux0,from which we can deduce that g is continuous at x0. Hence, g(x)∈C[a,b].
Sinceg(x)∈C[a,b]∩BVp[a,b], it follows from Lemma 5 that g(x) is a constant c. Thus f(x)=hf(x)+c=∑t∈A‍ht~(x)+c. The proof of Theorem 3 is complete.



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