# The Composition Operator and the Space of the Functions of Bounded Variation in Schramm-Korenblum's Sense

Authors: Z. Jesús; O. Mejía; N. Merentes; S. Rivas
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We show that the composition operatorH, associated with h:[a,b]→ℝ, maps the spaces Lip[a,b] on to the space κBVϕa,b of functions of bounded variation in Schramm-Korenblum's sense if and only if h is locally Lipschitz. Also, verify that if the composition operator generated by h:[a,b]×ℝ→ℝ maps this space into itself and is uniformly bounded, then regularization of h is affine in the second variable.


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## Body

## 1. Introduction

The composition operator problem (or COP, for short) refers to determining the conditions on a functionh:ℝ→ℝ, such that the composition operator, associated with the function h, maps a space 𝕏 of functions u:[a,b]→ℝ into itself [1, 2]. There are several spaces where the COP has been resolved. For example, in 1961, Babaev [3] showed that the composition operator H, associated with the function h:ℝ→ℝ, maps the space Lip[a,b] of the Lipschitz functions into itself if and only if h is locally Lipschitz; in 1967, Mukhtarov [4] obtained the same result for the space Lipα[a,b] of the Hölder functions of order α(0<α<1).The first work on the COP in the space of functions of bounded variationBV[a,b] was made by Josephy in 1981 [5]. In 1986, Ciemnoczołowski and Orlicz [6] got the same result for the space of the functions of bounded φ-variation in Wiener’s sense. In 1974, Chaika and Waterman [7] reached a similar result for the space of functions of bounded harmonic variation HBV[a,b]. In the years 1991 and 1995 Merentes showed a similar result for the spaces of absolutely continuous functions AC[a,b] and the space of function of bounded φ-variation in Riesz’s sense RVφ[a,b] (see [8, 9]), and in 1998 Merentes and Rivas achieved the same result when the composition operator maps the space RVp[a,b] of the functions of bounded p-variation in Riesz’s sense (1<p<∞) into the space BV[a,b] [10]. In 2003, Pierce and Waterman solved the COP for the spaces ϕBV[a,b] and ΛBV[a,b] [11]. More recently, in 2011, Appell et al. [1] conclude the same results verifying when the composition operator maps Lip[a,b] into BV[a,b]. Finally, Appell and Merentes verify the same result for the space of functions of bounded κ-variation [12].There exist spaces𝕏 of real functions defined on an interval [a,b], such that H maps 𝕏 into itself and h is not locally Lipschitz. For example, in the case the space of continuous functions C[a,b] it follows from the Tietze-Urysohn theorem that the composition operator H acts from C[a,b] into itself and the function h must be continuous; that is, h does not need to be Lipschitz. A similar result was obtained in the space of regulated functions [13].A first objective of this work is to demonstrate that the composition operator, associated with the functionh, maps the space Lip[a,b] of the Lipschitz functions into the space κBVϕ[a,b] of functions of bounded variation in Schramm-Korenblum’s sense or into the space κBV[a,b] of functions of bounded variation in Korenblum’s sense if and only if h is locally Lipschitz. We also extend this result to function spaces 𝕏,𝕐, such that Lip[a,b]⊂𝕏⊂𝕐, where 𝕐⊂κBV[a,b] or 𝕐⊂BVϕ[a,b].In a seminal article of 1982, Matkowski [14] showed that if the composition operator H, associated with the function h:[a,b]×ℝ→ℝ, maps the space Lip[a,b] of the Lipschitzian functions into itself and is a globally Lipschitzian map, then the function  h has the form
(1)h(t,x)=α(t)x+β(t),t∈[a,b],x∈ℝ,
for some α,β∈Lip[a,b].There are a variety of spaces besidesLip[a,b] that verify this result [15]. The spaces of Banach (𝕏,∥·∥) that fulfill this property are said to satisfy the Matkowski property [1].In 1984, Matkowski and Miś [16] considered the same hypotheses on the operator H for the space BV[a,b] of the function of bounded variation and concluded that (1) is true for the regularization h- of the function h with respect of the first variable; that is,
(2)h-(t,x)=α(t)x+β(t),t∈[a,b],x∈ℝ,
where α,β∈BV-[a,b]. Spaces that satisfy this conditions said to be verified Weak Matkowski Property [1].A second objective of this paper is to show that if functionh(t,·) is continuous in the second variable, for each t∈[a,b], and the composition operator H, associated with the function h, is uniformly bounded, then h satisfies (2).


## 2. Preliminaries

Let[a,b] be a closed interval of the real line ℝ(a,b∈ℝ,a<b). From now on, for a function u:[a,b]→ℝ denote by Lab(u) the Lipschitz constant of u; that is,
(3)Lab(u)=sup{|u(t)-u(s)||s-t|:t,s∈[a,b],s≠t}.ByLip[a,b]={u:[a,b]→ℝ:Lab(u)<∞} we will denote the space of the Lipschitz functions. It is well known that the space Lip[a,b] is a Banach space endowed with the norm
(4)∥u∥Lip:=|u(a)|+Lab(u)(u∈Lip[a,b]).Ever since the notion of a function of bounded variation appeared, it has led to an incredible number of generalizations. In 1881, Jordan [17] introduced the definition of function of bounded variation for a function u:[a,b]→ℝ and showed that these kinds of function can be decomposed as the difference of two monotone functions. As a consequence of this result we have that those functions satisfy the Dirichlet criterion, that is, the functions that have pointwise convergent Fourier series.Jordan defined such functions in the following way.Definition 1.
Letu:[a,b]→ℝ and π:a≤t1<⋯<tn≤b be a partition of the interval [a,b]. Consider
(5)σ(u,π):=∑j=1n-1‍|u(tj+1)-u(tj)|,V(u;[a,b])=supπσ(u,π),
where the supremum is taken over all partitions π of the interval [a,b].
IfV(u;[a,b])<∞, then u has bounded variation on the interval [a,b] and this number is called the variation in Jordan’s sense on [a,b]. This space of function is denoted by BV[a,b].The concept of bounded variation has been the subject of intensive research, and many applications, generalizations, and improvements of them can be found in the literature (see for instance [18–20]). Some generalizations have been introduced by De La Vallée Poussin, F. Riesz, N. Wiener, L. C. Young, Yu. T. Medvedev, D. Waterman, B. Korenblum and M. Schramm.In 1975, Korenblum in [21] considered a new kind of variation, called κ-variation, introducing a function κ for distorting the expression |tj-tj-1| in the partition itself rather than the expression |f(tj)-f(tj-1)| in the range. Subsequently, this class of functions has been studied in detail by Cyphert and Kelingos [22]. One advantage of this alternate approach is that a function of bounded κ-variation may be decomposed into the difference of two simpler functions called κ-decreasing functions (for the precise definition see the following).Definition 2.
A functionκ:[0,1]→[0,1] is said to be a κ-function or distortion function if it satisfies the following properties:(1)
κis continuous with κ(0)=0 and κ(1)=1,(2)
κ is concave, increasing, and(3)
limt→0+(κ(t)/t)=∞.Simple examples of distortion functions are(6)κ(t)=tα(0<α<1),κ(t)=t(1-logt).From Definition2 we can see that κ is subadditive; that is,
(7)κ(s+t)≤κ(s)+κ(t)(0≤s,t≤1),
and since limt→0+(κ(t)/t)=∞, then without loss of generality we can assume that
(8)κ(t)≥t(t∈[0,1]).Furthermore Korenblum introduces the following concept of variation.Definition 3.
Letκ:[0,1]→[0,1] be a distortion function and u:[a,b]→ℝ and π:a≤t1<⋯<tn≤b a partition of the interval [a,b]. Consider
(9)κ(u,π):=∑j=1n-1|u(tj+1)-u(tj)|∑j=1n-1κ((tj+1-tj)/(b-a)),κV(u)=κV(u;[a,b]):=supπκ(u,π),
where the supremum is taken over all partitions π of the interval [a,b], called the κ-variation of u on [a,b]. In the case, κV(u;[a,b])<∞ we say that u has κ-variation on [a,b] and we will denote by κBV[a,b] the space of functions of κ-variation on [a,b].In the case that[a,b]=[0,1] the equality (9) becomes
(10)κ(u,π):=∑j=1n-1|u(tj+1)-u(tj)|∑j=1n-1κ(tj+1-tj).Some properties of the functions with boundedκ-variation are summarized in the following theorem.Theorem 4 (see [22]).
Letκ:[0,1]→[0,1] be a distortion function. Then(1)
κBV[a,b] is a Banach space endowed with the norm
(11)∥u∥κ=|u(a)|+κV(u;[a,b])(u∈κBV[a,b]).(2)
If the functionu is monotone, then κV(u;[a,b])=|u(b)-u(a)|.(3)
Ifu∈κBV[a,b], then u is bounded and ∥u∥∞≤2∥u∥κ.(4)
Lip[a,b]⊂BV[a,b]⊂κBV[a,b]⊂R[a,b], where R[a,b] denotes the space of regulated functions.(5)
Ifu∈κBV[a,b], then u can be decomposed as difference of two κ-decreasing functions, that is, there exist functions v:[a,b]→ℝ such that sups<t((v(t)-v(s))/κ((t-s)/(b-a)))<∞.It is easy to show that iflimt→0(k(t)/t) is finite, then κBV[a,b]=BV[a,b]. In [1] it is shown that inclusions (4) of Theorem 4 are strict.Throughout this paper aφ-function φ is a continuous increasing function φ:[0,∞)→ℝ, such that φ(0)=0 and limt→∞φ(t)=∞.Aϕ-sequence is a sequence of decreasing ϕ={φn}n≥1 of convex φ-function that satisfies ∑n≥1‍φn(t) diverge for t>0.We denote byFℕ[a,b] the collection of finite or numerable family of nonoverlapping interval {[an,bn]}n≥1, such that ⋃n≥1‍[an,bn]=[a,b].In 1985, Schramm [23] introduced a new concept of variation as follows.Definition 5.
Letϕ={φn}n≥1 be a ϕ-sequence, {In=[an,bn]}n≥1∈Fℕ[a,b], and u:[a,b]→ℝ. We define
(12)σϕ(u,In):=∑n=1∞φn(|u(bn)-u(an)|),Vϕ(u)=Vϕ(u;[a,b]):=supIn∈Fℕ[a,b]σϕ(u,In).
If Vϕ(u;[a,b])<∞, we say that u has bounded ϕ-variation in the interval [a,b] and this number denotes the ϕ-variation of u in Schramm’s sense in [a,b]. The class of functions that have bounded ϕ-variation in the interval [a,b] is denoted by Vϕ[a,b]. The vectorial space generated by this class is denoted by BVϕ[a,b].The next lemma is useful for building the space generated by several classes of functions.Lemma 6.
Let𝕏 be a vector space and A⊂𝕏 a nonempty and symmetric set. Then (1)
0∈A.(2)
The vector space generated forA is equal to
(13)〈A〉={x∈𝕏:∃λ>0suchthatλx∈A}=⋃λ>0‍λA.Some properties of functions of boundedϕ-variation in Schramm’s sense are given in the following theorem.Theorem 7 (see [23]).
Letϕ={φn}n≥1 be ϕ-sequence then (1)
BVϕ[a,b] is a Banach space endowed with the norm
(14)∥u∥ϕ:=|u(a)|+μ(u)(u∈BVϕ[a,b]),whereμ(u):=infλ>0{λ>0:Vϕ(u/λ)≤1}.(2)
Ifu is monotone, then Vϕ(u)=φ1(|u(b)-u(a)|).(3)
Ifu∈Vϕ[a,b], then u is bounded and ∥u∥∞≤|u(a)|+2φ1-1(Vϕ(u)).(4)
Vϕ[a,b] is a symmetrical and convex set.(5)
BVϕ[a,b]={u:[a,b]→ℝ:∃λ>0suchthatVϕ(λu)<∞}.(6)
u∈Vϕ[a,b], then u has lateral limits at each point of [a,b].In 1986, S. K. Kim and J. Kim [24] combined the concepts of κ-variation and ϕ-variation introduced by Korenblum and Schramm to create the concept of κϕ-variation or variation in Schramm-Korenblum’s sense.Definition 8.
Letκ:[0,1]→[0,1] be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, {In=[an,bn]}n≥1∈Fℕ[a,b], and u:[a,b]→ℝ. We define
(15)κσϕ(u,In):=∑n=1∞φn(|u(bn)-u(an)|)∑n=1∞κ((bn-an)/(b-a)),κVϕ(u)=κVϕ(u;[a,b]):=supIn∈Fℕ[a,b]κσϕ(u,In).
If κVϕ(u;[a,b])<∞, we say that u has bounded κϕ-variation in the interval [a,b] and this number denotes the κϕ-variation of u in Schramm-Korenblum’s sense in [a,b]. The class of functions that have bounded κϕ-variation in the interval [a,b] is denoted by κVϕ[a,b]. The vectorial space generated by this class is denoted by κBVϕ[a,b].A particular case ofϕ-sequence is when all the functions φn,n∈ℕ are equal to a fixed φ-function φ. In this situation the class κVφ[a,b] is the class of the functions that have bounded κφ-variation in Wiener-Korenblum’s sense. This class of functions is denoted by κVφ[a,b] and the vectorial space generated by this class of function is denoted by κBVφ[a,b].Some properties of functions of boundedκϕ-variation in Schramm-Korenblum’s sense are given in the following theorem.Theorem 9.
Letκ:[0,1]→[0,1] be a distortion function and ϕ={φn}n≥1 a ϕ-sequence, then (1)
κBVϕ[a,b] is a Banach space endowed with the norm
(16)∥u∥κϕ:=|u(a)|+μϕ(u)(u∈κBVϕ[a,b]),whereμϕ(u):=infλ>0{λ>0:κVϕ(u/λ)≤1}.(2)
Ifu is monotone, then κVϕ(u)=φ1(|u(b)-u(a)|).(3)
Ifu∈κVϕ[a,b], then u is bounded and ∥u∥∞≤|u(a)|+2φ1-1(κVϕ(u)).(4)
κVϕ[a,b] is a symmetrical and convex set.(5)
κBVϕ[a,b]={u:[a,b]→ℝ:∃λ>0such thatκVϕ(λu)<∞}.(6)
BV[a,b]⊂Vϕ[a,b]⊂κVϕ[a,b], and therefore BV[a,b]⊂BVϕ[a,b]⊂κBVϕ[a,b].(7)
κBV[a,b]⊂κBVϕ[a,b].(8)
u∈κVϕ[a,b], then u has lateral limits at each point of [a,b].ProofPart(1). See [24].Part(2). We take {[an,bn]}n≥1∈Fℕ[a,b], then since the functions 0<t→φn(t)/t are increasing, we obtain
(17)∑n=1∞φn(|u(bn)-u(an)|)∑n=1∞κ((bn-an)/(b-a))=∑n=1∞φn(|u(bn)-u(an)|)|u(bn)-u(an)||u(bn)-u(an)|×(∑n=1∞κ(bn-anb-a))-1≤∑n=1∞φn(|u(b)-u(a)|)|u(b)-u(a)||u(bn)-u(an)|×(∑n=1∞κ(bn-anb-a))-1≤φ1(|u(b)-u(a)|)|u(b)-u(a)|∑n=1∞‍|u(bn)-u(an)|=φ1(|u(b)-u(a)|),from which is obtained(18)κVϕ(u)≤φ1(|u(b)-u(a)|).From the Definition ofkVϕ(u) we have the reciprocal inequality.Part (3). We consider a<t≤b, then
(19)φ1(|u(t)-u(a)|)κ((t-a)/(b-a))+κ((b-t)/(b-a))≤κVϕ(u).
Then φ1(|u(t)-u(a)|)≤2κVϕ(u) and from this inequality we have the required relation.Part (4). We get Part (4) by the convexity of functions φn,n∈ℕ and the definition of κϕ-variation.Part (5). It follows from part (4) and Lemma 6.Part (6). Let u∈BV[a,b] and consider {[an,bn]}n≥1∈Fℕ[a,b]. Define
(20)A:={n∈ℕ:|u(bn)-u(an)|≤1}.Then(21)∑n=1∞φn(|u(bn)-u(an)|)≤φ1(1)∑n∈A‍|u(bn)-u(an)|+∑n∉A‍φn(|u(bn)-u(an)|).The last sum has at most[V(u)] terms, where [x]=max{n:n≤x}. Because otherwise it has at least [V(u)]+1 summands.Accordingly(22)V(u)≥∑n∉A‍|u(bn)-u(an)|≥[V(u)]+1.Which is a contradiction. Therefore(23)∑n=1∞φn(|u(bn)-u(an)|)≤φ1(1)V(u)+∑n∉A‍φn(|u(bn)-u(an)|)≤φ1(1)V(u)+φ1(2∥u∥∞)[V(u)].This concludes thatBV[a,b]⊂Vϕ[a,b].Let us show thatVϕ[a,b]⊂κVϕ[a,b]. In Fact let u∈Vϕ[a,b] and {[an,bn]}n≥1∈Fℕ[a,b], then
(24)∑n=1∞‍φn(|u(bn)-u(an)|)≤Vϕ(u)κ(1)≤Vϕ(u)κ(∑n=1∞‍bn-anb-a).Hence, we get thatκVϕ(u)≤Vϕ(u).Part (7). Let u∈κBV[a,b], then by part (3) u is bounded in [a,b]. Let us fix λ>0, such that ∥λu∥∞≤1/2. Let {[an,bn]}n≥1∈Fℕ[a,b], then from the convexity of the functions φn,n∈ℕ, we have
(25)∑n=1∞φn(|λu(bn)-λu(an)|)∑n=1∞κ((bn-an)/(b-a))≤∑n=1∞φn(1)λ|u(bn)-u(an)|∑n=1∞κ((bn-an)/(b-a))≤φ1(1)λκV(u).Thus Lemma6 concludes that u∈κBVϕ[a,b].Part (8). Suppose that there is t*∈(a,b] such that limt↑t*u(t) does not exist.By part (3)u is bounded then
(26)α=limt↑t*infu(t)<limt↑t*supu(t)=β,(α,β∈ℝ).For each integern (large enough) we can choose tn,tn′ such that
(27)t*-1n<tn<tn′<t*,|u(tn)-u(tn′)|≥β-α.Using the definition ofκϕ-variation, we have
(28)φ1(|u(tn)-u(tn′)|)×(κ(tn-ab-a)+κ(tn′-tnb-a)+κ(b-tn′b-a))-1≤κVϕ(u).Therefore,(29)φ1(β-α)≤κVϕ(u)(2+κ(1n(b-a))).By taking limit whenn→∞, we obtain φ1(β-α)=0, which is absurd. From each it follows that
(30)limt↑t*infu(t)=limt↑t*supu(t).By a similar argument it follows that there existlimt↓t*u(t),t*∈[a,b).Assuming thatlimt→0(k(t)/t) is finite, then Vϕ[a,b]=κV[a,b]. For the last part of Theorem 9, we can give the definition of left and right regularizations of the function u∈κVϕ[a,b].Definition 10.
Letu∈κBVϕ[a,b], then
(31)u-(t):={lims↑tu(s),t∈(a,b]u(a),t=a,u+(t):={lims↓tu(s),t∈[a,b)u(b),t=b.The functionu- is called the left regularization of the function u and the function u+ the right regularization of the function u.Applying the previous definition and the last part of Theorem9, we can define
(32)κBVϕ-[a,b]:={u-:u∈κBVϕ[a,b]}.Similarly, we definedκBVϕ+[a,b].Recently Castillo et al. [25] introduced the concept of κp-variation in Riesz-Korenblum’s sense in the following way.Definition 11.
Let1<p<∞, κ:[0,1]→[0,1] be a distortion function and u:[a,b]→ℝ and π:a≤t1<⋯<tn≤b a partition of the interval [a,b]. We define
(33)κVpR(u)=κVpR(u;[a,b]):=supπκσp(u,π),
where
(34)κσp(u,π):=(∑j=1n-1|u(tj+1)-u(tj)|p|tj+1-tj|p-1)×(∑j=1n-1κ(tj+1-tjb-a))-1
and the supremum is taken on the set of all partitions of [a,b]. If κVpR(u;[a,b])<∞, we say that u has bounded κp-variation in the interval [a,b]. The number κVpR(u;[a,b]) denoted the κp-variation of u in Riesz-Korenblum’s sense in [a,b]. The space of functions that have bounded κp-variation in the interval [a,b] is denoted by κRVp[a,b].Some properties of these functions are exposed in the following theorem.Theorem 12 (see [25]).
Let1<p<∞ and let κ:[0,1]→[0,1] be a distortion function, then (1)
κRV[a,b] is a Banach space endowed with the norm
(35)∥u∥κp:=|u(a)|+(κVpR(u))1/p(u∈κRVp[a,b]).(2)
Lip[a,b]⊂RVp[a,b]⊂κRVp[a,b]⊂κBV[a,b], where RVp[a,b] denote the space of the functions u that have bounded p-variation in Riesz’s sense [20].(3)
κRVp[a,b] is an algebra.This concept was generalized by Castillo et al. [26] as stated in the following definition.Definition 13.
Letφ be a φ-function, κ:[0,1]→[0,1] be a distortion function, and u:[a,b]→ℝ and π:a≤t1<⋯<tn≤b a partition of the interval [a,b]. We define
(36)κVφR(u)=κVφR(u;[a,b]):=supπκσφ(u,π),
where
(37)κσφ(u,π):=∑j=1n-1φ(|u(tj+1)-u(tj)|/|tj+1-tj|)|tj+1-tj|∑j=1n-1κ((tj+1-tj)/(b-a))
and the supremum is taken over all partitions of [a,b]. If κVφR(u;[a,b])<∞, we say that u has bounded κφ-variation in the interval [a,b] and this number denotes the κφ-variation of u in Riesz-Korenblum’s sense in [a,b]. The class of functions that have bounded κφ-variation in the interval [a,b] is denoted by κVφ[a,b]. The vectorial space generate by this class is denoted by κRVφ[a,b].The space of all functions that have boundedκφ-variation on [a,b] is denoted by κRVφ[a,b]. Some properties of these functions are exposed in the following theorem.Theorem 14 (see [26]).
Letφ be a convex φ-function and κ:[0,1]→[0,1] a distortion function, then (1)
Lip[a,b]⊂κVφR[a,b]⊂κBV[a,b].(2)
RVφ[a,b]⊂κRVφ[a,b]⊂κBV[a,b], where RVφ[a,b] denote the space generated by the class of functions of bounded φ-variation in Riesz’s sense [20].(3)
Iflimt→∞(φ(t)/t)<∞, then κVφR[a,b]=κBV[a,b].(4)
Ifu∈κVφR[a,b], then u is bounded.(5)
κVφR[a,b] is a convex and symmetric set.(6)
κRVφ[a,b]={u:[a,b]→ℝ:∃λ>0 such that  κVϕR(λu)<∞}.(7)
κRVφ[a,b] is a Banach space endowed with the norm
(38)∥u∥κφ:=|u(a)|+μ(u)(u∈κRVφ[a,b]),whereμφ(u):=infλ>0{λ>0:κVφR(u/λ)≤1}.


## 3. Composition Operator betweenLip[a,b] and κBV[a,b] or κBVϕ[a,b]

Given a functionh:ℝ→ℝ, the composition operator H, associated to the function h (case autonomous), maps each function u:[a,b]→ℝ into the composition function Hu:[a,b]→ℝ defined by
(39)Hu(t):=h(u(t)),(t∈[a,b]).More generally, givenh:[a,b]×ℝ→ℝ, we consider operator H defined by
(40)Hu(t):=h(t,u(t)),(t∈[a,b]).This operator is also calledsuperposition operator or substitution operator or Nemytskii operator. In what follows, will refer to (39) as the autonomous case and to (40) as the nonautonomous case.A problem related with this operator is to establish necessary and sufficient conditions of functionh so that the operators H map the space 𝕏 of real functions defined on [a,b] into itself, that is, H(𝕏)⊂𝕏, or in more general way that operator H maps the space 𝕏 into space of functions 𝕐(H(𝕏)⊂𝕐). This problem is sometimes referred to as the composition operator problem (or COP). The solution to this problem for given 𝕏 is sometimes very easy and sometimes highly nontrivial. As we mentioned in the introduction of this paper in a variety of spaces the required condition is that function h is locally Lipschitz. Another interesting problem is to determine the smallest space of functions 𝕏 and the bigger space 𝕐 such that H(𝕏)⊂𝕐.In order to obtain the main result of this section, we will use a function of the zig-zig type such as the employed by Appell et al. in [1, 15]. In this section we will show that the locally Lipschitz condition of the function h is a necessary and sufficient condition such that H(Lip[a,b])⊂κBV[a,b] and that in this situation H is bounded.The following lemma will be useful in the proof of our main theorem (Theorem17).Lemma 15.
Letu:[a,b]→ℝ, a≤s<η<t≤b, then
(41)|u(t)-u(s)|t-s≤|u(η)-u(s)|η-s+|u(t)-u(η)|t-η.Proof.
Leta≤s<η<t≤b. Then
(42)|u(t)-u(s)|t-s≤|u(η)-u(s)|t-s+|u(t)-u(η)|t-s=|u(η)-u(s)|η-s(η-st-s)+|u(t)-u(η)|t-η(t-ηt-s)≤|u(η)-u(s)|η-s+|u(t)-u(η)|t-η.Lemma 16.
Letκ:[0,1]→[0,1] be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, u∈κBVϕ[a,b], and λ>0. Then μϕ(u)<λ if and only if κVϕ(u/λ)<1.Proof.
Letu∈κBVϕ[a,b]. Suppose that μϕ(u)<λ; then, by definition of μϕ(u) there exists k such that λ>k>μϕ(u) and κVϕ(u/k)≤1. Hence, by the convexity of the functions φn, we have
(43)κVϕ(uλ)=κVϕ(ukkλ)≤kλκVϕ(uk)≤kλ≤1.
Conversely, assumeκVϕ(u/λ)<1, then λ∈{λ>0:κVϕ(u/λ)≤1}; hence μϕ(u)<λ.Theorem 17.
Letκ:[0,1]→[0,1] be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, h:ℝ→ℝ, and H the composition operator associated to h. H maps the space Lip[0,1] into the space κBVϕ[0,1] or κBV[0,1] if and only if h is locally Lipschitz. Furthermore operator H is bounded.Proof.
Letu∈Lip[0,1], r=∥u∥∞ and suppose that h is locally Lipschitz, then there exist k=k(r), such that
(44)|h(t)-h(s)|≤k(r)|s-t|,(s,t∈ℝ,|s|≤r,|t|≤r).
Let{[an,bn]}n≥1∈Fℕ[a,b] and λ>0, such that λ<1/(2k(r)∥u∥Lip∥u∥∞+1), then
(45)∑n=1∞φn(λ|h(u(bn))-h(u(an))|)∑n=1∞κ((bn-an)/(b-a))≤∑n=1∞φn(λk(r)|u(bn)-u(an)|)κ(1)≤λk(r)φ1(1)∥u∥Lip∑n=1∞‍(bn-an)<∞.
Then by Lemma15 and Theorem 9, we have H(u)∈κBVϕ[a,b].
The proof of the only if direction will be by contradiction, that is, we assumeH(Lip[0,1])⊂κBVϕ[0,1] and h is not locally Lipschitz. Since the identity function Id:[0,1]→[0,1] belongs to Lip[0,1], then h∘Id∈κBVϕ[0,1] and therefore h is bounded in the interval [0,1]. Without loss of generality we may assume that
(46)∥h∣[0,1]∥∞≤14.
Sinceh is not locally Lipschitz in ℝ, there is a closed interval I such that h does not satisfy any Lipschitz condition. In order to simplify the proof we can assume that I=[0,1]. In this way for any increasing sequence of positive real numbers {kn}n≥1 that converge to infinite that we will define later, we can choose sequences {an}n≥1, {bn}n≥1, such that
(47)|h(bn)-h(an)|>kn|bn-an|,(n∈ℕ).
In addition we can choosean,bn such that
(48)an<bn,(n∈ℕ).
Considering subsequences if necessary, we can assume that the sequence{an}n≥1 is monotone. We can assume without loss of generality that sequence {an}n≥1 is increasing.
Since[0,1] is compact, from inequality (47) we have that there exist subsequences of {an}n≥1 and {bn}n≥1 that we will denote in the same way, and that converge to a∞∈[0,1].
Since the sequence{an}n≥1 is a Cauchy sequence, we can assume (taking subsequence if necessary) that
(49)|am-an|<1kn,(m>n).
Again considering subsequences if needed using the properties of the functionκ we can assume that
(50)max{κ(bn-an),κ(am-an)}<1kn,(n∈ℕ,m≥n).
Consider the new sequence{mn}n≥1 defined by
(51)mn:=1kn(bn-an),(n∈ℕ).
From inequalities (46) and (47) it follows that mn>2; therefore
(52)mn2<[mn]≤mn,(n∈ℕ).
Consider the sequence defined recursively{tn}n≥1 by
(53)t1:=0,tn+1:=tn+an+1-an+2[mn](bn-an),(n∈ℕ).
This sequence is strictly increasing and from the relations (49) and (50), we get
(54)tn⟶t∞:=∑‍(tn+1-tn)=∑n=1∞‍(an+1-an)+2∑n=1∞‍[mn](bn-an)≤3∑n=1∞‍1kn.
Then to ensure thatt∞∈[0,1], it is sufficient to suppose that ∑n=1∞(1/kn)≤1/3.
We define the continuous zig-zag functionu:[0,1]→ℝ, as shown in the following:
(55)u(t):={an,t=tn+2i(bn-an),i=0,…,[mn]bn,t=tn+(2i+1)(bn-an),i=0,…,[mn]-1a∞,t∞≤t≤1affine,othercase.
Put(56)tn,i:=tn+i(bn-an),n∈ℕ,i=0,…,2[mn].
We can write each intervalIn=[tn,tn+1],n∈ℕ, as the union of the family of nonoverlapping intervals
(57)In,i:=[tn.i,tn,i+1],i=0,…,[mn]-1,In,2[mn]:=[tn,2[mn],tn+1].
And functionu is defined on In,i,i=0,…,2[mn] as follows:
(58)u(t)=t-(tn+2i(bn-an))+an,(t∈In,2i),(59)u(t)=t-tn+(2i+1)(bn-an)+bn,(t∈In,2i+1),(60)u(t)=t-tn+1+an,(t∈In,2[m]).
In all these situations, the slopes of the segments of lines are equal to 1.
Hence, we have forn∈ℕ, the absolute value of the slope of the line segments in these ranges are bounded by 1, as shown below
(61)2-n|bn-an|κ-1(bn-an)≤2-nkn(bn-an)≤1,2-(n+1)an+1-antn+an+1-an≤1.
We will show thatu∈Lip[0,1].
Let0≤s<t≤1, and then there are the following possibilities for the location of s and t on [0,1].
Case
1. If s,t∈In,(n∈ℕ) and are in the same interval In,i,i=0,…,2[mn].
From relations (58), (59), and (60) it follows that |u(t)-u(s)|/|t-s|=1.
Case
2. If s,t∈In,(n∈ℕ) and are in two different intervals In,i,i=0,…,2[mn].
There are several possibilities.(a)
s∈In.i,t∈In,j,i<j<2[mn].
(a1)
j=i+1. By Lemma 15 and relations (58) and (59) we have
(62)|u(t)-u(s)||t-s|≤|u(tn,i+1)-u(s)|tn,i+1-s+|u(t)-u(tn,i+1)|t-tn,i+1≤2.(a2)
j>i+1. Then
(63)|u(t)-u(s)||t-s|≤bn-antn,i+2-tn,,i+1=1.(b)
s∈In.i,t∈In,j,i<j=2[mn].
Ifj=i+1, proceed as (a1).Ifj>i+1, again using the Lemma 15 and relations (58), (59), and (60) we obtain
(64)|u(t)-u(s)||t-s|≤|u(tn,2[mn])-u(s)|tn,2[mn]-s+|u(t)-u(tn,2[mn])|tn,2[mn]-t≤bn-antn,2[mn]-tn,2[mn]-1+1≤2.
Case
3. If s∈In,t∈Im,n,m∈ℕ,n<m.
From Lemma15 and Case 2, we conclude that
(65)|u(t)-u(s)|t-s≤|u(tn+1)-u(s)|tn+1-s+|u(t)-u(tm)|t-tm≤4.
Case
4. If s∈In,n∈ℕ,t=t∞.
Then from Lemma15(66)|u(t∞)-u(s)|t∞-s≤|u(tn,i+1)-u(s)|tn,i+1-s+|a∞-u(tn,i+1)|bn-an≤1+a∞-anbn-an≤2.
Case
5. If s<t∞<t≤1.
From Lemma15 and Case 4,
(67)|u(t)-u(s)|t-s≤|u(t∞)-u(s)|t∞-s≤2.
Case
6. If t∞≤s<t≤1.
In this circumstanceu(s)=u(t)=a∞ and the situation is trivial. Therefore we have that
(68)|u(t)-u(s)|≤|t-s|,(s,t∈[0,1]).
Sou is Lipschitz in [0,1]. Moreover, for each partition of the interval [0,1] of the form
(69)π:0=t1<t1+(b1-a1)<⋯<t1+2[m1](b1-a2)<t2<t2+(b2-a2)<⋯<tk<⋯<tk+2[mk](bk-ak)<1
and c>0, using the inequality (47), convexity of the function φn,n≥1, and definition of mn,n∈ℕ, we have
(70)κVϕ(c(h∘u);[0,1])≥∑n=1k2[mn]φn(c)|h(bn)-h(an)|∑n=1k[2[mn]κ(bn-an)+κ(an+1-an)]≥∑n=1k2[mn]φn(c)kn(bn-an)∑n=1k[2[mn]κ(bn-an)+κ(an+1-an)]≥∑n=1k2[mn]kn(bn-an)φn(c)∑n=1k(1/kn)≥∑n=1kφn(c).
As the series ∑n=1∞φn(c) diverge, ch∘u∉κBVϕ[0,1], which is a contradiction.
Let us see thath∘u∉κBV[a,b]. In fact, as in the case of κ-variation we have
(71)Vκ(h∘u;[0,1])≥∑n=1k‍2[mk]kn(bn-an)≥k.
Therefore,h∘u∉κBV[0,1].
To prove that the operatorH is bounded, let r>0 and u∈Lip[0,1], such that ∥u∥Lip≤r, then from the definition of ∥·∥Lip[0,1], we have
(72)∥u∥∞≤|u(0)|+L01(u)≤r.
Without loss of generality assume thatr=1. As h is locally Lipschitz, there exists k(1)>0, such that
(73)|h(x)-h(y)|≤k(1)|x-y|,|x|≤1,|y|≤1.
As the identity functionId:[0,1]→[0,1] belongs to Lip[0,1], then h∘Id∈κBVϕ[0,1]. By Theorem 9 we have that ∥h∘Id∥∞<∞.
Let{[an,bn]}n≥1∈Fℕ[0,1] and choose λ such that k(1)κVϕ(Id)<λ, then
(74)∑n=1∞φn(|h(u(bn))-h(u(an))|/λ)∑n=1∞κ(bn-an)≤∑n=1∞φn(k(1)|u(bn)-u(an)|λ)≤∑n=1∞φn(k(1)(bn-an)λ)≤∑n=1∞k(1)λφn(bn-an)≤k(1)λκVϕ(Id)<1.
From Lemma6 we have μ(h∘u)<λ(r). Thus we conclude that
(75)∥H(u)∥κϕ=|h(u(a))|+μ(h∘u)≤∥h∘u∥∞+λ(r).
And so operatorH is bounded. In the case that H(Lip[0,1])⊂κBV[0,1], proceed similarly.In the following result we give a Lemma of invariance.Lemma 18.
Letκ be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, and ν:[0,1]→[a,b] affine function that maps [0,1] on [a,b](v(t):=(b-a)t+a,t∈[0,1]). (1)
u∈κBVϕ[a,b] if and only if u∘ν∈κBVϕ[0,1].(2)
u∈κBV[a,b] if and only if u∘ν∈κBVϕ[0,1].Proof.
(1) Letu∈κBVϕ[a,b] and {[an,bn]}n≥1∈Fℕ[0,1], then {[ν(an),ν(bn)]}n≥1∈Fℕ[a,b] and
(76)∑n=1∞|u(ν(bn))-u(ν(an))|∑n=1∞κ(bn-an)=∑n=1∞|u(ν(bn))-u(ν(an))|∑n=1n-1κ((ν(bn)-ν(an))/(b-a))≤κVϕ(u;[a,b]).
Henceu∘ν∈κBVϕ[0,1].
Reciprocally, let us suppose thatu∘ν∈κBVϕ[0,1] and let {[an,bn]}n≥1∈Fℕ[a,b] be a partition of the interval [a,b]. Define
(77)In:=[ν-1(bn),ν-1(an)](n∈ℕ).
We have {In}n≥1∈Fℕ[0,1] and
(78)∑j=1n-1|u(bn)-u(an)|∑n=1∞κ((bn-an)/(b-a))=(∑n=1∞|u(ν((bn-a)/(b-a)))-u(ν((an-a)/(b-a)))|{∑n=1∞|u(ν((bn-a)/(b-a)))})×(∑n=1∞κ((bn-a)/(b-a)-(an-a)/(b-a)){∑n=1∞κ(((bn-a)/(b-a))})-1≤κVϕ(u∘ν;[0,1]),
so u∈κBVϕ[a,b].
(2) The proof is similar from part (1).As consequence of Lemma18 we have the following resultsLemma 19.
Letκ be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, h:ℝ→ℝ, and H the composition operator associated to the function h. (1)
H(Lip[a,b])⊂κBVϕ[a,b] if and only if H(Lip[0,1])⊂κBVϕ[0,1].(2)
H(Lip[a,b])⊂κBV[a,b] if and only if H(Lip[0,1])⊂κBV[0,1].Corollary 20.
Letκ be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, and h:ℝ→ℝ. Then the composition operator H, associated with function h, maps the space Lip[a,b] on κBVϕ[a,b] or on κBV[a,b] if and only if h is locally Lipschitz.Corollary 21.
Letκ be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, h:ℝ→ℝ, and 𝕏,𝕐 normed spaces such that Lip[a,b]⊂𝕏⊂𝕐, where 𝕐⊂κBVϕ[a,b] or 𝕐⊂κBV[a,b]. Then the composition operator H associated with the function h maps the space 𝕏 in the space 𝕐 if and only if h is locally Lipschitz.Some particular cases of Corollary21 are the following(1)
𝕏=𝕐, where 𝕏 is one of the following spaces Lip[a,b] [3], Hα[a,b](0<α<1) [4], BVφ[a,b] [6], BV[a,b] [5], HBV [7], AC[a,b] [8], RVφ[a,b] [9], ϕBV [23], ΛBV [11], and κBV[a,b] [12].(2)
𝕏=RVφ[a,b], 𝕐=BV[a,b]. See [10].(3)
𝕏=Lip[a,b], 𝕐=BV[a,b]. See [1].From Corollary21 we get the following new cases(1)
𝕏 is one of the following spaces: Lip[a,b], Hα(0<α<1), RVφ[a,b],AC[a,b],BV[a,b], BVϕ[a,b], and κRVφ[a,b]; 𝕐 is one of the following spaces: κBV[a,b], κBVϕ[a,b].(2)
𝕏=κRVp[a,b],𝕐=κRVq[a,b], 1<p,q<∞.(3)
𝕏=κBV[a,b], 𝕐=κBVϕ[a,b],More generally(1)
𝕏=κ1BV[a,b],𝕐=κ2RVϕ[a,b];(2)
𝕏=κ1RVφ[a,b],𝕐=κ2RVψ[a,b];(3)
𝕏=κ1RVϕ[a,b],𝕐=κ2RVϕ[a,b];whereκ1,κ2 are distortion functions.


## 4. Uniformly Continuous Composition Operator in the SpaceκBVϕ[a,b]

In many problems solving equation where the composition operator appears to guarantee the existence of solution it is necessary to apply a Fixed Point Theorem. To ensure the application this type of results is necessary to request the condition of global Lipschitz operatorH. In several works Matkowski and Mís have shown that this condition implies that the function h has the form (1) or (2) (see, e.g., [16, 27]). This means that we may apply the Banach contraction mapping principle only if the underlying problems are actually linear and therefore are not interesting.More recently, Matkowski and other researchers have replaced the condition of global Lipschitz by uniform continuity conditions or uniform boundedness composition operator (see e.g., [14]).In this section we present results in this direction for the spaceκBVϕ[a,b].Theorem 22.
Letκ:[0,1]→[0,1] be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, and h:[a,b]×ℝ→ℝ such that the function h(t,·):ℝ→ℝ is continuous with respect to the second variable, for each t∈(a,b]. If the composition operator H, associated with h, maps κBVϕ[a,b] into itself and satisfies the inequality
(79)∥H(u)-H(v)∥κϕ≤γ(∥u-v∥κϕ),(u,v∈κBVϕ[a,b])
then there exists α,β∈κBVϕ-[a,b], such that
(80)h-(t,x)=α(t)x+β(t),(t∈[a,b],x∈ℝ).Proof.
As for eachx∈ℝ fixed the function u(t)=x,t∈[a,b] is in κBVϕ[a,b], then H(u)=h(·,x)∈κBVϕ[a,b]. There is left regularization of h-(·,x).
From inequality (79) and Lemma 16, we get
(81)κVϕ(H(u)-H(v)γ(∥u-v∥κϕ))<1,(u,v∈κBVϕ[a,b]).
Leta<r<a1<b1<a2<⋯<am<bm=s<b,x1,x2∈ℝ,x1≠x2 and put the zig-zag continuous functions uk:[a,b]→ℝ,k=1,2, as(82)uk(t)={xk+x22,a≤t≤a1ort=ai,i=1,…,mxk+x12,bm≤t≤bort=bi,i=1,…,m,k=1,2affine,othercase.
The functionsuk,k=1,2 are Lipschitz and therefore belongs to κBVϕ[a,b]. Furthermore
(83)(u1-u2)(t)=x1-x22,(t∈[a,b]).
From inequality (81) and the definition of kϕ-variation, we have
(84)(∑n=1mφn({(γ(∥u1-u2∥κϕ))-1}(|h(bn,u1(bn))-h(bn,u2(bn))-h(an,u1(an))+h(an,u2(an))|)×(γ(∥u1-u2∥κϕ))-1){∑n=1mφn})×(κ(a1-ab-a)+∑n=1mκ(bn-anb-a)+∑n=1m-1κ(an+1-bnb-a)+κ(b-bmb-a))-1≤1.
By the construction of theuk,k=1,2, we get
(85)(∑n=1mφn((2h(bn,x1)-h(bn,x1+x22)-h(an,x1+x22)+h(an,x2))×(γ(2-1|x1-x2|))-1{(bn,x1+x22)}){∑n=1mφn})×(κ(a1-ab-a)+∑n=1mκ(bn-anb-a)+∑n=1m-1κ(an+1-bnb-a)+κ(b-bmb-a))-1≤1.
Letr tend to s in the above inequality; we obtain
(86)∑n=1mφn(|h-(s,x1)-2h-(s,(x1+x2)/2)+h-(s,x2)|γ(2-1|x1-x2|))≤κ(s-ab-a)+κ(b-sb-a),(s∈(a,b]).
Passing the limit asm→∞,
(87)∑n=1∞φn(|h-(s,x1)-2h-(s,(x1+x2)/2)+h-(s,x2)|γ(2-1|x1-x2|))≤κ(s-ab-a)+κ(b-sb-a),(s∈(a,b]).
As the series∑n=1∞φn(x) is divergent for each x>0, necessarily
(88)|h-(s,x1)-2h-(s,x1+x22)+h-(s,x2)|=0,(s∈(a,b]).
So we conclude thath-(s,·) satisfies the Jensen equation in ℝ (see [28], page 315). The continuity of h with respect of the second variable implies that for every t∈[a,b] there exist α,β:[a,b]→ℝ, such that
(89)h-(t,x)=α(t)x+β(t),(t∈[a,b],x∈ℝ).
Sinceβ(t)=h-(t,0),t∈[a,b],α(t)=h-(t,1)-β(t) and h-(·,x)∈κBVϕ-[a,b], for each x∈ℝ, we obtain that α,β∈κBVϕ[a,b].Corollary 23.
Letκ:[0,1]→[0,1] be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, and h:[a,b]×ℝ→ℝ. If the composition operator H, associated with h, maps κBVϕ[a,b] in itself and satisfies the inequality
(90)∥H(u)-H(v)∥κϕ≤γ(∥u-v∥κϕ),(u,v∈κBVϕ[a,b]),
for any function γ:[0,∞)→[0,∞), verifying γ(t)→γ(0)=0, when t↓0, then there exists α,β∈κBVϕ-[a,b], such that
(91)h-(t,x)=α(t)x+β(t),(t∈[a,b],x∈ℝ).Proof.
Let us fixx,y∈ℝ,x≠y, and define u(t):=x,v(t):=y,t∈[a,b]. Then u,v∈κBVϕ[a,b] and ∥u-v∥κϕ=|x-y|. Then from inequality (90) and Lemma 16, we have
(92)φ1(|h(t,x)-h(t,y)-h(a,x)+h(a,y)|γ(|x-y|))≤κ(t-ab-a)+κ(b-tb-a),(t∈(a,b]).
And therefore(93)|h(t,x)-h(t,y)-h(a,x)+h(a,y)|≤φ1-1(2)γ(|x-y|).
Proceeding as in the proof of Theorem22 we have that there is left regularization of h-(·,x), for each x∈ℝ and if
a
≤
r
<
a
1
<
b
1
<
a
2
<
⋯
<
a
m
<
b
m
=
s
≤
b,  x,y∈ℝ,x≠y we defined the zig-zag continuous functions u,v:[a,b]→ℝ, as show below
(94)u(t)={x,a≤t≤a1ort=ai,i=1,…,my,bm≤t≤bort=bi,i=1,…,maffine,othercase,u(t)={y,a≤t≤a1ort=ai,i=1,…,mx,bm≤t≤bort=bi,i=1,…,maffine,othercase,
thenu,v∈κBVϕ[a,b], (u-v)(t)=x-y,t∈[a,b] and for s∈(a,b](95)φm(2|h-(s,x)-h-(s,y)|γ(|x-y|))≤∑n=1mφn(2|h-(s,x)-h-(s,y)|γ(|x-y|))≤κ(s-ab-a)+κ(b-sb-a)≤2.
Hence,(96)|h-(s,x)-h-(s,y)|≤φm-1(2)2γ(|x-y|),(s∈(a,b]).
And soh-(·,x) is continuous in (a,b]. Now the result is a consequence of Theorem 22.Corollary 24.
Letκ:[0,1]→[0,1] be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, h:[a,b]×ℝ→ℝ, and H the composition operator associated with h. Suppose that H maps κBVϕ[a,b] into itself and is uniformly continuous, then there exist α,β∈κBVϕ-[a,b], such that
(97)h-(t,x)=α(t)x+β(t),(t∈[a,b],x∈ℝ),
where h-(·,x) is the left regularization of h(·,x) for all x∈ℝ.Proof.
Consider the modulus of continuity associated withH; that is,
(98)γ(t):=sup{∥H(u)-H(v)∥κϕ:∥u-v∥κϕ≤t,u,v∈κBVϕ[a,b]},(t≥0).
Thenγ(t)≥0,γ(0)=0. Furthermore, if t>0 and ∥u-v∥κϕ≤t, we obtain
(99)∥H(u)-H(v)∥κϕ≤γ(t).
Particularly, in the case wheret=∥u-v∥κϕ, we have
(100)∥H(u)-H(v)∥κϕ≤γ(∥u-v∥κv),(u,v∈κBVϕ[a,b]).
From Corollary23 we obtain the conclusion.Matkowski [27] introduced the notion of a uniformly bounded operator and proved that the generator of any uniformly bounded composition operator acting between general Lipschitz function normed spaces must be affine with respect to the function variable.Definition 25.
Let𝕐 and ℤ be two metric (or normed) spaces. We say that a mapping F:𝕐→ℤ is uniformly bounded, if for any t>0, there exists a nonnegative real number γ(t) such that for any nonempty set B⊂𝕐 we have
(101)diamB≤t⟹diamF(B)≤γ(t).Remark 26.
Every uniformly continuous operator or Lipschitzian operator is uniformly bounded.Corollary 27.
Letκ:[0,1]→[0,1] be a distortion function, ϕ={φn}n≥1 a ϕ-sequence, and h:[a,b]×ℝ→ℝ, such that function h(t,·):ℝ→ℝ is continuous in respect to the second variable, for each t∈(a,b]. If the composition operator H, associated with h, maps κBVϕ[a,b] in itself and is uniformly bounded, then exists α,β∈κBVϕ-[a,b], such that
(102)h-(t,x)=α(t)x+β(t),(t∈[a,b],x∈ℝ).Proof.
Take anyt>0 and u,v∈κBVϕ[a,b] such that ∥u-v∥κϕ≤t. Since diam{u,v}≤t, by uniform boundedness of H, we have diamH({u,v})≤γ(t); that is,
(103)∥H(u)-H(v)∥κϕ=diamH({u,v})≤γ(∥u-v∥κϕ).
From Theorem22 we have (102).



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