Functions of Generalized Bounded Variation605177/FULLTEXT01.pdf · Such functions are called p-continuous, and the class of all p-continuous func-tions is denoted by C p. For f2Lp([0;1])(1

Functions of Generalized Bounded Variation

Martin Lind

DISSERTATION | Karlstad University Studies | 2013:11

Mathematics

Faculty of Health, Science and Technology



Martin Lind

Distribution:Karlstad University Faculty of Health, Science and TechnologyDepartment of Mathematics and Computer ScienceSE-651 88 Karlstad, Sweden+46 54 700 10 00

© The author

ISBN 978-91-7063-486-4

Print: Universitetstryckeriet, Karlstad 2013

ISSN 1403-8099

Karlstad University Studies | 2013:11

DISSERTATION

Martin Lind


WWW.KAU.SE

ii

AbstractThis thesis is devoted to the study of different generalizations of the

classical conception of a function of bounded variation.First, we study the functions of bounded p-variation introduced by Wiener

in 1924. We obtain estimates of the total p-variation (1 < p <∞) and otherrelated functionals for a periodic function f ∈ Lp([0, 1]) in terms of its Lp-modulus of continuity ω(f ; δ)p. These estimates are sharp for any rate ofdecay of ω(f ; δ)p. Moreover, the constant coefficients in them depend onparameters in an optimal way.

Inspired by these results, we consider the relationship between the Riesztype generalized variation vp,α(f) (1 < p < ∞, 0 ≤ α ≤ 1 − 1/p) andthe modulus of p-continuity ω1−1/p(f ; δ). These functionals generate scalesof spaces that connect the space of functions of bounded p-variation andthe Sobolev space W 1

p . We prove sharp estimates of vp,α(f) in terms ofω1−1/p(f ; δ).

In the same direction, we study relations between moduli of p-continuityand q-continuity for 1 < p < q <∞. We prove an inequality that estimatesω1−1/p(f ; δ) in terms of ω1−1/q(f ; δ). The inequality is sharp for any order ofdecay of ω1−1/q(f ; δ).

Next, we study another generalization of bounded variation: the so-calledbounded Λ-variation, introduced by Waterman in 1972. We investigate re-lations between the space ΛBV of functions of bounded Λ-variation, andclasses of functions defined via integral smoothness properties. In particular,we obtain the necessary and sufficient condition for the embedding of theclass Lip(α; p) into ΛBV . This solves a problem of Wang (2009).

We consider also functions of two variables. Applying our one-dimensionalresult, we obtain sharp estimates of the Hardy-Vitali type p-variation of abivariate function in terms of its mixed modulus of continuity in Lp([0, 1]2).

Further, we investigate Fubini-type properties of the space H(2)p of functions

of bounded Hardy-Vitali p-variation. This leads us to consider the symmetricmixed norm space Vp [Vp ]sym of functions of bounded iterated p-variation.

For p > 1, we prove that H(2)p 6⊂ Vp [Vp ]sym and Vp [Vp ]sym 6⊂ H

(2)p . In

other words, Fubini-type properties completely fail in the class of functionsof bounded Hardy-Vitali type p-variation for p > 1.

iii

Basis of the thesis

This thesis is mainly based on the following works.

Published/accepted papers

[1] M. Lind, Functions of bounded Λ-variation and integral smoothness,to appear in Forum Math., 15 pages.

[2] M. Lind, Estimates of the total p-variation of bivariate functions, J.Math. Anal. Appl. 401(2013), no. 1, 218–231.

[3] M. Lind, On fractional smoothness of functions related to p-variation,Math. Inequal. Appl. 16(2013), no. 1, 21–39.

[4] V.I. Kolyada and M. Lind, On moduli of p-continuity, Acta Math.Hungar. 137 (2012), no. 3, 191–213.

[5] V.I. Kolyada and M. Lind, On functions of bounded p-variation, J.Math. Anal. Appl. 356 (2009), no. 2, 582–604.

Submitted papers

[6] M. Lind, Fubini-type properties of bivariate functions of bounded p-variation, 8 pages.

iv

Acknowledgements

I am deeply grateful to my supervisor Professor Viktor Kolyada for hisguidance and encouragement. Throughout my studies, Viktor has alwaysbeen willing to take time to discuss mathematics with me, generously sharinghis ideas and deep knowledge of the subject. His ability to explain difficulttopics in a clear way is remarkable, and I have benefited greatly from workingwith him.

Moreover, I thank Professor Alexander Bobylev for his crucial supportwhen I wanted to begin my doctoral studies. Further, I thank Martin Brundinfor encouraging me to study mathematics in the first place.

Thanks also to past and present colleagues at the Department of Math-ematics, Karlstad University, especially Ilie Barza, Sorina Barza, MartinKrepela and Asa Windfall.

The support of the Graduate School of Mathematics and Computing(FMB) is gratefully acknowledged.

Contents

1 Introduction 1

2 Auxiliary statements 13

2.1 Lp-moduli of continuity . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Properties related to p-variation . . . . . . . . . . . . . . . . . 16

2.2.1 Local p-variation . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 The modulus of p-continuity . . . . . . . . . . . . . . . 19

2.3 On γ-moduli of continuity . . . . . . . . . . . . . . . . . . . . 22

3 Integral smoothness and p-variation 25

3.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Estimates of L∞-norm and p-variation . . . . . . . . . . . . . 31

3.3 Estimates of the modulus of p-continuity . . . . . . . . . . . . 38

3.4 The classes V αp . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 On classes Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Fractional smoothness via p-variation 55

4.1 Approximation with Steklov averages . . . . . . . . . . . . . . 55

4.2 Limiting relations . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Estimates of the Riesz-type variation . . . . . . . . . . . . . . 64

5 Embeddings within the scale Vp 71

5.1 Some known results and statement of problem . . . . . . . . . 71

5.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Embeddings of the space V ωq . . . . . . . . . . . . . . . . . . . 80

5.4 Sharpness of the main estimate . . . . . . . . . . . . . . . . . 84

v

vi Contents

6 On functions of bounded Λ-variation 916.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 926.2 Embedding of Lipschitz classes . . . . . . . . . . . . . . . . . 946.3 A Perlman-type theorem . . . . . . . . . . . . . . . . . . . . . 101

7 Multidimensional results 1057.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2 Estimates of the L∞-norm . . . . . . . . . . . . . . . . . . . . 1097.3 Estimates of the Vitali type p-variation . . . . . . . . . . . . . 1127.4 Fubini-type properties of H

(2)p . . . . . . . . . . . . . . . . . . 122

Chapter 1

Introduction

General description of the area

The notion of total variation of a function was introduced by Jordan in 1881in connection with his investigation of convergence of Fourier series. As iswell-known, bounded variation is a very important concept with many appli-cations, for example in the study Stieltjes integration and rectifiable curves.Subsequently, several extensions of Jordan’s notion of bounded variation wereconsidered. Two well-known generalizations are the functions of bounded p-variation and the functions of bounded Φ-variation, due to Wiener [82] andL.C. Young [84] respectively. These notions of generalized bounded variationhave attracted much interest, e.g., [83, 45, 19, 20, 50, 23, 24, 25]. For a moreextensive list of works related to generalized bounded variation, see [15, PartIV].

In [83], L.C. Young obtained an estimate of the Lp-modulus of continuityof a function in terms of its total p-variation. Conversely, sharp estimates ofthe total p-variation of a function in terms of the Lp-modulus of continuitywere first obtained by Terehin [71].

Soon after Jordan’s work, many mathematicians began to study notionsof bounded variation for functions of several variables. In the multivariatecase, there is no unique concept of bounded variation. One of the mostknown approaches is based on ideas of Tonelli (it can be expressed in adifferent but equivalent form in terms of partial moduli of continuity in L1).Another definition, due to Vitali, is based on mixed differences. This is moresimilar to the original definition of Jordan. Later, Hardy restricted Vitali’sclass by adding certain conditions on the sections of the functions. For a

1

2 Chapter 1. Introduction

survey of classical notions of multivariate functions of bounded variation, see[1]. Further, Golubov [26] introduced the total p-variation for multivariatefunctions that for p = 1 corresponds to the Hardy-Vitali type total variation.

In the seventies, Waterman [79, 80] considered a completely different ex-tension of bounded variation for univariate functions, the so-called functionsof bounded Λ-variation. These classes have been studied by many authors,see, e.g., the surveys [67, 81]. There are also extensions to multivariate func-tions, see [16] and the references given there.

Functions of generalized bounded variation are important in several dif-ferent areas of mathematics, such as Fourier analysis (e.g, [53, 79, 16, 43, 44])and operator theory (e.g., [15, 9, 10]). In [15, Part IV], many works relatedto applications of the concept of p-variation in probability are listed (wemention also [68], which will be considered below). Further, [55] surveys therelevance of Vitali-type variation in certain areas of numerical analysis.

Main objectives and methods

1. The main objective of this thesis is to study various properties offunctions of generalized bounded variation. In particular, we consider thefollowing:

• sharp relations between spaces of generalized bounded variation andspaces of functions defined by integral smoothness conditions (e.g.,Sobolev and Besov spaces);

• optimal properties of certain scales of function spaces of fractionalsmoothness generated by functionals of variational type;

• sharp embeddings within the scale of spaces of functions of boundedp-variation;

• bivariate functions of bounded p-variation, in particular sharp estimatesof total variation in terms of the mixed Lp-modulus of continuity, andFubini-type properties.

2. Two central methods in our work are approximations with Steklovaverages and a decomposition technique for moduli of continuity. We alsodevelop a scheme for constructing relevant counterexamples that is based ona sort of accumulation of piecewise linear functions. Such constructions areused to prove the sharpness of our results.

3

Summary

We shall give a summary of the thesis and some antecedent results.Chapter 2 contains general results and definitions, in particular related

to p-variation and moduli of continuity. It is convenient to introduce thesenotions right now.

Let f be a 1-periodic function on the real line R and let 1 ≤ p <∞. Anyset Π = x0, x1, ..., xn of points

x0 < x1 < ... < xn, where xn = x0 + 1,

will be called a partition of a period (or simply a partition). We also denote‖Π‖ = maxk(xk+1 − xk). For any partition Π, set

vp(f ; Π) =

(n−1∑k=0

|f(xk+1)− f(xk)|p)1/p

.

We say that f is a function of bounded p-variation (written f ∈ Vp) if

vp(f) = supΠvp(f ; Π) <∞, (1.0.1)

where the supremum is taken over all partitions Π. For 1 < p < ∞, we alsoconsider the function

ω1−1/p(f ; δ) = sup‖Π‖≤δ

vp(f ; Π) (0 ≤ δ ≤ 1), (1.0.2)

where the supremum is taken over all partitions Π with ‖Π‖ ≤ δ. As it wasmentioned above, for p = 1, the definiton (1.0.1) was given by Jordan, andfor p > 1, both (1.0.1) and (1.0.2) are due to Wiener [82]. Following Terehin[70], we call the function (1.0.2) the modulus of p-continuity of f . For p > 1,there are non-constant functions f such that

limδ→0+

ω1−1/p(f ; δ) = 0.

Such functions are called p-continuous, and the class of all p-continuous func-tions is denoted by Cp.

For f ∈ Lp([0, 1]) (1 ≤ p <∞), the Lp-modulus of continuity of f is givenby

ω(f ; δ)p = sup0≤|h|≤δ

(∫ 1

0

|f(x+ h)− f(x)|pdx)1/p

(0 ≤ δ ≤ 1). (1.0.3)


In Chapter 3, we study relations between bounded p-variation and in-tegral smoothness of univariate functions.

Terehin [71] obtained sharp estimates of the total p-variation of a functionin terms of its second order modulus of continuity (see (3.2.19) below). Thesame result was obtained by Peetre [57] with the use of interpolation methods.In particular, the following was proved in [71]: if

Jp(f) =

∫ 1

0

t−1/pω(f ; t)pdt

t<∞ (1 < p <∞), (1.0.4)

then f is equivalent1 to a continuous 1-periodic function f ∈ Vp, and

vp(f) ≤ AJp(f). (1.0.5)

The first part of this statement was proved in 1958 by Geronimus [21]. Moreexactly, it was shown in [21] that any function f ∈ Lp([0, 1]) satisfying (1.0.4)is equivalent a continuous 1-periodic function, and

‖f‖∞ ≤ A(‖f‖p + Jp(f)), (1.0.6)

where A is an absolute constant. However, it can easily be shown that theconstant coefficients of (1.0.5) and (1.0.6) should depend on p.

We prove that the following stronger versions of (1.0.5) and (1.0.6) hold:

‖f‖∞ ≤ A

(‖f‖p +

1

pp′Jp(f)

), (1.0.7)

and

vp(f) ≤ A

(ω(f ; 1)p +

1

pp′Jp(f)

), (1.0.8)

where A is an absolute constant (as usual, p′ = p/(p− 1)).

We show that the asymptotic behaviour of the constant A/(pp′) in (1.0.7)and (1.0.8) is optimal in a sense.

It was shown in [83, 71] that for 1 < p <∞,

ω(f ; δ)p ≤ δ1/pω1−1/p(f ; δ) (0 ≤ δ ≤ 1). (1.0.9)

1Two functions are said to be equivalent if they coincide almost everywhere.

5

Reverse estimates were obtained in [71]. There it was shown that if (1.0.4)holds and the function f is modified on a set of measure zero so as to becomecontinuous, then we have the following inequality

ω1−1/p(f ; δ) ≤ A

[pδ−1/pω(f ; δ)p +

∫ δ

0

t−1/pω(f ; t)pdt

t

], (1.0.10)

where A is an absolute constant.Applying (1.0.8), we obtain that the constant coefficients in (1.0.10) can

be improved. Namely, we show that the following estimate holds.For 1 < p <∞, we have

ω1−1/p(f ; δ) ≤ A

[δ−1/pω(f ; δ)p +

1

pp′

∫ δ

0

t−1/pω(f ; t)pdt

t

](1.0.11)

for any δ ∈ (0, 1], where A is an absolute constant.Our main result here is the sharpness of the estimate (1.0.11). More

exactly, we construct a function with an arbitrary prescribed order of themodulus of continuity in Lp([0, 1]), for which the opposite inequality holdsfor all δ ∈ [0, 1].

Assume that 1 < p <∞ and α ≥ 0. Let f be an 1-periodic function andlet Π = x0, x1, ..., xn be a partition. Set

vp,α(f ; Π) =

(n−1∑k=0

|f(xk+1)− f(xk)|p

(xk+1 − xk)αp

)1/p

. (1.0.12)

We denote by V αp the class of all 1-periodic functions f such that

vp,α(f) = supΠvp,α(f ; Π) <∞, (1.0.13)

where Π runs over all partitions of a period (see [57], p. 114). Obviously,V 0p = Vp and V β

p ⊂ V αp if 0 ≤ α < β.

Denote by W 1p the class of all 1-periodic, absolutely continuous functions

f with f ′ ∈ Lp([0, 1]). By a theorem of F. Riesz (see Theorem 2.2 below), we

have V1/p′p = W 1

p for any 1 1/p′, then any function f ∈ V αp

is constant.We obtain the following sharp estimate of the Riesz-type variation vp,α(f)

in terms of ω(f ; δ)p.


If 1 < p <∞, 0 < α < 1/p′, and

Kp,α(f) =

(∫ 1

0

t−αp−1ω(f ; t)ppdt

t

)1/p

<∞, (1.0.14)

then f is equivalent to a continuous function f ∈ V αp , and

vp,α(f) ≤ Aα−1/p′(1/p′ − α)1/pKp,α(f), (1.0.15)

where A is an absolute constant.Further, we show that the condition (1.0.14) is sharp for any rate of the

decay of the modulus of continuity, and the order of the constant in (1.0.15)is optimal as α→ 0 or α→ 1/p′.

In Chapter 4, we continue our study of the modulus of p-continuity(1.0.2) and Riesz-type variation (1.0.13). One of the main results of thechapter is the following sharp estimate of the Riesz-type variation vp,α(f) interms of ω1−1/p(f ; δ) (similar to (1.0.15)).

If 1 < p <∞, 0 < α < 1/p′ and

Ip,α(f) =

(∫ 1

0

[t−αω1−1/p(f ; t)]pdt

t

)1/p

<∞, (1.0.16)

then f ∈ V αp and

vp,α(f) ≤ A(vp(f) + p′α1/p(1/p′ − α)1/p′Ip,α(f)

), (1.0.17)

where A is an absolute constant.We prove that the constant coefficient of (1.0.17) has optimal order as

α → 0 or α → 1/p′. Observe that an estimate of the type (1.0.17) followsimmediately from (1.0.15) and (1.0.9). However, the constant obtained inthis way is not optimal. We also show that the condition (1.0.16) is sharpfor any rate of decay of the modulus of p-continuity.

Further, we obtain several limiting relations for the functionals ω1−1/p(f ; δ)and vp,α(f). In particular, we prove the following.

If f ∈ Vp (1 < p <∞) and

sup0<s<1/p′

(1/p′ − s)∫ 1

0

[t−sω1−1/p(f ; t)]pdt

t<∞,

7

then f ∈ W 1p , and

lims→1/p′−

(1/p′ − s)∫ 1

0


t= p−1‖f ′‖pp.

This result is similar to limiting relations for Besov spaces, first studied byBourgain, Brezis and Mironescu [11, 12, 13] (for further results, see [46, 33,49]).

In Chapter 5 we study intrinsic embeddings in the scale of functionsof bounded p-variation. Let 1 < p < q < ∞ and let f ∈ Vq. By Jensen’sinequality, we have that

ω1−1/q(f ; δ) ≤ ω1−1/p(f ; δ) (0 ≤ δ ≤ 1).

We obtain the following sharp reverse inequality.Let 1 < p < q <∞, then

ω1−1/p(f ; δ) ≤ 4

(∫ δ

0

(t−θω1−1/q(f ; t))qdt

t

)1/q

(0 ≤ δ ≤ 1), (1.0.18)

where 1 < p < q <∞ and θ = 1/p− 1/q.For 1 < q <∞, denote

V ωq = f ∈ Vq : ω1−1/q(f ; δ) = O(ω(δ)), (1.0.19)

where ω ∈ Ω1/q′ (see Definition 2.10) is an arbitrary majorant of the modulusof q-continuity. We prove the following embedding theorem.

V ωq ⊂ Vp ⇐⇒

∫ 1

0

(t−θω(t))qdt

t<∞, (1.0.20)

whatever be 1 < p < q <∞ and ω ∈ Ω1/q′ .Let 1 < q <∞ and let ω ∈ Ω1/q′ . Set

Vω

q = f ∈ Vq : ω1−1/q(f ; δ) ≤ ω(δ), δ ∈ [0, 1], (1.0.21)

The main result of Chapter 5 is that the estimate (1.0.18) is sharp in thefollowing strong sense.

Let 1 < p < q, there exists a constant c = c(p, q) such that for anyω ∈ Ω1/q′, there exists a function f ∈ V ω

q for which the inequality

ω1−1/p(f ; δ) ≥ c

(∫ δ

0

(t−θω(t))qdt

t

)1/q

= cρp,q,ω(δ),


holds for all δ > 0.The previous result can also be formulated in the following way.Let 1 < p < q <∞ and ω ∈ Ω1/q′, then

c(p, q) ≤ supf∈V ωq

inf0<δ≤1

ω1−1/p(f ; δ)

ρp,q,ω(δ)≤ sup

f∈V ωq

sup0<δ≤1

ω1−1/p(f ; δ)

ρp,q,ω(δ)≤ 4, (1.0.22)

where c(p, q) only depends on p and q.The estimate (1.0.18) and the embedding theorem (1.0.20) are analogous

to results for Lp-moduli of continuity studied by Ul’yanov [74] – [76] (seealso Chapter 5.1 below). However, the analogy fails to be complete, since noresult of the type (1.0.22) holds for Lp-moduli of continuity.

In Chapter 6 we study some properties of the so-called functions ofbounded Λ-variation, introduced by Waterman [79, 80].

Let Λ = λn be any nondecreasing sequence of positive numbers suchthat λn →∞ and

∞∑n=1

1

λn=∞.

A 1-periodic function f on the real line is said to be of bounded Λ-variationif

vΛ(f) = sup∞∑n=1

|f(bn)− f(an)|λn

<∞,

where the supremum is taken over all sequences [an, bn]∞n=1 of nonover-lapping intervals contained in a period. The class of functions of boundedΛ-variation is denoted ΛBV .

Let 1 ≤ p <∞ and 0 < α ≤ 1, we recall the notation

Lip(α; p) = f ∈ Lp([0, 1]) : ω(f ; δ)p = O(δα).

Relations between integral smoothness properties of functions and ΛBV haveattracted some interest in recent years, see, e.g. [69, 78, 38, 22, 29, 77]. Inparticular, H. Wang [77] observed that a necessary condition for the embed-ding

Lip(α; p) ⊂ ΛBV (1 1/p in (1.0.23) is essential; for α ≤ 1/p,the class Lip(α; p) contains unbounded functions and (1.0.23) cannot hold.The main result of Chapter 6 is the following.

Let 1 < p <∞ and 1/p < α < 1, and set

r =1

α− 1/pand r′ =

1

1 + 1/p− α.

Then the embedding (1.0.23) holds if and only if

∞∑n=0

(2n+1∑k=2n

(1

kα−1/pλk

)p′)r′/p′

<∞.

It follows that the conjecture of Wang is not true.In Chapter 7, we study problems related to p-variation of functions of

several variables. A set N = (xi, yj) : 0 ≤ i ≤ m, 0 ≤ j ≤ n of points inR2 such that

x0 < x1 < ... < xm = x0 + 1, y0 < y1 < ... < yn = y0 + 1,

will be called a net. Let 1 ≤ p <∞ and let the function f(x, y) be 1-periodicin both variables. For a fixed net N , we denote

∆f(xi, yj) = f(xi+1, yj+1)− f(xi+1, yj)− f(xi, yj+1) + f(xi, yj),

for 0 ≤ i ≤ m− 1, 0 ≤ j ≤ n− 1, and

v(2)p (f ;N ) =

(m−1∑i=0

n−1∑j=0

|∆f(xi, yj)|p)1/p

.

The space V(2)p consists of all functions that satisfy

v(2)p (f) = sup

Nv(2)p (f ;N ) <∞, (1.0.25)

where the supremum is taken over all nets N .If f(x, y) is 1-periodic in both variables and x ∈ R is fixed, then the

x-section of f is the 1-periodic function fx defined by

fx(y) = f(x, y), y ∈ R. (1.0.26)


The y-sections of f are defined analogously.The space H

(2)p ⊂ V

(2)p consists of all functions that, in addition to

(1.0.25), also satisfy the following conditions on their sections: for any x, y ∈R, we have fx, fy ∈ Vp. Observe that the class H

(2)p contains only bounded

functions, while a function in V(2)p may be unbounded.

For p = 1, the definition (1.0.25) was given by Vitali, and Hardy was the

first who considered the class H(2)1 (see, e.g., [1]). For p > 1, the classes V

(2)p

and H(2)p were first defined and studied by Golubov [26].

Let s, t ∈ R and set

∆(s, t)f(x, y) =

f(x+ s, y + t)− f(x+ s, y)− f(x, y + t) + f(x, y). (1.0.27)

For 1 ≤ p < ∞ and f ∈ Lp([0, 1]2), the mixed Lp-modulus of continuity isdefined by

ω(f ;u, v)p = sup0≤s≤u, 0≤t≤v

‖∆(s, t)f‖p.

It was proved by Golubov [26] that if f ∈ V (2)p (1 ≤ p <∞), then

ω(f ;u, v)p ≤ v(2)p (f)u1/pv1/p. (1.0.28)

We obtain sharp estimates of the total Vitali-type p-variation (1.0.25)of a bivariate function in terms of its mixed Lp-modulus of continuity for1 ≤ p <∞. For p = 1, we have the following converse of (1.0.28).

Let f ∈ L1([0, 1]2) and assume that

ω(f ;u, v)1 = O(uv).

Then there exists a function f ∈ V (2)1 such that f = f a.e., and

v(2)1 (f) = sup

u,v>0

ω(f ;u, v)1

uv.

This is a two-dimensional analogue of a classical result of Hardy and Little-wood (see Theorem 2.1 below).

For p > 1, we have the following result.Let 1 < p <∞ and f ∈ Lp([0, 1]2) and assume that∫∫

[0,1]2(uv)−1/pω(f ;u, v)p

du

u

dv

v<∞. (1.0.29)

11

Then there exists a function f ∈ V (2)p such that f = f a.e. and

v(2)p (f) ≤ A

[ω(f ; 1, 1)p +

1

pp′

∫ 1

0

t−1/p[ω(f ; 1, t)p + ω(f ; t, 1)p]dt

t

+1

(pp′)2

∫∫[0,1]2

(uv)−1/pω(f ;u, v)pdu

u

dv

v

], (1.0.30)

where A is an absolute constant.None of the terms at the right-hand side of (1.0.30) can be omitted, and

the constants 1/pp′ and 1/(pp′)2 have optimal asymptotic behaviour as p→ 1and p→∞. This is a two-dimensional analogue of (1.0.8).

Let X, Y be spaces of functions defined on the real line, with norms ‖ ·‖Xand ‖ · ‖Y . A function f(x, y) is said to belong to the symmetric mixed normspace X [Y ]sym if the functions

x 7→ ‖fx‖Y and y 7→ ‖fy‖Y

both belong to the space X (fx, fy denotes the sections of f , see (1.0.26)).The importance of mixed norm spaces that are invariant under permutationsof variables was first demonstrated by Fournier [18].

Recall that by the Fubini-Tonelli theorem, we have

Lp([0, 1]) [Lp([0, 1]) ]sym = Lp([0, 1]2) (0 < p ≤ ∞).

Fubini-type properties for the scale of Lorentz spaces were studied in [36].

In Chapter 7, we also investigate Fubini-type properties of the class H(2)p

for p ≥ 1. For this, we consider the symmetric mixed norm space Vp [Vp ]sym offunctions of bounded iterated p-variation. That is, if f is a bivariate functionsand we denote

ϕ(x) = vp(fx) and ψ(y) = vp(fy),

then f ∈ Vp [Vp ]sym if and only if ϕ, ψ ∈ Vp.For p = 1, it was proved in [1] that H

(2)1 ⊂ V1 [V1 ]sym. This is a Fubini-

type property ofH(2)1 (in one direction). We prove that Fubini-type properties

completely fail for H(2)p when p > 1. In other words, the following holds.

For p > 1,

H(2)p 6⊂ Vp [Vp ]sym and Vp [Vp ]sym 6⊂ H(2)

p .

Chapter 2

Auxiliary statements

In this chapter we collect some general results which are used throughoutthis thesis.

2.1 Lp-moduli of continuity

Let Lp([0, 1]n) denote the set of all measurable functions on Rn that are1-periodic in each variable and satisfy

‖f‖p =

(∫[0,1]n|f(x)|pdx

)1/p

<∞,

if p <∞, and ‖f‖∞ = ess supx∈[0,1]n |f(x)| <∞ for p =∞.Let f be a function on Rn that is 1-periodic in each variable. For h ∈ Rn,

we denote

∆(h)f(x) = f(x+ h)− f(x). (2.1.1)

Recall that for f ∈ Lp([0, 1]n) (1 ≤ p < ∞), the Lp-modulus of continuity isdefined by

ω(f ; δ)p = sup0≤|h|≤δ

‖∆(h)f‖p (0 < δ ≤ 1).

A modulus of continuity is a continuous functions ω defined on the interval[0, 1] such that ω(0) = 0 and ω is nonincreasing and subadditive. Denote byΩ the class of all moduli of continuity. For any ω ∈ Ω, we have

ω(2δ) ≤ 2ω(δ) (0 ≤ δ ≤ 1/2),

13

14 Chapter 2. Auxiliary statements

and it follows that

ω(µ)

µ≤ 2ω(h)

hfor 0 < h < µ ≤ 1. (2.1.2)

Whence, if ω is not identically 0, then for all δ ∈ [0, 1], we have

ω(δ) ≥ cωδ (cω = ω(1)/2).

Thus, the best order of decay for any modulus of continuity is ω(δ) = O(δ).

Clearly, if f ∈ Lp([0, 1]n), then ω(f ; ·)p ∈ Ω. It follows from Lebesgue’sdifferentiation theorem that ω(f ; ·)p is identically 0 if and only if f is equiv-alent to a constant.

For the rest of this chapter, we shall only consider the case n = 1. Recallthat W 1

p (1 ≤ p <∞) denotes the class of all 1-periodic absolutely continuousfunctions f with f ′ ∈ Lp([0, 1]). If f ∈ W 1

p (1 ≤ p <∞), then

ω(f ; δ)p ≤ ‖f ′‖pδ (0 ≤ δ ≤ 1). (2.1.3)

Moreover, we have the following theorem of Hardy and Littlewood [27].

Theorem 2.1. Let f ∈ Lp([0, 1]) (1 ≤ p < ∞). The following statementsare true.

(i) For 1 0

ω(f ; δ)pδ

.

(ii) For p = 1, we have ω(f ; δ)1 = O(δ) if and only if f is equivalent to afunction g ∈ V1. Also,

v1(g) = supδ>0

ω(f ; δ)1

δ.

For 1 < p <∞, we also have the following variational characterization ofW 1p , due to F. Riesz (see, e.g., [51]).

2.1. Lp-moduli of continuity 15

Theorem 2.2. Let 1 < p <∞, then f ∈ W 1p if and only if

supΠ

(n−1∑k=0

|f(xk+1)− f(xk)|p

(xk+1 − xk)p−1

)1/p

<∞,

where the supremum is taken over all partitions Π. In other words,

W 1p = V 1/p′

p (1 < p <∞).

The next lemma is well-known (see, e.g., [7]). For the sake of complete-ness, we give the proof.

Lemma 2.3. Let f ∈ Lp([0, 1]) (1 ≤ p <∞), then

ω(f ; δ)p ≤3

δ

∫ δ

0

‖∆(t)f‖pdt, δ ∈ (0, 1], (2.1.4)

where ∆(t)f is given by (2.1.1).

Proof. Fix δ > 0 and let h ∈ (0, δ]. Clearly, for any t ∈ (0, δ], we have

‖∆(h)f‖p ≤ ‖∆(h− t)f‖p + ‖∆(t)f‖p.

Integrating the previous inequality with respect to t in [0, δ] and using thefact that ‖∆(u)f‖p = ‖∆(−u)f‖p, we get

δ‖∆(h)f‖p ≤ 3

∫ δ

0

‖∆(t)f‖pdt,

and (2.3) follows.

One consequence of (2.1.4) that we shall use below is the following esti-mate: ∫ 1

0

t−1/pω(f ; t)pdt

t≤ 3

∫ 1

0

t−1/p‖∆(t)f‖pdt

t. (2.1.5)

For any function f ∈ Lp([0, 1]) (1 ≤ p <∞), set

Ωp(f) =

(∫ 1

0

∫ 1

0

|f(x)− f(y)|pdxdy)1/p


We haveΩp(f) ≤ ω(f ; 1)p ≤ 2Ωp(f). (2.1.6)

Indeed, since f is 1-periodic,

Ωp(f) =

(∫ 1

0

dh

∫ 1

0

|f(x)− f(x+ h)|pdx)1/p

≤(∫ 1

0

ω(f ;h)ppdh

)1/p

≤ ω(f ; 1)p

On the other hand, denoting I =∫ 1

0f(y)dy, we obtain

ω(f ; 1)p = ω(f − I; 1)p ≤ 2‖f − I‖p

= 2

(∫ 1

0

∣∣∣∣f(x)−∫ 1

0

f(y)dy

∣∣∣∣p dx)1/p

≤ 2Ωp(f).

Let f ∈ L1([0, 1]). For any 0 < h ≤ 1, let

fh(x) =1

h

∫ h

0

f(x+ t)dt (2.1.7)

be the Steklov average of the function f.

Lemma 2.4. If f ∈ Lp([0, 1]), 1 ≤ p <∞, then

‖f − fh‖p ≤ ω(f ;h)p (2.1.8)

and‖f ′h‖p ≤ ω(f ;h)p/h. (2.1.9)

These inequalities are well-known and their proofs are immediate.

2.2 Properties related to p-variation

2.2.1 Local p-variation

Let f be a 1-periodic function on the real line and let [a, b] ⊂ R be anyinterval. A partition of the interval [a, b] is a finite set of points Πa,b =x0, x1, ..., xn such that

a = x0 < x1 < ... < xn = b.

2.2. Properties related to p-variation 17

For 1 ≤ p <∞, the p-variation of f on [a, b] is defined as

vp(f ; [a, b]) = supΠa,b

(n−1∑k=0

|f(xk+1)− f(xk)|p)1/p

,

where the supremum is taken over all partitions Πa,b of [a, b]. Observe thatfor vp(f) defined by (1.0.1), we have

vp(f) = supa∈R

vp(f ; [a, a+ 1]).

Let 1 ≤ p < ∞ and assume that f is monotone on the interval [a, b].Then

vp(f ; [a, b]) = |f(b)− f(a)|, (2.2.1)

the proof of this is trivial.The next lemma is well-known (see, e.g., [45]).

Lemma 2.5. Let f be a 1-periodic function on the real line, let 1 ≤ p <∞and [a, b] ⊂ R. For any c ∈ (a, b), there holds

vp(f ; [a, c])p + vp(f ; [c, b])p ≤ vp(f ; [a, b])p, (2.2.2)

andvp(f ; [a, c]) + vp(f ; [c, b]) ≤ vp(f ; [a, b]). (2.2.3)

Moreover, if f attains global extremum at c ∈ (a, b), then

vp(f ; [a, b])p = vp(f ; [a, c])p + vp(f ; [c, b])p. (2.2.4)

Proof. The inequality (2.2.2) is obvious. To prove (2.2.3), we define thefunctions

g(x) = [f(x)− f(c)]χ[a,c](x) and h(x) = [f(x)− f(c)]χ[c,b](x).

Then f(x) = g(x) + h(x) + f(c) on [a, b], and

vp(f ; [a, b]) = vp(g + h; [a, b]) ≤ vp(g; [a, b]) + vp(h; [a, b]).

On the other hand, since g(x) = 0 for x ∈ [c, b], and g(x) = f(x) − f(c) forx ∈ [a, c], we have

vp(g; [a, b]) = vp(g; [a, c]) = vp(f ; [a, c]).


In the same way, vp(h; [a, b]) = vp(f ; [c, b]), and (2.2.3) follows. Finally, weprove (2.2.4). By (2.2.2), it is enough to show that the right-hand side of(2.2.4) is not smaller than the left-hand side. Let Πa,b = x0, x1, ..., xn beany partition of [a, b] and assume that c ∈ (xi, xi+1) for some i. Clearly,

|f(xi+1)− f(xi)|p ≤ max|f(xi+1)− f(c)|p, |f(c)− f(xi)|p,

and thus,

n−1∑k=0

|f(xk+1)− f(xk)|p ≤∑k 6=i

|f(xk+1)− f(xk)|p +

+ |f(xi+1)− f(c)|p + |f(c)− f(xi)|p

≤ vp(f ; [a, c])p + vp(f ; [c, b])p,

and (2.2.4) follows.

The next lemma is a consequence of (2.2.4).

Lemma 2.6. Let 1 ≤ p <∞ and assume that fn is a sequence of nonneg-ative 1-periodic functions such that

• (supp fn) ∩ [0, 1] = [an, bn] (n ∈ N);

• the intervals [an, bn] (n ∈ N) are nonoverlapping;

• fn(an) = fn(bn) = 0 (n ∈ N).

Set

f(x) =∞∑n=1

fn(x), x ∈ R,

then

vp(f) =

(∞∑n=1

vp(fn)p

)1/p

. (2.2.5)

Proof. Since fn ≥ 0 (n ∈ N), it follows that x = 0, 1 and x = an, bn (n ∈ N)are points of global minimum of f . Then, by (2.2.4), we have

vp(f) = vp(f ; [0, 1]) =

(∞∑n=1

vp(f ; [an, bn])p

)1/p

.

On the other hand, vp(f ; [an, bn]) = vp(fn; [an, bn]) = vp(fn).


2.2.2 The modulus of p-continuity

Recall the moduli of p-continuity (1.0.2). For 1 < p <∞ and I ⊂ R, we candefine the local modulus of p-continuity ω1−1/p(f, I; δ) in the obvious way.Clearly, analogues of (2.2.2)-(2.2.4) hold. In particular, if f attains globalextremum at c ∈ (a, b), then

ω1−1/p(f, [a, b]; δ)p = ω1−1/p(f, [a, c]; δ)

p + ω1−1/p(f, [c, b]; δ)p, (2.2.6)

for any δ ∈ (0, 1].It was proved in [70] that for 1 < p <∞ and any n ∈ N

ω1−1/p(f ;nδ) ≤ n1/p′ω1−1/p(f ; δ) (0 ≤ δ ≤ 1/n),

where p′ = p/(p− 1). Thus,

ω1−1/p(f ;µ)

µ1/p′≤ 21/p′ ω1−1/p(f ;h)

h1/p′for 0 < h < µ ≤ 1. (2.2.7)

It follows that if f is not a constant function, then

ω1−1/p(f ; δ) ≥ cδ1/p′ (c = vp(f)/21/p′).

Moreover, the best order of decay ω1−1/p(f ; δ) = O(δ1/p′) is attained if f ∈W 1p . Indeed, assume that f ∈ W 1

p (1 < p < ∞) and let Π = x0, x1, ..., xnbe any partition with ‖Π‖ ≤ δ. Applying Holder’s inequality, we have

vp(f ; Π) =

(n−1∑k=0

∣∣∣∣∫ xk+1

xk

f ′(t)dt

∣∣∣∣p)1/p

≤ δ1/p′

(n−1∑k=0

∫ xk+1

xk

|f ′(t)|pdt

)1/p

= δ1/p′||f ′||p..

Thus,ω1−1/p(f ; δ) ≤ δ1/p′‖f ′‖p, 0 ≤ δ ≤ 1. (2.2.8)

In [70] the converse was proved. That is, if ω1−1/p(f ; δ) = O(δ1/p′), thenf ∈ W 1

p (1 < p <∞).The next result is due to Terehin [71] for periodic functions (the easier

non-periodic case was proved in [83]). The argument presented in [71] is notsufficiently clear, therefore we give a complete proof.


Proposition 2.7. Let 1 < p <∞ and let f ∈ Vp. Then

ω(f ; δ)p ≤ δ1/pω1−1/p(f ; δ), 0 ≤ δ ≤ 1. (2.2.9)

Proof. We shall first prove (2.2.9) for functions in Vp that attain global max-imum. Let f ∈ Vp and suppose that f attains a global maximum at somepoint c. We may assume that c = 0, since both ω(f ; δ)p and ω1−1/p(f ; δ) areinvariant with respect to translation.

Fix δ ∈ (0, 1] and let k/m ∈ (0, δ] be a rational number. By the period-icity of f , we have

k‖∆(k/m)f‖p =

∫ k

0

|f(x+ k/m)− f(x)|pdx

=m−1∑j=0

∫ (j+1)k/m

jk/m

|f(u+ k/m)− f(u)|pdu

=

∫ k/m

0

m−1∑j=0

|f(u+ (j + 1)k/m)− f(u+ jk/m)|pdu

≤∫ δ

0

ω1−1/p(f, [u, u+ k]; δ)pdu. (2.2.10)

Since k ∈ N, it is a point of global maximum of f . By (2.2.6), we have forany u ∈ [0, δ]

ω1−1/p(f, [u, u+ k]; δ)p = ω1−1/p(f, [u, k]; δ)p + ω1−1/p(f, [k, u+ k]; δ)p.

Further, since f is 1-periodic,

ω1−1/p(f, [k, k + u]; δ) = ω1−1/p(f, [0, u]; δ).

Whence, by the two previous equations and (2.2.2),

ω1−1/p(f, [u, u+ k]; δ)p = ω1−1/p(f, [u, k]; δ)p + ω1−1/p(f, [0, u]; δ)p

≤ ω1−1/p(f, [0, k]; δ)p.

Applying (2.2.6) repetedly and using that f is 1-periodic, we have

ω1−1/p(f, [0, k]; δ)p = kω1−1/p(f, [0, 1]; δ)p ≤ kω1−1/p(f ; δ)p.


Whence,

ω1−1/p(f, [u, u+ k]; δ)p ≤ kω1−1/p(f ; δ)p

for any u ∈ [0, δ]. By the previous inequality and (2.2.10), we have

k‖∆(k/m)f‖pp ≤ k

∫ δ

0

ω1−1/p(f ; δ)pdu ≤ kδω1−1/p(f ; δ)p,

and it follows that ω(f ; δ)p ≤ δ1/pω1−1/p(f ; δ) for all functions f ∈ Vp thatattains global maximum.

Let now f be an arbitrary function in Vp and set M = sup f(x). Givenany ε > 0, there is a point c such that f(c) > M − ε. As above, we maysuppose that c = 0. Define the 1-periodic function fε by setting fε(x) = f(x)for x /∈ Z and fε(x) = M for x ∈ Z. Then fε attains global maximum andthus (2.2.9) holds for fε. On the other hand, since f = fε almost everywhere,we get ω(fε; δ)p = ω(f ; δ)p. Further, it is clear that vp(f − fε) ≤ 2ε, andtherefore

ω1−1/p(fε; δ) ≤ ω1−1/p(f ; δ) + 2ε (0 ≤ δ ≤ 1).

Then

ω(f ; δ)p = ω(fε; δ)p ≤ δ1/pω1−1/p(fε; δ) ≤ δ1/pω1−1/p(f ; δ) + 2δ1/pε,

and since ε > 0 was arbitrary, the inequality follows.

Remark 2.8. One can give a simple proof of the inequality (2.2.9) with aworse constant. Indeed, for any δ > 0 let 0 < h ≤ δ and take n ∈ N suchthat nh < 1 ≤ (n+ 1)h. Then

‖∆(h)f‖pp =

∫ h

0

n−1∑j=0

|f(x+ (j + 1)h)− f(x+ jh)|pdx+

+

∫ 1

nh

|f(x+ h)− f(x)|pdx

≤ 2δω1−1/p(f ; δ)p.

Thus,

ω(f ; δ)p ≤ 21/pδ1/pω1−1/p(f ; δ).

Recall that fh denotes the Steklov average function (2.1.7).


Lemma 2.9. Let f ∈ Vp, 1 < p <∞, then

ω1−1/p(fh, δ) ≤ ω1−1/p(f ; δ) (0 ≤ δ ≤ 1), (2.2.11)

and

‖f ′h‖p ≤ h−1/p′ω1−1/p(f ;h). (2.2.12)

The inequality (2.2.11) is immediate and (2.2.12) follows from (2.1.9) and(2.2.9).

Recall that Cp (1 < p < ∞) denotes the class of p-continuous functions,that is, functions such that

limδ→0+

ω1−1/p(f ; δ) = 0.

It is easy to see that if f ∈ Cp, then f ∈ Vp and f is continuous.Love [45] considered the following property of functions: for any ε > 0,

there exist δ > 0 such that if (ak, bk) is any finite collection of nonoverlap-ping intervals with(∑

k

(bk − ak)p)1/p

< δ, then

(∑k

|f(bk)− f(ak)|p)1/p

< ε.

For p = 1, the previous condition is just the definition of absolute continuity.For 1 < p < ∞, it is equivalent to f ∈ Cp. Thus, for 1 < p < ∞, we canview p-continuity as an intermediate property of functions, between absolutecontinuity and continuity. Let W be van der Waerden’s function, i.e.,

W (x) =∞∑n=1

2−nφ(2nx), where φ(x) = infk∈Z|x− k|, x ∈ R.

Then W is nowhere differentiable and thus not absolutely continuous. At thesame time, W ∈ Lip(α) for any 0 < α < 1, whence it follows that W ∈ Cpfor all p > 1.

2.3 On γ-moduli of continuity

We shall use the following scale of functions.

2.3. On γ-moduli of continuity 23

Definition 2.10. Let 0 < γ ≤ 1. We let Ωγ denote the class of all continuousfunctions ω defined on [0, 1] such that ω(0) = 0, ω(t) is nondecreasing andω(t)/tγ is nonincreasing.

For historical remarks and some new information concerning conditionsof this type (including the close relation to index numbers), we refer to thepaper [61] and the references given there.

For γ = 1, the class Ω1 is “almost” the same as the class of moduliof continuity (see, e.g., [14, p.41]), in the sense that for any modulus ofcontinuity η, there is ω ∈ Ω1 such that ω(t) ≤ η(t) ≤ 2ω(t), t ∈ [0, 1].

Similarly, Terehin [72] proved that for γ = 1/p′, the class Ω1/p′ “almostcoincides” with the class of all moduli of p-continuity for functions in Cp.Indeed, let f ∈ Cp and set

ω∗(t) = t1/p′

inf0<u≤t

ω1−1/p(f ;u)

u1/p′. (2.3.1)

Then clearly ω∗ ∈ Ω1/p′ and by (2.2.7)

ω∗(t) ≤ ω1−1/p(f ; t) ≤ 21/p′ω∗(t), 0 ≤ t ≤ 1. (2.3.2)

Conversely, for any ω ∈ Ω1/p′ , in [72] there was constructed a function f ∈ Cpsuch that

ω(t) ≤ ω1−1/p(f ; t) ≤ 9ω(t), 0 ≤ t ≤ 1.

For this reason, we shall call a function ω ∈ Ω1/p′ a modulus of p-continuity.Throughout this thesis, we will use the following construction. For any

ω ∈ Ωγ, we denote

ωn = ω(2−n) and ωn = 2nγω(2−n) (n ∈ N). (2.3.3)

Since ω(t) is nondecreasing and ω(t)/tγ is nonincreasing, we have

ωn+1 ≤ ωn ≤ 2γωn+1 (2.3.4)

andωn ≤ ωn+1 ≤ 2γωn (n ∈ N). (2.3.5)

Let ω ∈ Ωγ and assume that

limt→0+

ω(t)/tγ =∞. (2.3.6)


Then we define the sequence of natural numbers nk ≡ nk(ω, γ) as follows.Set n0 = 0 and

nk+1 = min

(n : max

(ωnωnk

,ωnkωn

)≤ 1

4

)(k = 0, 1, ...). (2.3.7)

Thus,4ωnk+1

≤ ωnk , 4ωnk ≤ ωnk+1(k = 0, 1, ...), (2.3.8)

and for each k = 0, 1, ... at least one of the inequalities

4ωnk+1−1 > ωnk or ωnk+1−1 < 4ωnk

holds. By (2.3.4) and (2.3.5), this implies that for each k = 0, 1, ... we haveat least one of the inequalities

ωnk < 8ωnk+1(2.3.9)

orωnk+1

< 8ωnk . (2.3.10)

Partitions (2.3.7) for moduli of continuity have been used for a long time,beginning from the works [3, 52, 75].

Chapter 3

Integral smoothness andp-variation

In this chapter, we study relations between variational properties of functionsand integral smoothness.

Recall that by Hardy-Littlewood’s theorem (Theorem 2.1 above), f ∈L1([0, 1]) coincides a.e. with a function g ∈ V1 if and only if ω(f ; δ)1 = O(δ).For p > 1, the class Vp does not admit any similar characterization in termsof Lp-modulus of continuity. It was shown in [83], [70] that

ω(f ; δ)p ≤ vp(f)δ1/p (0 ≤ δ ≤ 1). (3.0.1)

However, for p > 1, the condition ω(f ; δ)p = O(δ1/p) does not even implythat f ∈ L∞([0, 1]) (take e.g. f(x) = log(1/|x|), |x| ∈ (0, 1/2]).

As mentioned in the Introduction, Terehin [71] proved that if

Jp(f) =

∫ 1

0

t−1/pω(f ; t)pdt

t<∞,

then f is equivalent to a continuous function f ∈ Vp, and

vp(f) ≤ AJp(f). (3.0.2)

Simple arguments show that the constant coefficient in (3.0.2) should de-pend on p and vanish as p→ 1 or p→∞. For example, for any continuouslydifferentiable function f the left-hand side in (3.0.2) is bounded whilst theright-hand side tends to infinity as p → 1 (if f is not a constant). Further-more, if p→∞, then the right-hand side of (3.0.2) tends to the Dini integral,whilst the left-hand side tend to the oscillation of f .

25

26 Chapter 3. Integral smoothness and p-variation

The outline of this chapter is as follows. First, we determine the optimalasymptotics of the constant coefficient A = A(p) in (3.0.2) as p → 1 andp→∞. Using this version of (3.0.2) with improved constant, we obtain theestimate (1.0.11), which strengthens Terehin’s inequality (1.0.10). We alsoshow that (1.0.11) is sharp in a strong sense (see Theorem 3.12 below).

Next, we prove sharp estimates of the Riesz type variation (1.0.13) interms of Lp-moduli of continuity.

Finally, we give some remarks on certain spaces of functions defined interms of local oscillations.

3.1 Auxiliary results

We will use the next lemma at several places in this work.

Lemma 3.1. Let 0 < γ ≤ 1 and let ω ∈ Ωγ satisfy (2.3.6). Let 1 ≤ q < ∞and 0 < β < qγ be given numbers. Then

∞∑k=0

2nkβωqnk ≤ 2ωq0 +2qγ+2

qγβ(qγ − β)

∫ 1

0

t−βω(t)qdt

t.

Proof. By the first inequality of (2.3.8), we have

nk+1−1∑n=nk

2nβ(ωqn − ωqn+1) ≥ 2nkβ(ωqnk − ω

qnk+1

) ≥ 2nkβ−1ωqnk ,

for any k ≥ 0. This implies that

∞∑k=0

2nkβωqnk ≤ 2∞∑n=0

2nβ(ωqn − ωqn+1). (3.1.1)

Further, applying the second inequality of (2.3.8), we obtain

nk∑n=nk−1+1

2n(β−qγ)(ωqn − ωqn−1) ≥ 2nk(β−qγ)(ωqnk − ω

qnk−1

)

≥ 2nk(β−qγ)−1ωqnk ,

for any k ≥ 1. Thus,

∞∑k=1

2nkβωqnk ≤ 2∞∑n=1

2n(β−qγ)(ωqn − ωqn−1). (3.1.2)

3.1. Auxiliary results 27

Since∞∑n=0

2nβ(ωqn − ωqn+1) = ωq0 + (2β − 1)

∞∑n=1

2(n−1)βωqn,

and

(2β − 1)2(n−1)βωqn ≤ β

∫ 2−n+1

2−nt−βω(t)q

dt

t.

Whence,∞∑n=0

2nβ(ωqn − ωqn+1) ≤ ωq0 + β

∫ 1

0

t−βω(t)qdt

t. (3.1.3)

Further,

∞∑n=1

2n(β−qγ)(ωqn − ωqn−1) = (1− 2β−qγ)

∞∑n=1

2n(β−qγ)(ωqn − ωq0),

and by (2.3.5),∫ 2−n

2−n−1

t−βω(t)qdt

t≥

(ωn2γ

)q ∫ 2−n

2−n−1

t−β+qγ−1dt

= 2−qγωqn1− 2β−qγ

qγ − β2n(β−qγ).

Hence,

∞∑n=1

2n(β−qγ)(ωqn − ωqn−1) ≤ 2qγ(qγ − β)

∫ 1

0

t−βω(t)qdt

t. (3.1.4)

Denote

S =∞∑k=0

2nkβωqnk and J =

∫ 1

0

t−βω(t)qdt

t.

Then (3.1.1) and (3.1.3) imply that S ≤ 2(ωq0 + βJ). By (3.1.2) and (3.1.4),we have also that S ≤ ωq0 + 2qγ+1(qγ − β)J . Thus, we get

qγS ≤ 2qγωq0 + 2qγ+2β(qγ − β)J.


Lemma 3.2. Let f ∈ Lp([0, 1]) (1 ≤ p <∞) and let

ψh,µ(x) = fh(x)− fµ(x) (h, µ ∈ (0, 1]).

Then

‖ψh,µ‖∞ ≤ h1−1/pω(f ;µ)p

(1

h+

1

µ

)(3.1.5)

and

vp(ψh,µ) ≤ 5h1−1/pω(f ;µ)p

(1

h+

1

µ

). (3.1.6)

Proof. For any x we have

|ψh,µ(x)| ≤ 1

h

∫ x+h

x

|f(t)− fµ(x)|dt

≤ 1

h

∫ x+h

x

|f(t)− fµ(t)|dt+1

h

∫ x+h

x

|fµ(t)− fµ(x)|dt.

Further, for any t ∈ [x, x+ h]

|fµ(t)− fµ(x)| ≤∫ x+h

x

|f ′µ(u)|du.

Applying Holder’s inequality, we obtain

|ψh,µ(x)| ≤ h−1/p

(∫ x+h

x

|f(u)− fµ(u)|pdu)1/p

+ h1−1/p

(∫ x+h

x

|f ′µ(u)|pdu)1/p

. (3.1.7)

It follows that ‖ψh,µ‖∞ ≤ h1−1/p‖f ′µ‖p + h−1/p‖f − fµ‖p. Applying Lemma2.4, we obtain (3.1.5).

We shall now prove (3.1.6). Let Π = x0, x1, ..., xn be an arbitrarypartition and let K ′, K ′′ be defined by (4.1.2).

If j ∈ K ′, then, applying Holder’s inequality, we have

|ψh,µ(xj+1)− ψh,µ(xj)| ≤ |fh(xj+1)− fh(xj)|+ |fµ(xj+1)− fµ(xj)|

≤∫ xj+1

xj

(|f ′h(x)|+ |f ′µ(x)|)dx ≤ h1−1/p

(∫ xj+1

xj

(|f ′h(x)|+ |f ′µ(x)|)pdx

)1/p

.


Thus,

V ′ ≡

(∑j∈K′|ψh,µ(xj+1)− ψh,µ(xj)|p

)1/p

≤ h1−1/p(‖f ′h‖p + ‖f ′µ‖p). (3.1.8)

Further, let j ∈ K ′′. We have

|ψh,µ(xj+1)− ψh,µ(xj)| ≤ |ψh,µ(xj)|+ |ψh,µ(xj+1)|.

Using (3.1.7), we get

V ′′ ≡

(∑j∈K′′

|ψh,µ(xj+1)− ψh,µ(xj)|p)1/p

≤ h−1/p

(∑j∈K′′

∫ xj+h

xj

|f(t)− fµ(t)|pdt

)1/p

+h−1/p

(∑j∈K′′

∫ xj+1+h

xj+1

|f(t)− fµ(t)|pdt

)1/p

+h1−1/p

(∑j∈K′′

∫ xj+h

xj

|f ′µ(t)|pdt

)1/p

+h1−1/p

(∑j∈K′′

∫ xj+1+h

xj+1

|f ′µ(t)|pdt

)1/p

.

Observe that [xj, xj + h] ⊂ [xj, xj+1) for any j ∈ K ′′. Thus, if i < j andi, j ∈ K ′′, then [xi, xi + h] ∩ [xj, xj + h] = ∅ and⋃

j∈K′′[xj, xj + h] ⊂ [x0, xn] (xn = x0 + 1).

Further, if i < j and i, j ∈ K ′′, then xi+1 + h ≤ xj + h < xj+1. Thus,[xi+1, xi+1 + h] ∩ [xj+1, xj+1 + h] = ∅ and⋃

j∈K′′[xj+1, xj+1 + h] ⊂ [x0 + h, xn + h].


Taking into account these observations, we obtain

V ′′ ≤ 2h−1/p(‖f − fµ‖p + h‖f ′µ‖p). (3.1.9)

Using (3.1.8), (3.1.9), and Lemma 2.4, we have

vp(ψh,µ) ≤ h−1/p[h‖f ′h‖p + 3h‖f ′µ‖p + 2‖f − fµ‖p

]≤ h1−1/p

[ω(f ;h)p

h+ ω(f ;µ)p

(2

h+

3

µ

)].

Applying (2.1.2), we obtain (3.1.6).

The following result is well known (see, e.g., [2], [5, p. 346]).

Lemma 3.3. Let 1 ≤ p < q <∞. Then for any function f ∈ Lp([0, 1]) andany δ ∈ [0, 1]

ω(f, δ)q ≤ A

∫ δ

0

t1/q−1/pω(f ; t)pdt

t, (3.1.10)

where A is an absolute constant.

Corollary 3.4. If Jp(f) <∞ for some 1 < p <∞, then for any p < q <∞

Jq(f) ≤ AqJp(f), (3.1.11)


We shall use the following Hardy type inequality (see [39]).

Lemma 3.5. Let λk∞k=0 and αk∞k=0 be non-negative sequences.Assume that

λk+1 ≥ dλk (k = 0, 1, ...), where d > 1.

Let 1 ≤ p <∞. Set λ−1 = 0. Then(∞∑k=0

(λk − λk−1)

(∞∑j=k

αj

)p)1/p

≤ p

(d

d− 1

)1/p′(∞∑k=0

λkαpk

)1/p

.

3.2. Estimates of L∞-norm and p-variation 31

3.2 Estimates of L∞-norm and p-variation

Theorem 3.6. Let f ∈ Lp([0, 1]), 1 < p <∞. Assume that

Jp(f) =

∫ 1

0

t−1/pω(f ; t)pdt

t<∞. (3.2.1)

Then f is equivalent to a continuous function f ∈ Vp. Moreover,

‖f‖∞ ≤ A

(‖f‖p +

1

pp′Jp(f)

)(3.2.2)

and

vp(f) ≤ A

(Ωp(f) +

1

pp′Jp(f)

), (3.2.3)


Proof. Let ω ∈ Ω1 be an arbitrary modulus of continuity such that

ω(f ; t)p ≤ ω(t), t ∈ [0, 1], (3.2.4)

limt→0+

ω(t)/t =∞, (3.2.5)

and ∫ 1

0

t−1/p−1ω(t)dt <∞. (3.2.6)

Let nk = nk(ω, 1) be defined by (2.3.7). Set

ϕk(x) = 2nk∫ x+2−nk

x

f(t)dt, k = 0, 1, .... (3.2.7)

Since n0 = 0, we have ϕ0(x) =∫ 1

0f(t)dt = I. By Lebesgue’s differentiation

theorem, for almost all x ∈ [0, 1]

f(x) = I +∞∑k=0

(ϕk+1(x)− ϕk(x)). (3.2.8)

Set ψk = ϕk+1−ϕk. Fix k ≥ 0. Assume that (2.3.9) holds. Applying Lemma3.2 with h = 2−nk+1 and µ = 2−nk , we obtain

‖ψk‖∞ ≤ 2nk+1/p+1ω(f ; 2−nk)p


and

vp(ψk) ≤ 2nk+1/p+4ω(f ; 2−nk)p.

Thus, by (3.2.4) and (2.3.9),

‖ψk‖∞ ≤ 2nk+1/p+4ωnk+1(3.2.9)

and

vp(ψk) ≤ 2nk+1/p+7ωnk+1. (3.2.10)

Now we assume that (2.3.10) is true. Then, applying Lemma 3.2 with h =2−nk and µ = 2−nk+1 , we obtain

‖ψk‖∞ ≤ 2nk(1/p−1)+12nk+1ω(f ; 2−nk+1)p

and

vp(ψk) ≤ 2nk(1/p−1)+42nk+1ω(f ; 2−nk+1)p.

Using (3.2.4) and (2.3.10), we have

‖ψk‖∞ ≤ 2nk/p+4ωnk (3.2.11)

and

vp(ψk) ≤ 2nk/p+7ωnk . (3.2.12)

It follows from (3.2.9), (3.2.11), and (3.2.6), that the series (3.2.8) convergesuniformly on [0, 1]. Thus, f is equivalent to a continuous 1-periodic function.Moreover,

‖f‖∞ ≤ |I|+ 32∞∑k=0

2nk/pωnk . (3.2.13)

We may assume that f is continuous. Then by (3.2.8)

vp(f) ≤∞∑k=0

vp(ψk).

Applying (3.2.10) and (3.2.12), we get

vp(f) ≤ 256∞∑k=0

2nk/pωnk . (3.2.14)


By Lemma 3.1 with q = 1, β = 1/p, γ = 1,

∞∑k=0

2nk/pωnk ≤ 2ω0 +8

pp′

∫ 1

0

t−1/p−1ω(t)dt. (3.2.15)

Using (3.2.13), (3.2.14), and (3.2.15), we obtain

‖f‖∞ ≤ |I|+ A

[ω(1) +

1

pp′Dp,ω

](3.2.16)

and

vp(f) ≤ A

[ω(1) +

1

pp′Dp,ω

], (3.2.17)

where

Dp,ω =

∫ 1

0

t−1/p−1ω(t)dt

and A is an absolute constant. Here ω is any modulus of continuity satisfying(3.2.4) – (3.2.6). If ω(f ; t)p satisfies (3.2.5), then we take ω(t) = ω(f ; t)p.Otherwise, we take ω(t) = ω(f ; t)p + εtγ, where 1/p < γ < 1 and ε is anarbitrary positive number. Clearly, (3.2.4) – (3.2.6) are satisfied. Let ε→ 0.Then (3.2.16), (3.2.17), and (2.1.6) imply (3.2.2) and (3.2.3).

Remark 3.7. The condition (3.2.1) can not be improved. Moreover, it wasshown by Ul’yanov [74] that if ω ∈ Ω, 1 < p <∞, and∫ 1

0

t−1/pω(t)dt

t=∞,

then there exists an essentially unbounded function f ∈ Lp([0, 1]) such thatω(f ; t)p ≤ ω(t) (see also Theorem 3.12 and Theorem 3.18 below).

Remark 3.8. The term Ωp(f) on the right-hand side of (3.2.3) can not beomitted. Indeed, let f(x) = sin(2πx). Then for all p ≥ 1 we have ω(f ; t)p ≤2πt (t ∈ [0, 1]), and

1

pp′

∫ 1

0

t−1/pω(f ; t)pdt

t≤ 2π

p.

Thus, the second term on the right-hand side of (3.2.3) tends to 0 as p→∞.On the other hand, vp(f) ≥ 21/p (p ≥ 1).


Set for f ∈ Lp([0, 1]) and δ ∈ [0, 1]

ω(2)(f ; δ)p = sup0≤h≤δ

(∫ 1

0

|f(x+ h)− 2f(x) + f(x− h)|pdx)1/p

.

Terehin [71] proved that if f ∈ Lp([0, 1]), 1 ≤ p <∞, and

J (2)p (f) =

∫ 1

0

t−1/pω(2)(f ; t)pdt

t<∞, (3.2.18)

then f is equivalent to a continuous 1-periodic function f ∈ Vp and

vp(f) ≤ AJ (2)p (f), (3.2.19)

where A is an absolute constant. We have the following improvement of theprevious estimate.

Corollary 3.9. Let f ∈ Lp([0, 1]), 1 < p < ∞ and assume that (3.2.18)holds. Then f is equivalent to a continuous 1-periodic function f ∈ Vp and

vp(f) ≤ A

(Ωp(f) +

1

pJ (2)p (f)

). (3.2.20)

Proof. By Marchaud’s inequality ([14], p. 47)

ω(f ; t)p ≤ ct

(∫ 1

t

ω(2)(f ;u)pu2

du+ Ωp(f)

)(3.2.21)

for all 0 < t ≤ 1, where c is an absolute constant. Applying (3.2.21) andFubini’s theorem, we get

Jp(f) ≤ c

[p′Ωp(f) +

∫ 1

0

t−1/p

∫ 1

t

ω(2)(f ;u)pu2

dudt

]= cp′

[Ωp(f) +

∫ 1

0

u−1/pω(2)(f ;u)pdu

u

].

This estimate and (3.2.3) imply (3.2.20).

As above (see Remark 3.8), the term Ωp(f) on the right-hand side of(3.2.20) can not be omitted. However, we have that

Ωp(f) ≤ ω(2)(f ; 1/2)p ≤ 2

∫ 1

1/2


t. (3.2.22)


Indeed, by the periodicity of f ,

Ωp(f) ≤ 2

(∫ 1

0

∣∣∣∣f(x)−∫ 1

0

f(t)dt

∣∣∣∣p dx)1/p

= 2

(∫ 1

0

∣∣∣∣∣f(x)−∫ 1/2

0

[f(x+ u) + f(x− u)]du

∣∣∣∣∣p

dx

)1/p

≤ 21/p

(∫ 1/2

0

du

∫ 1

0

|f(x+ u) + f(x− u)− 2f(x)|pdx

)1/p

≤ ω(2)(f ; 1/2)p ≤ 2

∫ 1

1/2


t.

Inequality (3.2.19) follows from (3.2.20) and (3.2.22). We emphasize that,in comparison with (3.2.19), the right-hand side of (3.2.20) contains the factor1/p. This factor may play an essential role.

We shall consider trigonometric polynomials

Tn(x) =a0

2+

n∑k=1

(ak cos 2πkx+ bk sin 2πkx). (3.2.23)

Terehin [71] observed that (3.2.19) yields that for every trigonometric poly-nomial Tn of degree n and any 1 ≤ p <∞

vp(Tn) ≤ Apn1/p‖Tn‖p, (3.2.24)

where A is an absolute constant. Oskolkov [53, 54] proved that the coeffi-cient p on the right-hand side of (3.2.24) can be omitted. That is, for anytrigonometric polynomial of degree n and any 1 ≤ p <∞

vp(Tn) ≤ An1/p‖Tn‖p, (3.2.25)

where A is an absolute constant. Oskolkov’s proof was based on the use ofinterpolation methods. We note that (3.2.25) can be obtained from (3.2.20)or, more directly, from (3.2.3). Indeed, we have

ω(Tn; t)p ≤ min(t‖T ′n‖p, 2‖Tn‖p), t ∈ [0, 1].


Thus,

Jp(Tn) =

∫ 1

0

t−1/pω(Tn; t)pdt

t≤ ‖T ′n‖p

∫ 1/n

0

t−1/pdt

+ 2‖Tn‖p∫ 1

1/n

t−1/p−1dt ≤ p′‖T ′n‖pn1/p−1 + 2pn1/p‖Tn‖p.

By the Bernstein inequality [14, p. 97], ‖T ′n‖p ≤ 2πn‖Tn‖p, and we obtainJp(Tn) ≤ 2πpp′n1/p‖Tn‖p. We have also Ωp(Tn) ≤ 2‖Tn‖p. Applying theseestimates and (3.2.3), we obtain (3.2.25).

We give one more observation concerning the behaviour of the right-handside of (3.2.3) as p→∞.

It is easy to see that if f ∈ Vq for some q ≥ 1, then

limp→∞

vp(f) = osc(f) (3.2.26)

where osc(f) is the oscillation of f on [0, 1].The essential oscillation ess osc(f) of a measurable 1-periodic function f

is defined as the difference

ess supx∈[0,1]

f(x)− ess infx∈[0,1]

f(x).

Proposition 3.10. Let f be a 1-periodic measurable function. Assume thatJp0(f) <∞ for some 1 < p0 <∞. Then

limp→∞

Jp(f)

p≤ ess osc(f). (3.2.27)

Proof. Set ω0 = ess osc(f). Let p > p0. Then ω(f ; t)p ≤ ω0 and for any0 < h < 1 we have

1

p

∫ 1

h

t−1/pω(f ; t)pdt

t≤ ω0

p

∫ 1

h

t−1/p−1dt ≤ ω0h−1/p.

Further, applying Lemma 3.3 and Fubini’s theorem, we obtain

1

p

∫ h

0

t−1/pω(f ; t)pdt

t≤ A

p

∫ h

0

t−1/p

∫ t

0

u1/p−1/p0ω(f ; t)p0du

u

dt

t

≤ A

∫ h

0

u−1/p0ω(f ;u)p0du

u.


Let ε > 0. Then there exists h > 0 such that

A

∫ h

0

u−1/p0ω(f ;u)p0du

u< ε.

Thus, we have1

pJp(f) < ω0h

−1/p + ε

and therefore

limp→∞

1

pJp(f) ≤ ω0 + ε.

This implies (3.2.27).

Applying (3.2.26) and (3.2.27), we see that the behaviour of the right-hand side of (3.2.3) as p→∞ agrees with that of the left-hand side. Namely,the left-hand side of (3.2.3) tends to ω0 = ess osc(f) and the upper limit ofthe right-hand side does not exceed Aω0. Note that the right-hand side of(1.0.5) may tend to infinity as p→∞.

The results given above show that the constant factor A/(pp′) in (3.2.3)has an optimal order as p→∞. We observe now that its order is optimal asp→ 1, too.

Indeed, assume that f ∈ W 1q for some q > 1. Then, by (2.1.3),

Jp(f) ≤ ||f ′||p∫ 1

0

t−1/p dt = p′||f ′||p for any 1 < p ≤ q.

Thus,

limp→1

Jp(f)

p′≤ ||f ′||1 = v(f).

Further, for the first term on the right-hand side of (3.2.3) we have

limp→1

Ωp(f) ≤ ω(f ; 1)1

and for the left-hand side

limp→1

vp(f) = v(f).

It follows that the factor A/(pp′) in (3.2.3) can not be replaced by any factorα(p) such that limp→1 p

′α(p) = 0. Indeed, otherwise the inequality v(f) ≤Aω(f ; 1)1 would be true for any f ∈ W 1

q .


3.3 Estimates of the modulus of p-continuity

Let C denote the class of all continuous 1-periodic functions on R. Themodulus of continuity of a function f ∈ C is defined by

ω(f ; δ) = sup|x−y|≤δ

|f(x)− f(y)|, 0 ≤ δ ≤ 1.

Let f ∈ Lp([0, 1]) (1 < p <∞) and let

Jp(f) =

∫ 1

0

t−1/pω(f ; t)pdt

t<∞. (3.3.1)

Then we may assume that f ∈ C. Moreover,

ω(f ; δ) ≤ A

∫ δ

0

t−1/pω(f ; t)pdt

t, (3.3.2)

where A is an absolute constant (see [2],[56]). It was proved in [2] that (3.3.2)is sharp for any order of the modulus of continuity ω(f ; t)p.

Let ω ∈ Ω be a modulus of continuity and let 1 ≤ p <∞. Denote by Hωp

the class of all functions f ∈ Lp([0, 1]) such that

ω(f ; t)p ≤ ω(t), t ∈ [0, 1]. (3.3.3)

The result obtained in [2] can be formulated in the following equivalentway. Let 1 < p <∞. Then there exist positive constants c and c′ such thatfor any modulus of continuity ω and any δ ∈ (0, 1]

c′ξp,ω(δ) ≤ supf∈Hω

p

ω(f ; δ) ≤ cξp,ω(δ),

where

ξp,ω(δ) =

∫ δ

0

t−1/pω(t)dt

t.

Thus, for each separate value of δ it is impossible to strengthen (3.3.2).Namely, for any δ ∈ (0, 1] there exists a function fδ ∈ Hω

p such that

ω(fδ; δ) ≥ c′ξp,ω(δ).

3.3. Estimates of the modulus of p-continuity 39

However, a function f ∈ Hωp fitting all values δ may not exist. Moreover, it

was proved in [31] that the following refinement of (3.3.2) is true:(∫ 1

δ

t−pω(f ; t)pdt

)1/p

≤ cδ1/p−1

∫ δ

0

t−1/pω(f ; t)pdt

t(3.3.4)

for any δ ∈ (0, 1]. In particular, if ω(t) = t and f ∈ Hωp , then by (3.3.4)∫ 1

0

t−pω(f ; t)pdt <∞.

At the same time, ξp,ω(δ) = p′δ1−1/p, and the latter integral would diverge ifthe inequality ω(f ; δ) ≥ c′ξp,ω(δ) was true for all δ ∈ [0, 1].

In this section we study estimates of the modulus of p-continuity (1 < p <∞). As we have already mentioned above, such estimates were first obtainedby Terehin (see (1.0.10)). First we shall show that the constant coefficientsin (1.0.10) can be improved.

Theorem 3.11. Let f ∈ Lp([0, 1]) (1 < p <∞) and assume that Jp(f) <∞.Then f can be modified on a set of measure zero so as to become continuousand

ω1−1/p(f ; δ) ≤ A

[δ−1/pω(f ; δ)p +

1

pp′

∫ δ

0

t−1/pω(f ; t)pdt

t

], (3.3.5)

for any δ ∈ (0, 1], where A is an absolute constant.

Proof. By Theorem 3.6, we may assume that f is continuous. Let 0 < δ ≤ 1.We have (see (2.1.7))

ω1−1/p(f ; δ) ≤ ω1−1/p(fδ; δ) + vp(f − fδ). (3.3.6)

By (2.2.8) and (2.1.9),

ω1−1/p(fδ; δ) ≤ δ1−1/p||f ′δ||p ≤ δ−1/pω(f ; δ)p. (3.3.7)

Further, by Theorem 3.6,

vp(f − fδ) ≤ A

(Ωp(f − fδ) +

1

pp′Jp(f − fδ)

). (3.3.8)


First, by (2.1.8)

Ωp(f − fδ) ≤ 2||f − fδ||p ≤ 2ω(f ; δ)p (3.3.9)

and

ω(f − fδ; t)p ≤ 2||f − fδ||p ≤ 2ω(f ; δ)p (0 < t ≤ 1). (3.3.10)

Besides, we have ω(f − fδ; t)p ≤ ω(f ; t)p + ω(fδ; t)p. It is easy to see thatω(fδ; δ)p ≤ ω(f ; t)p. Thus,

ω(f − fδ; t)p ≤ 2ω(f ; t)p. (0 < t ≤ 1). (3.3.11)

Using estimates (3.3.10) and (3.3.11), we get

Jp(f − fδ) ≤ 2

[pδ−1/pω(f ; δ)p +

∫ δ

0

t−1/pω(f ; t)pdt

t

]. (3.3.12)

Applying (3.3.6) – (3.3.9) and (3.3.12), we obtain (3.3.5).

It is clear that ω(f ; δ) ≤ ω1−1/p(f ; δ) for any 1 < p < ∞. As we haveobserved, the estimate (3.3.2) for ω(f ; δ) can be strengthened (see (3.3.4)).Now we shall show that, in contrast to (3.3.2), the estimate (3.3.5) is sharpin the following strong sense.

Theorem 3.12. There exists a constant A > 0 such that for any 1 < p <∞and any ω ∈ Ω satisfying the condition∫ 1

0

t−1/pω(t)dt

t<∞, (3.3.13)

there is a continuous function f ∈ Hωp for which the inequality

ω1−1/p(f ; δ) ≥ A

[δ−1/pω(δ) +

1

pp′

∫ δ

0

t−1/pω(t)dt

t

](3.3.14)

holds for all δ ∈ (0, 1].

Proof. First we assume that (2.3.6) holds. Let nk = nk(ω) (see (2.3.7)). Weset εn = ωnk if n = nk for some k ∈ N and εn = 0 otherwise (n ∈ N). Then

∞∑n=ν

εn ≤ 2ων andν∑

n=1

2nεn ≤ 2ν+1ων (3.3.15)


for any ν ∈ N. Indeed, let j be the least natural number such that nj ≥ ν.Then by the first inequality in (2.3.8)

∞∑n=ν

εn =∞∑k=j

ωnk ≤ 2ωnj ≤ 2ων .

Similarly, denoting by s the greatest natural number such that ns ≤ ν andapplying the second inequality in (2.3.8), we get

ν∑n=1

2nεn =s∑

k=1

2nkωnk ≤ 2ns+1ωns ≤ 2ν+1ων .

Let ε(t) = εn for t ∈ (2−n−1, 2−n] (n ∈ N, n ≥ 2), and ε(t) = 0 for1/4 < t ≤ 1. Set

f(x) =

∫ 1/2

|x|ε(t)t−1−1/pdt (3.3.16)

if |x| ≤ 1/2, and extend f to the real line with period 1 (this constructionwas used before in [31, 32]).

We first estimate ω(f ; δ)p. Since f is 1-periodic, even and monotonicallydecreasing on [0, 1/2], we easily get that for any 0 < h ≤ 1/4,∫ 1

0

|f(x)− f(x+ h)|pdx ≤ 4

∫ 1/2

0

|f(x)− f(x+ h)|pdx

= 4∞∑n=2

∫ 2−n

2−n−1

(∫ x+h

x

ε(t)t−1−1/pdt

)pdx ≡ 4

∞∑n=2

Jn(h).

Let 2−ν−1 < h ≤ 2−ν (ν ∈ N, ν ≥ 2). If 2 ≤ n ≤ ν, then

Jn(h) ≤ 2(n+1)php(εn + εn−1)p.

Thus, by the second inequality in (3.3.15) and (2.1.2),

ν∑n=2

Jn(h) ≤ 23php

(ν∑

n=1

2nεn

)p

≤ 25pω(h)p. (3.3.17)

Let now n ≥ ν + 1. Then

Jn(h) ≤ 2−n−1

(∫ 2−ν+1

2−n−1

ε(t)t−1−1/pdt

)p

≤ 2−n

(n∑

k=ν−1

εk2k/p

)p

.



(n∑

k=ν−1

εk2k/p

)p

≤n∑

k=ν−1

2kεk

(∞∑

k=ν−1

εk

)p−1

.

Thus, by the first inequality in (3.3.15) and (2.1.2),

∞∑n=ν+1

Jn(h) ≤

(∞∑

k=ν−1

εk

)p−1 ∞∑n=ν−1

2−nn∑

k=ν−1

2kεk

= 2

(∞∑

k=ν−1

εk

)p

≤ 2p+1ωpν−1 ≤ 23p+1ω(h)p.

Using this estimate and (3.3.17), we obtain

ω(f ;h)p ≤ 64ω(h), 0 ≤ h ≤ 1/4. (3.3.18)

Now we estimate ω1−1/p(f ; δ) from below. Let s ≥ 3 and nν−1 < s ≤nν (ν = ν(s) ≥ 1). Set xj = 2−sj (j = 0, 1, ..., 2s−2) and

Vs =

(2s−2−1∑j=0

|f(xj+1)− f(xj)|p)1/p

.

First, we have

|f(x1)− f(x0)| =∫ 2−s

0

t−1−1/pε(t)dt =∞∑n=s

εn

∫ 2−n

2−n−1

t−1−1/pdt

≥ 1

2

∞∑n=s

εn2n/p =1

2

∞∑k=ν

2nk/pωnk .


Further,

2s−2−1∑j=1

|f(xj+1)− f(xj)|p =s−3∑m=0

2m+1−1∑j=2m

|f(xj+1)− f(xj)|p

=s−3∑m=0

2m+1−1∑j=2m

(∫ (j+1)2−s

j2−st−1−1/pε(t)dt

)p

=s−3∑m=0

εps−m−1

2m+1−1∑j=2m

(∫ (j+1)2−s

j2−st−1−1/pdt

)p

≥ 2s−1−ps−3∑m=0

εps−m−12−mp = 2s(1−p)−1

s−1∑n=2

εpn2np

≥ 2s(1−p)−12nν−1pωpnν−1.

Thus, we obtain

Vs ≥1

2Ds for nν−1 < s ≤ nν , (3.3.19)

where

Ds =∞∑k=ν

2nk/pωnk + 2s(1/p−1)2nν−1ωnν−1 .

Denote

ξ(δ) = δ−1/pω(δ) +1

pp′

∫ δ

0

t−1/pω(t)dt

t.

We shall show that there exists an absolute constant c such that

ξ(δ) ≤ cDs for 2−s ≤ δ < 2−s+1. (3.3.20)

Set

ξ1(δ) = δ−1/pω(δ) +1

pp′

∫ δ

2−nνt−1/pω(t)

dt

t.

We have at least one of the inequalities

ωnν−1 ≤ 8ωnν or ωnν ≤ 8ωnν−1 (3.3.21)

(see Chapter 2). If the first inequality holds, then

δ−1/pω(δ) ≤ δ−1/pωnν−1 ≤ 8δ−1/pωnν ≤ 2nν/p+3ωnν


and1

p

∫ δ

2−nνt−1/pω(t)

dt

t≤ 8

pωnν

∫ δ

2−nνt−1/pdt

t≤ 2nν/p+3ωnν .

Thus, ξ1(δ) ≤ 8Ds. Let now the second inequality in (3.3.21) hold. Then(see (2.1.2)) for 2−nν ≤ t ≤ 2−nν−1 we have ω(t)/t ≤ 2ωnν ≤ 16ωnν−1 . Hence,

ξ1(δ) ≤ 16ωnν−1

(δ1−1/p +

1

p′

∫ δ

0

t−1/pdt

)= 32δ1−1/pωnν−1 ≤ 2s(1/p−1)+6ωnν−1 ≤ 64Ds.

Further, let

ξ2(δ) =1

pp′

∫ 2−nν

0

t−1/pω(t)dt

t=

1

pp′

∞∑k=ν

∫ 2−nk

2−nk+1

t−1/pω(t)dt

t.

Fix k ≥ ν. If (2.3.9) holds, then

1

p

∫ 2−nk

2−nk+1

t−1/pω(t)dt

t≤ 8ωnk+1

2nk+1/p.

If (2.3.10) holds, then

1

p′

∫ 2−nk

2−nk+1

t−1/pω(t)dt

t≤ 2

p′ωnk+1

∫ 2−nk

0

t−1/pdt

≤ 16ωnk2nk(1/p−1) = 16ωnk2

nk/p.

Thus,

ξ2(δ) ≤ 16∞∑k=ν

2nk/pωnk ≤ 16Ds.

We have proved (3.3.20). It follows from (3.3.19) and (3.3.20) that

ω1−1/p(f ; δ) ≥ Aξ(δ)

for all 0 < δ ≤ 1/4, where A = 2−7. It remains to observe that for 1/4 < δ ≤1 we have ξ(δ) ≤ 8ξ(1/4).

Our theorem is proved if ω satisfies (2.3.6). Let now ω(t) = O(t). As in theproof of Theorem 3.6, take 1/p < γ < 1 and set ωn(t) = ω(t)+tγ/n (n ∈ N).Clearly, ωn satisfies (2.3.6) and (3.3.13). As we have proved, there exists a


constant A > 0 and a sequence of continuous 1-periodic functions fn suchthat ‖fn‖∞ ≤ Jp(ω1),

ω(fn; t)p ≤ ωn(t) ≤ ω(t) + tγ (0 ≤ t ≤ 1), (3.3.22)

and for 2−s ≤ δ < 2−s+1

vp(fn; Πs) ≥ A

[δ−1/pω(δ) +

1

pp′

∫ δ

0

t−1/pω(t)dt

t

], (3.3.23)

where Πs is the partition of [0, 1] by points 2−sj, (j = 0, 1, ..., 2s). Thefunctions fn are equibounded and equicontinuous (see (3.3.22) and (3.3.2)).Thus, by the Ascoli–Arzela theorem, there exists a subsequence fnk thatconverges uniformly to some function f ∈ C. It follows from (3.3.22) thatf ∈ Hω

p . Furthermore, (3.3.23) implies (3.3.14) for all δ ∈ [0, 1].

Remark 3.13. For 1 ≤ p, q < ∞ and 0 < s < 1, the Besov space Bsp,q

consists of all functions f ∈ Lp([0, 1]) such that

‖f‖bsp,q =

(∫ 1

0

(t−sω(f ; t)p)q dt

t

)1/q

<∞.

For 1 ≤ p <∞, the dyadic p-variation of a 1-periodic function f is given by

vp,k(f) =

2k−1∑j=0

|f((j + 1)2−k)− f(j2−k)|p1/p

(k ≥ 0).

It was shown in [68] that for 1 < p, q < ∞ and 1/p < s < 1, there areconstants c′, c′′ > 0 such that for any function f ∈ Bs

p,q, there holds

c′‖f‖bsp,q ≤

(∞∑k=0

2k(s−1/p)qvp,k(f)q

)1/q

≤ c′′‖f‖bsp,q . (3.3.24)

The result (3.3.24) has applications in probability.The left inequality of (3.3.24) follows easily from an alternative descrip-

tion of Besov spaces due to Ciesielski et al. (see [68] for references). Theproof of the right inequality given in [68] is more complicated. We observe


that the right inequality can be obtained directly from (3.3.5). Indeed, sincevp,k(f) ≤ ω1−1/p(f ; 2−k), we have

∞∑k=0

2k(s−1/p)qvp,k(f)q ≤∞∑k=0

(2k(s−1/p)ω1−1/p(f ; 2−k))q

≤ A∞∑k=0

2k(s−1/p)q

(∞∑j=k

2j/pω(f ; 2−j)p

)q

.

By the previous estimates and Hardy’s inequality (see, e.g., [39]) we have

∞∑k=0

2k(s−1/p)qvp,k(f)q ≤ cp,q,s

∞∑k=0

2ksqω(f ; 2−k)qp ≤ cp,q,s‖f‖qbsp,q .

3.4 The classes V αp

In this section we obtain sharp estimates of vp,α(f) (see (1.0.12)) in terms ofthe modulus of continuity ω(f ; δ)p.

Theorem 3.14. Let 1 < p < ∞ and 0 < α < 1/p′. Let f ∈ Lp([0, 1]) andassume that

Kp,α(f) =

(∫ 1

0

t−αp−1ω(f ; t)ppdt

t

)1/p

<∞. (3.4.1)

Then f is equivalent to a continuous function f ∈ V αp and

vp,α(f) ≤ cp,αKp,α(f), (3.4.2)

wherecp,α = Aα−1/p′(1/p′ − α)1/p (3.4.3)

and A is an absolute constant.

Proof. By Theorem 3.6, we may assume that f is continuous (indeed, (3.4.1)implies (3.2.1)). Set ω(δ) = ω(f ; δ)p. As in the proof of Theorem 3.6, wemay suppose that ω satisfies (2.3.6). Let nk be defined by (2.3.7). Fix apartition Π = x0, x1, ..., xn (xn = x0 + 1). Let

σk = j : 2−nk+1 < xj+1 − xj ≤ 2−nk (k = 0, 1, ...).

3.4. The classes V αp 47

For any function ϕ, set

Rk(ϕ) =

(∑j∈σk

|ϕ(xj+1)− ϕ(xj)|p

(xj+1 − xj)αp

)1/p

and

Sk(ϕ) =

(∑j∈σk

|ϕ(xj+1)− ϕ(xj)|p)1/p

.

For an integer k ≥ 0 we set µ(k) = k if (2.3.9) holds and µ(k) = k + 1 if(2.3.9) does not hold (in the latter case, (2.3.10) holds). Let

gk(x) = 2nµ(k)

∫ 2−nµ(k)

0

f(x+ t) dt.


|gk(xj+1)− gk(xj)|p

(xj+1 − xj)αp= (xj+1 − xj)−αp

∣∣∣∣∣∫ xj+1

xj

g′k(t)dt

∣∣∣∣∣p

≤ (xj+1 − xj)p−1−αp∫ xj+1

xj

|g′k(t)|pdt

≤ 2−nk(p−1−αp)∫ xj+1

xj

|g′k(t)|pdt

for any j ∈ σk. Thus, by (2.1.9),

Rk(gk) ≤ 2−nk(1/p′−α)||g′k||p ≤ 2−nk(1/p′−α)2nµ(k)ωnµ(k).

If (2.3.9) holds, then µ(k) = k and

Rk(gk) ≤ 2nk(α+1/p)ωnk .

If (2.3.9) does not hold, then (2.3.10) holds. In this case µ(k) = k + 1 andby (2.3.10)

Rk(gk) ≤ 2nk(α+1/p)+3ωnk .

We have alsoRk(f − gk) ≤ 2nk+1αSk(f − gk).


Using these estimates and denoting γk = Sk(f − gk), we obtain

vp,α(f ; Π) ≤

(∞∑ν=0

Rν(gν)p

)1/p

+

(∞∑ν=0

Rν(f − gν)p)1/p

≤ 8

(∞∑ν=0

2nν(αp+1)ωpnν

)1/p

+

(∞∑ν=0

2nν+1αpγpν

)1/p

. (3.4.4)

We estimate the latter sum. Applying Abel’s transform, we have

∞∑ν=0

2nν+1αpγpν =∞∑k=0

γpk +∞∑ν=0

(2nν+1αp − 2nναp)∞∑k=ν

γpk . (3.4.5)

Further, reasoning as in Theorem 3.6 (see (3.2.10) and (3.2.12)), we obtain

γk ≤ vp(f − gk) ≤ A∞∑

j=µ(k)

2nj/pωnj .

If (2.3.9) holds, then µ(k) = k and ωnk ≤ 8ωnk+1. If (2.3.9) does not hold,

then µ(k) = k + 1. Thus,

γk ≤ 8A∞∑

j=k+1

2nj/pωnj (k = 0, 1, ...). (3.4.6)

On the other hand, γk ≤ Sk(f) + Sk(gk). By (2.2.8) and (2.1.9),

Sk(gk) ≤ 2−nk/p′‖g′k‖p ≤ 2−nk/p

′ωnµ(k)

,

and we obtain, as above

Sk(gk) ≤ 2nk/p+3ωnk . (3.4.7)

Further, applying (3.3.5), we have

∞∑k=m

Sk(f)p =∑

j∈⋃∞k=m σk

|f(xj+1)− f(xj)|p ≤ ω1−1/p(f ; 2−nm)p

≤ Ap

[2nm/pωnm +

1

pp′

∫ 2−nm

0

t−1/pω(t)dt

t

]p.


Considering cases (2.3.9) and (2.3.10), we obtain that

1

pp′

∫ 2−nk

2−nk+1

t−1/pω(t)dt

t≤ 2nk/p+3ωnk + 2nk+1/p+3ωnk+1

.

Thus,∞∑k=m

Sk(f)p ≤

(A∞∑k=m

2nk/pωnk

)p

(m = 0, 1, ...).

Using this inequality, (3.4.6), and (3.4.7), we obtain

∞∑k=ν

γpk ≤ γpν + 2p

(∞∑

k=ν+1

Sk(gk)p +

∞∑k=ν+1

Sk(f)p

)

≤

(A′

∞∑k=ν+1

2nk/pωnk

)p

.

Thus, applying (3.4.5), we get

∞∑ν=0

2nν+1αpγpν ≤ Ap

[(∞∑k=0

2nk/pωnk

)p

+

+∞∑ν=1

(2nναp − 2nν−1αp)

(∞∑k=ν

2nk/pωnk

)p]. (3.4.8)

First we assume that αp < 4. Set λk = 2nkαp. Then λk+1 ≥ 2αpλk. ApplyingLemma 3.5 to the right-hand side of (3.4.8), we get(

∞∑ν=0

2nν+1αpγpν

)1/p

≤ Ap,α

(∞∑ν=0

2nν(αp+1)ωpnν

)1/p

,

where

Ap,α = Ap

(2αp

2αp − 1

)1/p′

≤ A1p(αp)−1/p′ = A1p

1/pα−1/p′ ≤ 2A1α−1/p′

(A1 is an absolute constant). Let now αp ≥ 4. Then, by Holder’s inequality(∞∑k=ν

2nk/pωnk

)p

≤∞∑k=ν

2nk(1+αp/2)ωpnk

(∞∑k=ν

2−nkαp′/2

)p−1

≤(

8

α

)p−1

2−nναp/2∞∑k=ν

2nk(1+αp/2)ωpnk


(we have used the condition αp′ < 1). Thus, applying (3.4.8), changing theorder of summations, and taking into account that αp ≥ 4, we obtain(

∞∑ν=0

2nν+1αpγpν

)1/p

≤ A

α1/p′

(∞∑ν=0

2nναp/2∞∑k=ν

2nk(1+αp/2)ωpnk

)1/p

≤ A1

α1/p′

(∞∑ν=0

2nν(αp+1)ωpnν

)1/p

.

These estimates and (3.4.4) yield that

vp,α(f ; Π) ≤ A

α1/p′

(∞∑ν=0

2nν(αp+1)ωpnν

)1/p

. (3.4.9)

Applying Lemma 3.1 with q = p, β = αp+ 1 and γ = 1, we have(∞∑ν=0

2nν(αp+1)ωpnν

)1/p

≤ 8[ω(1) + (αp+ 1)1/p(1/p′ − α)1/pKp,α(f)].

Further, (αp+ 1)1/p ≤ p1/p ≤ 2, and by (2.1.2)

(1/p′ − α)

∫ 1

0

t−αp−1ω(t)pdt

t≥ ω(1)p(1/p′ − α)

2p

∫ 1

0

tp(1−α)−2dt =ω(1)p

p2p.

Thus, (∞∑ν=0

2nν(αp+1)ωpnν

)1/p

≤ 48(1/p′ − α)1/pKp,α(f).

From here and (3.4.9) it follows that

vp,α(f ; Π) ≤ Aα−1/p′(1/p′ − α)1/pKp,α(f),


Remark 3.15. Assume that f ∈ W 1p (1 < p < ∞). It was proved in [11]

(see also [13]) that in this case

lims→1−

(1− s)1/p

(∫ 1

0

[t−sω(f ; t)p]pdt

t

)1/p

=

(1

p

)1/p

‖f ′‖p.

Thus, if α → 1/p′, the right hand side of (3.4.2) tends to cp‖f ′‖p. Thisagrees with Theorem 2.2 of F. Riesz. Besides, it shows that the order of theconstant cp,α in (3.4.2) as α→ 1/p′ is optimal.


Remark 3.16. We observe that the order of the constant (3.4.3) as α → 0also is optimal. Indeed, let 1 < p < ∞, 0 < α < 1/(2p′). Set f(x) =| sin πx|2α. Then vp,α(f) ≥ vp(f) ≥ 1. Further, it is easy to see thatω(f ; δ)p ≤ cpαδ

2α+1/p. Thus

α−1/p′Kp,α(f) ≤ cpα1/p

(∫ 1

0

tαp−1dt

)1/p

≤ cp.

This implies that the constant cp,α in (3.4.2) can not replaced by cp,α suchthat limα→0 cp,αα

1/p′ = 0.

Now we shall show that for 0 < α < 1/p′ the condition (3.4.1) is sharp.

Theorem 3.17. Let 1 < p <∞ and 0 < α < 1/p′. Assume that ω ∈ Ω is amodulus of continuity such that∫ 1

0

t−αp−1ω(t)pdt

t=∞. (3.4.10)

Then there exists a function f ∈ Hωp which is not equivalent to a function in

V αp .

Proof. The condition (3.4.10) implies (2.3.6). We define the function f as inTheorem 3.12 (see (3.3.16)). Then we have the estimate (3.3.18).

Let n ∈ N and ξk = 2−n+k−1 (k = 0, 1, ..., n). Then

n−1∑k=0

|f(ξk+1)− f(ξk)|p

(ξk+1 − ξk)αp=

n−1∑k=0

2(n+1−k)αpεpn−k

(∫ 2−n+k

2−n+k−1

t−1−1/pdt

)p

≥ 2−pn∑j=1

2j(αp+1)εpj .

This implies that

vp,α(f) ≥ 2−p∞∑j=1

2j(αp+1)εpj = 2−p∞∑k=1

2nk(αp+1)ωpnk .

It remains to show that the series at the right-hand side diverges.If (2.3.9) holds, then∫ 2−nk

2−nk+1

t−(αp+1)ω(t)pdt

t≤ 8p

αp+ 12nk+1(αp+1)ωpnk+1

.


If (2.3.10) holds, then∫ 2−nk

2−nk+1

t−(αp+1)ω(t)pdt

t≤ 8p

p(1− α)− 12nk(αp+1)ωpnk .


∞∑k=1

2nk(αp+1)ωpnk =∞.

3.5 On classes Up

It was recently shown in [10, 9] that the classes Vp play an important role inproblems of boundedness of superposition operators. In [10, 9, 8] there werealso studied classes Up defined in terms of local oscillations. We consider onecounterexample concerning these classes.

For any 1-periodic measurable and almost everywhere finite function f wedenote by ω(f ;x, δ) the essential oscillation of f on the interval (x−δ, x+δ),that is

ω(f ;x, δ) = ess sup|y−x|<δ

f(y)− ess inf|y−x|<δ

f(y), 0 < δ ≤ 1. (3.5.1)

It is easy to see that ω(f ;x, δ) is a measurable function of x for any fixed0 < δ ≤ 1.

For 1 ≤ p < ∞ we let Up denote the class of all measurable a.e. finite1-periodic functions f such that

supδ>0

(1

δ

∫ 1

0

ω(f ;x, δ)pdx

)1/p

<∞. (3.5.2)

This class was introduced in a slightly different but equivalent way in [8].Clearly, Up ⊂ L∞([0, 1]) (since if ω(f ;x0, δ) = ∞ for some x0 and someδ ∈ (0, 1/2], then ω(f ;x, 2δ) =∞ for any x ∈ (x0− δ, x0 + δ)). It was shownin [9] that

Vp ⊂ Up (1 ≤ p <∞) (3.5.3)

3.5. On classes Up 53

and for any 1 < p < ∞ there exists a function f ∈ Up which can not bemodified on a set of measure 0 so as to belong Vp.

Denote by Lip(1/p; p) the class of all measurable 1-periodic functionsf ∈ Lp([0, 1]) (1 ≤ p < ∞) such that ω(f ; δ)p = O(δ1/p). It follows from thedefinitions that

Up ⊂ Lip(1/p; p) (1 ≤ p <∞). (3.5.4)

By (3.5.3), Hardy-Littlewood’s Theorem 2.1, and (3.5.4), U1 = Lip(1; 1).For p > 1 the inclusion (3.5.4) is strict. Indeed, if f0 is the 1-periodicextension of log(1/|t|), |t| ∈ (0, 1/2], to the real line, then f0 ∈ Lip(1/p; p)for any p > 1; however, f0 /∈ Up since f0 is unbounded. With the use ofwavelet decompositions of Besov spaces, it was shown in [9] that there existsa bounded function in Lip(1/p; p) \ Up.

We observe that the latter result can be obtained from the followingtheorem proved by direct methods in [30].

Theorem 3.18. Let 1 0.

Since (3.5.5) holds with ω(t) = t1/p, there is a bounded function f ∈Lip(1/p; p) such that ω(f ;x, δ) ≥ 1 for any x ∈ R and any δ > 0. Clearly,f /∈ Up.

Chapter 4

Fractional smoothness offunctions via p-variation

Let 1 < p <∞, recall the definition (1.0.13) of the class V αp (0 ≤ α ≤ 1/p′).

For α = 1/p′, Theorem 2.2 states that a function f ∈ V1/p′p if and only if

f ∈ W 1p . For α = 0, we clearly have V 0

p = Vp. Thus, V αp (0 < α < 1/p′)

form a scale of spaces of fractional smoothness between Vp and W 1p .

Another characterization of W 1p is given by moduli of p-continuity. In-

deed, f ∈ W 1p if and only if ω1−1/p(f ; δ) = O(δ1/p′) (see Chapter 2).

Obviously, if f ∈ V αp (0 < α ≤ 1/p′), then

ω1−1/p(f ; δ) = O(δα). (4.0.1)

However, for 0 < α < 1/p′, the condition (4.0.1) does not imply that f ∈ V αp .

On the other hand, it is in general impossible to improve (4.0.1). The mainobjectives of this chapter are twofold:

(i) to obtain sharp relations between vp,α(f) and moduli of p-continuity;

(ii) to study limits in the scales generated by vp,α(f) and ω1−1/p(f ; δ).

4.1 Approximation with Steklov averages

We shall prove that the modulus of p-continuity “controls” the error of ap-proximation by Steklov averages in Vp.

55

56 Chapter 4. Fractional smoothness via p-variation

Lemma 4.1. Let 1 < p <∞ and f ∈ Vp. Then

vp(f − fh) ≤ 6ω1−1/p(f ;h). (4.1.1)

Proof. Let Π = x0, x1, ..., xn be any partition and set

K ′ = j : xj+1 − xj ≤ h, K ′′ = 0, 1, ..., n− 1 \K ′ (4.1.2)

Set also gh = f − fh and

V ′ =

(∑j∈K′|gh(xj+1)− gh(xj)|p

)1/p

and

V ′′ =

(∑j∈K′′

|gh(xj+1)− gh(xj)|p)1/p

.

Then vp(gh; Π) ≤ V ′ + V ′′. By Minkowski’s inequality

V ′ ≤

(∑j∈K′|f(xj+1)− f(xj)|p

)1/p

+

(∑j∈K′|fh(xj+1)− fh(xj)|p

)1/p

≤ ω1−1/p(f ;h) + ω1−1/p(fh;h).

Using (2.2.11), we getV ′ ≤ 2ω1−1/p(f ;h). (4.1.3)

We now estimate V ′′. We have

(V ′′)p = h−p∑j∈K′′

∣∣∣∣∫ h

0

[f(xj+1)− f(xj+1 + t)− f(xj) + f(xj + t)]dt

∣∣∣∣p .Applying the trivial inequality |a+b|p ≤ 2p(|a|p+|b|p) and Holder’s inequality,we obtain

(V ′′)p ≤ 2ph−1

∫ h

0

[∑j∈K′′

|f(xj+1 + t)− f(xj+1)|p+

+∑j∈K′′

|f(xj + t)− f(xj)|p]dt.

4.1. Approximation with Steklov averages 57

For t ∈ [0, h] and j ∈ K ′′ we have [xj, xj + t] ⊂ [xj, xj+1), and hence [xj, xj +t]∩ [xi, xi+ t] = ∅ for i, j ∈ K ′′, i 6= j. Moreover, since j ≤ n−1 and j ∈ K ′′,we have that xj + t ≤ xj+1 ≤ xn. Thus,⋃

j∈K′′[xj, xj + t] ⊂ [x0, xn],

and ∑j∈K′′

|f(xj + t)− f(xj)|p ≤ ω1−1/p(f ;h)p

for each t ∈ [0, h]. Furthermore, if i, j ∈ K ′′ and i < j, then xi+1 + t ≤xj + t ≤ xj+1. Whence, [xi+1, xi+1 + t] ∩ [xj+1, xj+1 + t] = ∅, i < j, and⋃

j∈K′′[xj+1, xj+1 + t] ⊂ [x0 + t, xn + t].

Thus, ∑j∈K′′

|f(xj+1 + t)− f(xj+1)|p ≤ ω1−1/p(f ;h)p

for each t ∈ [0, h]. It follows that

V ′′ ≤ 21+1/pω1−1/p(f ;h). (4.1.4)

By (4.1.3) and (4.1.4) we obtain

vp(f − fh) ≤ 6ω1−1/p(f ;h).

This completes the proof.

Remark 4.2. Applying Lemma 4.1, we can show that the PeetreK-functionalK(f, t;Vp,W

1p ) is equivalent to ω1−1/p(f ; tp

′).

Set ‖f‖Vp = |f(0)|+ vp(f) for f ∈ Vp. It is simple to show that ‖ · ‖Vp isa norm on Vp and that Vp is a Banach space with respect to this norm.

As in [14, p.172], we define the K-functional for the pair (Vp,W1p ) by the

equalityK(f, t;Vp,W

1p ) = inf

g∈W 1p

(‖f − g‖Vp + t‖g′‖p).

We emphasize that the second term on the right-hand side is only a seminormon W 1

p . We shall now prove that

ω1−1/p(f ; tp′) ≤ K(f, t;Vp,W

1p ) ≤ 8ω1−1/p(f ; tp

′). (4.1.5)


Fix an arbitrary t ∈ (0, 1] and set h = tp′. Let g = fh be the Steklov average

(2.1.7), then g ∈ W 1p . By (2.2.12) and (4.1.1), we have that

|f(0)− g(0)|+ vp(f − g) + h1/p′‖g′‖p ≤ 8ω1−1/p(f ;h).

Substituting h = tp′

above yields

‖f − g‖Vp + t‖g′‖p ≤ 8ω1−1/p(f ; tp′),

and therefore,

K(f, t;Vp,W1p ) ≤ 8ω1−1/p(f ; tp

′).

On the other hand, for any g ∈ W 1p , we have by (2.2.8) that

ω1−1/p(f ; tp′) ≤ ω1−1/p(f − g; tp

′) + ω1−1/p(g; tp

′)

≤ vp(f − g) + t‖g′‖p.

Taking infimum over all g ∈ W 1p , we obtain that

ω1−1/p(f ; tp′) ≤ K(f, t;Vp,W

1p ).

Thus, (4.1.5) is proved.

4.2 Limiting relations

Let f ∈ Lp([0, 1]) (1 < p <∞). It was proved in [11] that if

sup0<s<1

(1− s)∫ 1

0

(t−sω(f ; t)p)pdt

t<∞,

then f ∈ W 1p and

lims→1−

(1− s)1/p

(∫ 1

0

(t−sω(f ; t)p)pdt

t

)1/p

=

(1

p

)1/p

‖f ′‖p.

We shall consider a similar limiting relation involving the modulus of p-continuity instead of Lp-modulus of continuity. We begin with the followingproposition.

4.2. Limiting relations 59

Proposition 4.3. Let f ∈ W 1p (1 < p <∞). Then

limh→0+

ω1−1/p(f ;h)

h1/p′= ‖f ′‖p. (4.2.1)

Proof. It is a direct consequence of (2.2.8) that

limh→0+

ω1−1/p(f ;h)

h1/p′≤ ‖f ′‖p.

For h ∈ (0, 1], denote ∆hf(x) = f(x+ h)− f(x) and set

µ(h) = ‖f ′ − (∆hf)/h‖p.

Then

‖f ′‖p ≤ µ(h) +‖∆hf‖p

h≤ µ(h) +

ω(f ;h)ph

.

From here and (2.2.9), we obtain that

‖f ′‖p ≤ µ(h) +ω1−1/p(f ;h)

h1/p′, (4.2.2)

for any 0 < h ≤ 1. Further,

∆hf(x) =

∫ h

0

f ′(x+ t)dt.

Thus, applying Holder’s inequality and Fubini’s theorem, we have

µ(h) =

(∫ 1

0

∣∣∣∣f ′(x)− 1

h

∫ h

0

f ′(x+ t)dt

∣∣∣∣p dx)1/p

≤(

1

h

∫ h

0

(∫ 1

0

|f ′(x)− f ′(x+ t)|pdx)dt

)1/p

≤ ω(f ′;h)p.

Since f ′ ∈ Lp([0, 1]), ω(f ′;h)p → 0 as h→ 0. Thus, µ(h)→ 0 as h→ 0 andwe get from (4.2.2) that

limh→0+

ω1−1/p(f ;h)

h1/p′≥ ‖f ′‖p.

This completes the proof.


Theorem 4.4. Let f be an 1-periodic function. Then the following state-ments hold:

(i) if f ∈ W 1p (1 < p <∞), then

lims→1/p′−

(1/p′ − s)1/p

(∫ 1

0


t

)1/p

=

(1

p

)1/p

‖f ′‖p; (4.2.3)

(ii) if f ∈ Cp (1 < p <∞) and

lims→1/p′−

(1/p′ − s)∫ 1

0


t<∞,

then f ∈ W 1p .

Proof. We first prove the statement (i). Let f ∈ W 1p and s ∈ (0, 1/p′). Set

J(s, h) = p(1/p′ − s)∫ h

0


t, 0 ≤ h ≤ 1,

then we shall prove that

lims→1/p′−

J(s, 1) = ‖f ′‖pp.

By (4.2.1) we have that for any ε > 0, there is a number δ = δ(ε) > 0 suchthat for 0 < t < δ

‖f ′‖pp − ε <ω1−1/p(f ; t)p

tp−1< ‖f ′‖pp + ε. (4.2.4)

Multiplying (4.2.4) by tp−2−sp, integrating over [0, δ] and taking into accountthat p− 1− sp = p(1/p′ − s) yield the inequalities

δp−1−sp(‖f ′‖pp − ε) ≤ J(s, δ) ≤ δp−1−sp(‖f ′‖pp + ε).

It follows that

(1− δp−1−sp)‖f ′‖pp − εδp−1−sp ≤ ‖f ′‖pp − J(s, δ) ≤≤ (1− δp−1−sp)‖f ′‖pp + εδp−1−sp.


Furthermore, since f ∈ W 1p , we also have f ∈ Vp and

p(1/p′ − s)∫ 1

δ


t≤ p(1/p′ − s)δ−sp−1vp(f)p.

Therefore, ∣∣J(s, 1)− ‖f ′‖pp∣∣ ≤ (1− δp−1−sp)‖f ′‖p + εδp−1−sp

+ p(1/p′ − s)δ−sp−1vp(f)p.

As s → 1/p′−, the limit of the right hand side of this inequality is equal toε. Since ε > 0 is arbitrary, the proof of (i) is complete.

Let now f ∈ Cp. For any 0 < h < 1, let fh be the Steklov average off given by (2.1.7). Then fh ∈ W 1

p and f ′h(x) = [f(x + h) − f(x)]/h a.e.Applying (4.2.3) to the function fh and using (2.2.11), we have

1

p‖f ′h‖pp = lim

s→1/p′−(1/p′ − s)

∫ 1

0

[t−sω1−1/p(fh; t)]pdt

t

≤ lims→1/p′−

(1/p′ − s)∫ 1

0


t= C <∞.

On the other hand,

‖f ′h‖pp = h−p∫ 1

0

|f(x+ h)− f(x)|pdx.

Thus, (∫ 1

0

|f(x+ h)− f(x)|pdx)1/p

≤ Ch, h ∈ (0, 1].

Since f is continuous, Theorem 2.1 implies that f ∈ W 1p .

Remark 4.5. Milman [49] studied continuity properties of interpolationscales at the endpoints. In particular, it follows from his results that forany f ∈ W 1

p ,

lims→1−

(1− s)1/p

(∫ 1

0

(t−sK(f, t;Vp,W1p ))p

dt

t

)1/p

=

(1

p

)1/p

‖f ′‖p

Together with (4.1.5), this provides another look on (4.2.3).


We shall also give some limiting relations for the functionals vp,α(f) de-fined by (1.0.13).

Theorem 4.6. Let f be an 1-periodic function and let 1 0, then

limα→0+

vp,α(f) = vp(f). (4.2.6)

Proof. To prove (i), we first observe that

vp,α(f) ≤ vp,1/p′(f), 0 < α < 1/p′.

Further, let Π = x0, x1, ..., xn be any partition. Then, since

vp,α(f) ≥

(n−1∑k=0

|f(xk+1)− f(xk)|p

(xk+1 − xk)αp

)1/p

,

we get

limα→1/p′−

vp,α(f) ≥

(n−1∑k=0

|f(xk+1)− f(xk)|p

(xk+1 − xk)p−1

)1/p

.

Taking supremum over all partitions, we obtain

limα→1/p′−

vp,α(f) ≥ vp,1/p′(f).

Thus, (4.2.5) holds.We proceed to prove (ii). Since

vp(f) ≤ vp,α(f)

for any α > 0, it is sufficient to show that

limα→0+

vp,α(f) ≤ vp(f).


For any partition Π = x0, x1, ..., xn, we set

σk = j : 2−k−1 < xj+1 − xj ≤ 2−k,

and

Sk(f) =

(∑j∈σk

|f(xj+1)− f(xj)|p)1/p

.

Then

vp,α(f ; Π) ≤ 2α

(∞∑k=0

2kαpSk(f)p

)1/p

. (4.2.7)

Furthermore, by applying the Abel transform we have

∞∑k=0

2kαpSk(f)p =∞∑k=0

2kαp

[∞∑j=k

Sj(f)p −∞∑

j=k+1

Sj(f)p

]

=∞∑k=0

Sk(f)p + (1− 2−αp)∞∑k=1

2kαp∞∑j=k

Sj(f)p.

It is easy to see that∞∑j=k

Sj(f)p ≤ vp,α0(f)p2−kα0p.

Whence, for 0 < α < α0

∞∑k=0

2kαpSk(f)p ≤ vp(f)p + vp,α0(f)pαp∞∑k=1

2−k(α0−α)p.

Thus, by (4.2.7)

vp,α(f) ≤ 2α

(vp(f) + α1/pvp,α0(f)

(p

2(α0−α)p − 1

)1/p)

and it follows thatlimα→0+

vp,α(f) ≤ vp(f),

which concludes the proof.

Remark 4.7. The condition that f ∈ V α0p for some α0 > 0 in (ii) cannot be

omitted. Indeed, if f ∈ Vp has a discontinuity at some point, then vp,α(f) =∞ for all α > 0 whence lim vp,α(f) =∞, while vp(f) <∞.


4.3 Estimates of the Riesz-type variation

In this section we obtain a sharp estimate of vp,α(f) (see (1.0.13)) in termsof the modulus of p-continuity ω1−1/p(f ; δ).

Theorem 4.8. Let 1 < p < ∞ and let 0 < α < 1/p′. Assume that f ∈ Vpand that

Ip,α(f) =

(∫ 1

0

[t−αω1−1/p(f ; t)]pdt

t

)1/p

<∞. (4.3.1)

Then f ∈ V αp and

vp,α(f) ≤ A[vp(f) + cp,αIp,α(f)], (4.3.2)

where A is an absolute constant and

cp,α = p′α1/p(1/p′ − α)1/p. (4.3.3)

Proof. The condition (4.3.1) implies that f ∈ Cp. Let ω∗(t) be given by(2.3.1) and take ω ∈ Ω1/p′ such that

ω∗(t) ≤ ω(t), t ∈ [0, 1] (4.3.4)

andlimt→0+

ω(t)/t1/p′

=∞. (4.3.5)

We specify later how such ω can be obtained. As before, set ωn = ω(2−n)and ωn = 2n/p

′ωn. Let the natural numbers nk ≡ nk(ω, 1/p

′), k = 0, 1, ..., bedefined by (2.3.7). Set µ(k) = k if (2.3.9) holds and µ(k) = k + 1 if (2.3.10)holds, and define

gk(x) = 2nµ(k)

∫ 2−nµ(k)

0

f(x+ t)dt.

Fix a partition Π = x0, x1, ..., xn and set

σk = j : 2−nk+1 < xj+1 − xj ≤ 2−nk.

For any function ϕ we define

Rk(ϕ) =

(∑j∈σk

|ϕ(xj+1)− ϕ(xj)|p

(xj+1 − xj)αp

)1/p

4.3. Estimates of the Riesz-type variation 65

and

Sk(ϕ) =

(∑j∈σk

|ϕ(xj+1)− ϕ(xj)|p)1/p

.

By Holder’s inequality we have for j ∈ σk

|gk(xj+1)− gk(xj)|p

(xj+1 − xj)αp=

1

(xj+1 − xj)αp

∣∣∣∣∣∫ xj+1

xj

g′k(t)dt

∣∣∣∣∣p

≤ (xj+1 − xj)p−1−αp∫ xj+1

xj

|g′k(t)|pdt

≤ 2−nk(p−1−αp)∫ xj+1

xj

|g′k(t)|pdt.

Thus, by (2.2.12) and (4.3.4),

Rk(gk) ≤ 2−nk(1/p′−α)‖g′k‖p≤ 2−nk(1/p′−α)2nµ(k)/p

′ω1−1/p(f ; 2−nµ(k))

≤ 2−nk(1/p′−α)2nµ(k)/p′ωnµ(k)

.

If µ(k) = k, then Rk(gk) ≤ 2nkαωnk . If µ(k) = k + 1, then ωnk+1< 8ωnk

andRk(gk) ≤ 2−nk(1/p′−α)ωnk+1

≤ 2nkα+3ωnk . (4.3.6)

Thus, (4.3.6) holds for each k ∈ N. Further,

Rk(f − gk) ≤ 2nk+1αSk(f − gk). (4.3.7)


vp,α(f ; Π) ≤

(∞∑k=0

Rk(gk)p

)1/p

+

(∞∑k=0

Rk(f − gk)p)1/p

≤ 8

(∞∑k=0

2nkαpωpnk

)1/p

+

(∞∑k=0

2nk+1αpSk(f − gk)p)1/p

. (4.3.8)

We estimate the latter sum. Clearly, Sk(f − gk) ≤ vp(f − gk). Applying(4.1.1), we obtain

Sk(f − gk) ≤ 6ωnµ(k).


If µ(k) = k, then ωnk < 8ωnk+1and

2nk+1αSk(f − gk) ≤ 2nk+1α+3ωnk ≤ 2nk+1α+6ωnk+1.

If µ(k) = k + 1, then

2nk+1αSk(f − gk) ≤ 2nk+1α+3ωnk+1.

Thus (∞∑k=0

2nk+1αpSk(f − gk)p)1/p

≤ 64

(∞∑k=0

2nkαpωpnk

)1/p

.

It follows from the previous estimate and (4.3.8) that

vp,α(f ; Π) ≤ 72

(∞∑k=0

2nkαpωpnk

)1/p

.

Applying Lemma 3.1 with γ = 1/p′, q = p and β = αp yields

vp,α(f) ≤ 72

(2ωp0 + 2ppp′α(1/p′ − α)

∫ 1

0

t−αpω(t)pdt

t

)1/p

.

Set

Dp,α(ω) =

(∫ 1

0

t−αpω(t)pdt

t

)1/p

.

Since p1/p ≤ 2 and (p′)1/p ≤ p′, we obtain

vp,α(f) ≤ 300[ω0 + p′α1/p(1/p′ − α)1/pDp,α(ω)

]. (4.3.9)

If there holds

limt→0+

ω∗(t)

t1/p′=∞, (4.3.10)

then we take ω(t) = ω∗(t). In this case

Dp,α(ω) ≤ Ip,α(f)

and ω0 ≤ vp(f), by (2.3.2). Thus, (4.3.2) is proved in this case.If (4.3.10) does not hold, we take ωε(t) = ω∗(t) + εtγ where α < γ <

1/p′. Then ωε ∈ Ω1/p′ for each ε > 0 and ωε satisfies (4.3.4) and (4.3.5).Furthermore, by (2.3.2) and a simple calculation we have

Dp,α(ωε) ≤ Ip,α(f) + ε(p(γ − α))1/p


and ωε(1) ≤ vp(f) + ε. Thus, we get from (4.3.9) that

vp,α(f) ≤ 300(vp(f) + ε+ p′α1/p(1/p′ − α)1/p[Ip,α(f) + ε(p(γ − α))1/p]).

Letting ε→ 0 yields (4.3.2).

Remark 4.9. Assume that f ∈ W 1p (1 < p <∞). By Theorem 4.4

limα→1/p′−

(1/p′ − α)1/pIp,α(f) = p−1/p‖f ′‖p.

Further, vp(f) ≤ ‖f ′‖p for f ∈ W 1p . Thus, the upper limit as α → 1/p′− of

the right-hand side of (4.3.2) does not exceed A‖f ′‖p (where A is an absoluteconstant). On the other hand, by Proposition 4.6, the left-hand side of (4.3.2)tends to vp,1/p′(f) as α→ 1/p′−. Thus,

vp,1/p′(f) ≤ A‖f ′‖p.

This agrees with Theorem 2.2, and shows that the order of the constant(4.3.3) is optimal as α→ 1/p′−.

Remark 4.10. Assume that Ip,α0(f) < ∞ for some 0 < α0 < 1/p′. SinceIp,α(f) ≤ Ip,α0(f) for 0 < α ≤ α0, we get that

limα→0+

α1/pIp,α(f) ≤ limα→0+

α1/pIp,α0(f) = 0.

Thus, as α→ 0+, the limit of the right-hand side of (4.3.2) does not exceedAvp(f) (where A is an absolute constant). On the other hand, if Ip,α0(f) <∞, then f ∈ V a0

p by Theorem 4.8. By (4.2.6), vp,α(f) → vp(f) as α → 0+.Thus, the behaviour of the left-hand side of (4.3.2) agrees with the behaviourof the right-hand side as α→ 0+.

Remark 4.11. We shall study the relationship between Theorem 4.8 andTheorem 3.14. In particular, we compare the estimates (3.4.2) and (4.3.2).For 1 < p <∞, 0 < α < 1/p′ and f ∈ Vp, we have

Kp,α(f) ≤ Ip,α(f) ≤ C

αKp,α(f), (4.3.11)

where C is an absolute constant. Indeed, the left inequality is an immediateconsequence of (2.2.9), while the right inequality follows from the estimateof Theorem 3.11 combined with Hardy’s inequality (see [37, p.7]).


By (3.4.2) and the left inequality of (4.3.11), we have

vp,α(f) ≤ Ac′p,αIp,α(f),

where A is an absolute constant and c′p,α = α−1/p′(1/p′ − α)1/p. Observethat for small α > 0, the constant c′p,α is much larger than the constant cp,αgiven by (4.3.3). Indeed, c′p,α → ∞ as α → 0+, while cp,α → 0 as α → 0+.Thus, (4.3.2) with the sharp constant (4.3.3) cannot be obtained from (3.4.2).However, as was observed, the order of the constant in (3.4.2) as α→ 0+ isoptimal.

Now we show that for 0 < α < 1/p′ the condition (4.3.1) is sharp.

Theorem 4.12. Let 1 < p < ∞ and 0 < α < 1/p′. Assume that ω ∈ Ω1/p′

is any modulus of p-continuity such that∫ 1

0

(t−αω(t))pdt

t=∞. (4.3.12)

Then there is a function f ∈ Vp such that ω1−1/p(f ; δ) ≤ ω(δ) but f /∈ V αp .

Proof. Define ωn, ωn by (2.3.3) with γ = 1/p′. The condition (4.3.12) impliesthat ω(δ) 6= O(δ1/p′), thus we may construct nk∞k=0 by (2.3.7).

For k = 1, 2, ..., set ξk = 2−nk , δk = 2−nk−2 and Ik = [ξk − δk, ξk + δk].Then Ik ⊂ (0, 1). Further, since nk+1 ≥ nk + 1, we have ξk+1 + δk+1 < ξk− δkand thus the intervals Ikk∈N are pairwise disjoint and ordered from theright to the left.

For k ∈ N, define ϕk as a continuous 1-periodic function such that ϕk(x) =0 for x ∈ [0, 1]\Ik, ϕk(ξk) = ωnk , and ϕk is linear on [ξk−δk, ξk] and [ξk, ξk+δk].Set

f(x) =∞∑k=1

ϕk(x).

We shall estimate ω1−1/p(f ; 2−s) for s ∈ N. Assume that nm ≤ s < nm+1 forsome m ≥ 1. Clearly, there holds

ω1−1/p(f ; 2−s) ≤∞∑k=1

ω1−1/p(ϕk; 2−s).

For each k ≥ m+ 1 we have the trivial estimate

ω1−1/p(ϕk; 2−s) ≤ v1(ϕk) = 2ωnk .


Fix 1 ≤ k ≤ m. Observe that

|ϕ′k(x)| = 2nk+2ωnk , x ∈ (ξk − δk, ξk) ∪ (ξk, ξk + δk),

andϕ′k(x) = 0, x ∈ [0, 1] \ Ik.

By (2.2.8), we have

ω1−1/p(ϕk; 2−s) ≤ 2−s/p′‖ϕ′k‖p = 2−s/p

′(∫

Ik

2(nk+2)pωpnkdx

)1/p

= 2−s/p′+2−1/pωnk .

By (2.3.8),

ω1−1/p(f ; 2−s) ≤ 4

[2−s/p

′m∑k=1

ωnk +∞∑

k=m+1

ωnk

]≤ 8(2−s/p

′ωnm + ωnm+1).

Further, since nm ≤ s < nm+1, we have ωnm ≤ ωs = 2s/p′ωs, and ωnm+1 ≤ ωs.

Thus, ω1−1/p(f ; 2−s) ≤ 16ωs. This implies that

ω1−1/p(f ; δ) ≤ 32ω(δ) for 0 ≤ δ ≤ 1.

We shall prove that f /∈ V αp . For any N ∈ N, consider the points

0 < ξN − δN < ξN < ξN−1 − δN−1 < .... < ξ1 − δ1 < ξ1 < 1.

Clearly

vp,α(f) ≥

(N∑k=1

|f(ξk)− f(ξk − δk)|p

δαpk

)1/p

= 4α

(N∑k=1

2nkαpωpnk

)1/p

.

Thus,

vp,α(f) ≥ 4α

(∞∑k=1

2nkαpωpnk

)1/p

.

It remains to show that the series at the right-hand side diverges.



2−nk+1

(t−αω(t))pdt

t≤ 8p

αp2nk+1αpωpnk+1

.


2−nk+1

(t−αω(t))pdt

t≤ 8p

p− 1− αp2nkαpωpnk .


∞∑k=1

2nkαpωpnk =∞.

Chapter 5

Embeddings within the scale Vp

Let 1 < p < q <∞ and let f ∈ Vq. By Jensen’s inequality, we have that

ω1−1/q(f ; δ) ≤ ω1−1/p(f ; δ) (0 ≤ δ ≤ 1).

The main objective of this chapter is to obtain sharp reverse inequalities,that is, estimates of ω1−1/p(f ; δ) in terms of ω1−1/q(f ; δ). Such problems fordifferent scales of function spaces of fractional smoothness have been stud-ied for a long time. We shall first discuss some previous results concerningLr−moduli of continuity.

5.1 Some known results and statement of prob-

lem

One of the origins of embedding theory is the classical Hardy-Littlewoodtheorem on Lipschitz classes [28]. This theorem states that if

ω(f ; δ)r = O(δα) (1 ≤ r <∞, 0 < α ≤ 1),

r < s < ∞, and θ = 1/r − 1/s < α, then ω(f ; δ)s = O(δα−θ). Problems ongeneral relations between moduli of continuity in different norms and theirsharpness were first posed and studied in the works by Ul’yanov [74] – [76].Therein, the conception of sharpness was formulated in terms of necessaryand sufficient conditions for embeddings of classes of functions

Hωr = f ∈ Lr([0, 1]) : ω(f ; δ)r = O(ω(δ)),

71

72 Chapter 5. Embeddings within the scale Vp

where ω is a given majorant (ω ∈ Ω1) (see Definiton 2.10).Ul’yanov [75] proved that

Hωr ⊂ Ls ⇐⇒

∫ 1

0

(t−θω(t))sdt

t<∞, (5.1.1)

where 1 ≤ r < s < ∞, θ = 1/r − 1/s, ω ∈ Ω1. Furthermore, he obtained ageneral estimate

ω(f ; δ)s ≤ c

(∫ δ

0

(t−θω(f ; t)r)sdt

t

)1/s

(0 ≤ δ ≤ 1), (5.1.2)

where 1 ≤ r < s <∞ and θ = 1/r−1/s (see also [56] and references in [31]).Andrienko [3] proved that this estimate is sharp in the following sense

Hωr ⊂ Hη

s ⇐⇒ µr,s,ω(δ) ≡(∫ δ

0

(t−θω(t))sdt

t

)1/s

= O(η(δ)), (5.1.3)

whatever be ω, η ∈ Ω1 and 1 ≤ r < s <∞. We observe that this result canbe expressed in an alternative form. Set

Hω

r = f ∈ Lr([0, 1]) : ω(f ; δ)r ≤ ω(δ).

It is easy to see that (5.1.3) is equivalent to the following statement: there isa constant c > 0 such that for every δ ∈ [0, 1] there exists a function fδ ∈ H

ω

r

for which

ω(fδ; δ)s ≥ c

(∫ δ

0

(t−θω(t))sdt

t

)1/s

. (5.1.4)

We stress that for different values of δ we get different functions fδ. If r > 1,it may not exist a single function f fitting all δ ∈ (0, 1] (see [31]). Moreover,it was proved in [31] that inequality (5.1.2) can be strengthened in a sense.

One of the questions that we consider in this chapter can be formulatedin the following way: if 1 < p < q < ∞ and f ∈ Vq, what is the necessaryand sufficient condition on the rate of decay of ω1−1/q(f ; δ) in order to havef ∈ Vp? Of course, this is related to the problem of finding estimates ofω1−1/p(f ; δ) in terms of ω1−1/q(f ; δ). We shall show that the answers to thesequestions are given by results that are formally analogous to (5.1.1) and(5.1.2). However, the analogy fails to be complete. We show that in contrastto (5.1.2), the corresponding inequality for moduli of p-continuity is sharpin a stronger sense. Namely, the extremal function (similar to the one in(5.1.4)) can be chosen independently of δ.



We shall use the following construction.

Definition 5.1. Let I = [a, b] ⊂ [0, 1] be an interval and N ∈ N. Seth = (b−a)/N , and let ξj = a+ jh for j = 0, 1, ..., N , and ξ∗j = a+ (j+ 1/2)hfor j = 0, 1, ..., N − 1.

The function F (I,N,H;x) is defined to be the continuous 1-periodic func-tion such that F (x) = 0 for x ∈ [0, 1] \ I, F (ξj) = 0 (j = 0, 1, ..., N),F (ξ∗j ) = H (j = 0, 1, ..., N − 1), and F is linear on each of the intervals[ξj, ξ

∗j ] and [ξ∗j , ξj+1] (j = 0, 1, ..., N − 1).

Thus, the graph of F consists of N congruent isosceles triangles withheight H and base h. Using (2.2.1) and Lemma 2.6, we have

vr(F ) = (2N)1/rH (1 ≤ r <∞), (5.2.1)

and‖F ′‖r = 2h−1/r′N1/rH (1 ≤ r <∞). (5.2.2)

Lemma 5.2. Let 1 < p < q <∞ and θ = 1/p− 1/q. Suppose that ω ∈ Ω1/q′

satisfies (2.3.6) and let nk = nk(ω) (k ∈ N) be defined by (2.3.7). Fix naturalnumbers ν < µ and a number γ ∈ (0, 1]. Let

σ =

(µ−1∑k=ν

2nkθqωqnk

)1/q

.

Then there exists a nonnegative continuous 1-periodic function f such that:

(i) (supp f) ∩ [0, 1] = [0, α], where

α ≤ 1

8

(γ +

µ−1∑k=ν

2−nk

); (5.2.3)

(ii) for any 0 < h ≤ 1,

ω1−1/q(f ;h) ≤ 16

µ−1∑k=ν

min(1, (2nkh)1−1/q)ωnk (5.2.4)

andω1−1/p(f ;h) ≤ 16 min(σ, h1−1/pωnµ−1); (5.2.5)


(iii) the following estimates from below hold:

ω1−1/p(f ; 2−nν ) ≥ 21/qγθσ, (5.2.6)

and

ω1−1/p(f ; 2−nµ−1) ≥ 21/qγθσ

(2nµ−1θωnµ−1

σ

)q/p. (5.2.7)

Proof. Set1

Nk = [2nkq/pωqnkσ−qγ] + 1 (k = ν, ..., µ− 1),

αν = 0, αm =m−1∑k=ν

Nk2−nk−3 (m = ν + 1, ..., µ),

Ik = [αk, αk+1] for k = ν, ..., µ− 1.

Further, for each ν ≤ k ≤ µ − 1, let Hk = ωnkN−1/qk and set Fk(x) =

F (Ik, Nk, Hk;x), and

f(x) =

µ−1∑k=ν

Fk(x).

Clearly, (supp f) ∩ [0, 1] = [0, αµ] and

αµ ≤1

8

(σ−qγ

µ−1∑k=ν

2nk(q/p−1)ωqnk +

µ−1∑k=ν

2−nk

)=

1

8

(γ +

µ−1∑k=ν

2−nk

).

This implies (i).Let now 0 < h ≤ 1. We have (see (5.2.1))

ω1−1/q(Fk;h) ≤ vq(Fk) = 21/qωnk ,

and by (2.2.8)

ω1−1/q(Fk;h) ≤ h1−1/q‖F ′k‖q ≤ 16h1−1/q2nk(1−1/q)ωnk .

These estimates imply (5.2.4).

1[x] denotes the integral part of a number x.


Further, each Fk is nonnegative and equals to 0 at the endpoints of theinterval Ik = suppFk. Moreover, the intervals Ik have disjoint interiors.Thus, taking into account Lemma 2.6 and (5.2.1), we get

vp(f)p =

µ−1∑k=ν

vp(Fk)p = 2

µ−1∑k=ν

HpkNk = 2

µ−1∑k=ν

ωpnkN1−p/qk .

Since Nk ≤ 2nkq/pωqnkσ−q + 1, we have, by applying the first inequality in

(2.3.8)

vp(f)p ≤ 2

(σp−q

µ−1∑k=ν

2nkθqωqnk +

µ−1∑k=ν

ωpnk

)≤ 2(σp + 2ωpnν ) ≤ 6σp.

Thus,ω1−1/p(f ;h) ≤ vp(f) ≤ 61/pσ, 0 < h ≤ 1. (5.2.8)

Further, by (5.2.2)

‖f ′‖pp =

µ−1∑k=ν

‖F ′k‖pp =

µ−1∑k=ν

(Hk2nk+4)pNk2

−nk−3

= 24p−3

µ−1∑k=ν

N1−p/qk 2nk(p−1)ωpnk ≤ 24p−3(A+B),

where

A =

µ−1∑k=ν

2nk(p−1)ωpnk

and

B = σp−qµ−1∑k=ν

2nkp(1−1/q)ωpnk(2nkθqωqnk)

1−p/q.

By the second inequality in (2.3.8), we have that

A ≤µ−1∑k=ν

2−nk(1−p/q)ωpnk ≤µ−1∑k=ν

ωpnk ≤ 2ωpnµ−1.

Further, applying Holder’s inequality and (2.3.8), we obtain

B ≤ σp−q

(µ−1∑k=ν

ωqnk

)p/q(µ−1∑k=ν

2nkθqωqnk

)1−p/q

≤ 2p/qωpnµ−1.


Thus, ‖f ′‖p ≤ 16ωnµ−1 , and by (2.2.8) we have that

ω1−1/p(f ;h) ≤ 16h1−1/pωnµ−1 , 0 < h ≤ 1. (5.2.9)

Estimates (5.2.8) and (5.2.9) imply (5.2.5).All extremal points of the functions Fk (k = ν, ..., µ− 1) subdivide [0, αµ]

into intervals with lengths smaller than 2−nν . This implies that

ω1−1/p(f ; 2−nν )p ≥ 2

µ−1∑k=ν

NkHpk = 2

µ−1∑k=ν

N1−p/qk ωpnk .

Since Nk ≥ 2nkq/p−1ωqnkσ−qγ, we have

ω1−1/p(f ; 2−nν )p ≥ 2p/qσp−qγ1−p/qµ−1∑k=ν

2nkθqωqnk = 2p/qγ1−p/qσp.

This proves (5.2.6). Finally, subdividing [0, αµ] into intervals of the length2−nµ−1−4, we take only the terms related to the interval [αµ−1, αµ]. Thus, weobtain

ω1−1/p(f ; 2−nµ−1)p ≥ 2Nµ−1Hpµ−1 ≥ 2p/qγ1−p/qσp−q2nµ−1θqωqnµ−1

.


We shall also use van der Waerden type functions to prove the followingstatement.

Lemma 5.3. Let 1 ≤ p < q < ∞ and θ = 1/p − 1/q. Suppose that ω ∈Ω1/q′ satisfies (2.3.6). Then there exists a nonnegative continuous 1-periodicfunction ψ such that

(suppψ) ∩ [0, 1] = [0, 1/2], (5.2.10)

ω1−1/q(ψ; δ) ≤ ω(δ), 0 ≤ δ ≤ 1, (5.2.11)

and

ω1−1/p(ψ; δ) ≥ Aδ−θω(δ), 0 < δ ≤ 1, (5.2.12)



Proof. Let nk = nk(ω) (see (2.3.7)). Denote

I = [0, 1/2], Nk = 2nk , Hk = ωnk2−nk/q.

Further, applying Definition 5.1, we set

gk(x) = F (I,Nk, Hk;x) (k ∈ N)

and

g(x) =∞∑k=1

gk(x).

Then g is a nonnegative, continuous and 1-periodic function satisfying (5.2.10).By (5.2.1), we have

ω1−1/q(gk;h) ≤ vq(gk) = 21/qωnk (5.2.13)

for any 0 < h ≤ 1. Besides,

|g′k(x)| = 2nk+2Hk = 4ωnk (k ∈ N) (5.2.14)

for almost all x ∈ I (g′k(x) = 0 for x ∈ (0, 1) \ I). By (2.2.8), it follows that

ω1−1/q(gk;h) ≤ 4h1−1/qωnk , 0 < h ≤ 1. (5.2.15)

Let 2−nj+1 < h ≤ 2−nj (j ∈ N). Then, by (5.2.13), (5.2.15), and (2.3.8),

ω1−1/q(g;h) ≤ 4

(∞∑

k=j+1

ωnk + h1−1/q

j∑k=1

ωnk

)≤ 8(ωnj+1

+ h1−1/qωnj) ≤ 16ω(h). (5.2.16)

Now we estimate ω1−1/p(g; 2−m) from below. We shall use the inequality

ω1−1/p(gk;h) ≤ 22−1/ph1−1/pωnk (0 < h ≤ 1), (5.2.17)

which follows directly from (2.2.8) and (5.2.14). Fix an integer m ≥ 0. Letnµ ≤ m < nµ+1. First we assume that

ωnµ+1 < 8ωnµ . (5.2.18)


Set hm = 2−m−2 and let Π be the partition of [0, 1] by the points xi = ihm (i =0, 1, ..., 2m+2). Then gk(xi) = 0 (i = 0, 1, ..., 2m+2) for all k ≥ µ+ 1. Further,if µ ≥ 2, then by (5.2.17) and (2.3.8),

µ−1∑k=1

vp(gk; Π) ≤ 22−1/ph1−1/pm

µ−1∑k=1

ωnk ≤ 23−1/ph1−1/pm ωnµ−1

≤ 21/p′h1−1/pm ωnµ .

On the other hand, the function gµ is linear on each of the intervals [xi, xi+1].Thus, by (5.2.14),

vp(gµ; Π) = 4ωnµhm2(m+1)/p = 21+1/p′h1−1/pm ωnµ .

Applying these estimates and (5.2.18), we obtain

vp(g; Π) ≥ 21/p′h1−1/pm ωnµ ≥ 21/p′−3h1−1/p

m ωm = 2−1/p′−32mθωm.

This implies thatω1−1/p(g; 2−m) ≥ 2mθ−4ωm (5.2.19)

for nµ ≤ m < nµ+1, provided that (5.2.18) holds.If (5.2.18) does not hold, then

8ωnµ ≤ ωnµ+1 (5.2.20)

and thus (see (2.3.9), (2.3.10) above)

8ωnµ+1 > ωnµ . (5.2.21)

Set tµ = 2−nµ+1−2 and let Π′ be the partition of [0, 1] by points xi = itµ (i =0, 1, ..., 2nµ+1+2). Then gj(xi) = 0 for all j ≥ µ+ 2. Further, by (5.2.17) and(5.2.20), we have

µ∑k=1

vp(gk; Π′) ≤ 22−1/pt1−1/pµ

µ∑k=1

ωnk ≤ 23−1/pt1−1/pµ ωnµ

≤ 2−1/pt1−1/pµ ωnµ+1 = 2−2+1/p2nµ+1θωnµ+1 .

On the other hand,

vp(gµ+1; Π′) = 2nµ+1θ+1/pωnµ+1 .


Applying (5.2.21), we obtain

vp(g; Π′) ≥ 2−1/p′2nµ+1θωnµ+1 ≥ 2−1/p′−32nµ+1θωm ≥ 2mθ−4ωm.

Thus, (5.2.19) is true for nµ ≤ m < nµ+1 also in the case when (5.2.18)does not hold. Now, the statement of the lemma follows from (5.2.16) and(5.2.19).

Lemma 5.4. Let 1 < p < q <∞ and θ = 1/p− 1/q. Assume that ω ∈ Ω1/q′

satisfies (2.3.7) and let nk = nk(ω) (k ∈ N). Then:

(i) the series∞∑m=1

2mθqωqm (5.2.22)

converges if and only if the series

∞∑k=1

2nkθqωqnk (5.2.23)

converges;

(ii) if

rn =

(∞∑m=n

2mθqωqm

)1/q

(n = 0, 1, ...)

and

ρ(ν) =

(∞∑k=ν

2nkθqωqnk

)1/q

(ν ∈ N),

then

rn ≤ c(2nθωn + ρ(ν + 1)) for nν ≤ n < nν+1 (ν ∈ N), (5.2.24)

where c = c(p, q) depends only on p and q.

Proof. Denote

Sk =

nk+1−1∑m=nk

2mθqωqm.


Assume that (2.3.9) holds. Then

Sk ≤ 8qωqnk+1

nk+1−1∑m=nk

2mθq ≤ c2nk+1θqωqnk+1. (5.2.25)

If (2.3.9) does not hold, then (2.3.10) holds. In this case

Sk =

nk+1−1∑m=nk

2−mq/p′ωqm ≤ 8qωqnk

∞∑m=nk

2−mq/p′

= cωqnk2−nkq/p′ = c2nkθqωqnk . (5.2.26)

Estimates (5.2.25) and (5.2.26) yield the statement (i). Similarly, we have

nν+1−1∑m=n

2mθqωqm ≤ cmax(2nθqωqn, 2nν+1θqωqnν+1

). (5.2.27)

Since

rqn =

nν+1−1∑m=n

2mθqωqm +∞∑

k=ν+1

Sk,

estimates (5.2.25) – (5.2.27) imply (5.2.24).

5.3 Embeddings of the space V ωq

Theorem 5.5. Let 1 < p < q <∞ and θ = 1/p− 1/q. Assume that f ∈ Vqand that ∫ 1

0


t<∞. (5.3.1)

Then f ∈ Vp and

ω1−1/p(f ; δ) ≤ 4

(∫ δ

0


t

)1/q

(5.3.2)

for all δ ∈ [0, 1].

5.3. Embeddings of the space V ωq 81

Proof. Let Π = x0, x1, ..., xn be any partition of a period. ApplyingHolder’s inequality with exponents q/p and (q/p)′ = 1/(pθ), we obtain

vp(f ; Π) =

(n−1∑j=0

(xj+1 − xj)pθ|f(xj+1)− f(xj)|p

(xj+1 − xj)pθ

)1/p

≤

(n−1∑j=0

|f(xj+1)− f(xj)|q

(xj+1 − xj)qθ

)1/q

. (5.3.3)

Denote

σk(Π) = j : 2−k−1 < xj+1 − xj ≤ 2−k (k = 0, 1, ...).

Set also

Sk(Π) =

∑j∈σk(Π)

|f(xj+1)− f(xj)|q1/q

if σk(Π) 6= ∅ and Sk(Π) = 0 otherwise. Then, by (5.3.3) we have that

vp(f ; Π) ≤

∞∑k=0

∑j∈σk(Π)

|f(xj+1)− f(xj)|q

(xj+1 − xj)qθ

≤

(∞∑k=0

2(k+1)θqSk(Π)q

)1/q

. (5.3.4)

Clearly,Sk(Π) ≤ ω1−1/q(f ; 2−k), (5.3.5)

for any partition Π. Using (5.3.4), (5.3.5), and (2.2.7), we obtain

vp(f) ≤

(∞∑k=0

2(k+1)θqω1−1/q(f ; 2−k)q

)1/q

≤ 4

(∫ 1

0


t

)1/q

.

Thus, f ∈ Vp. Further, let 2−ν < δ ≤ 2−ν+1, ν ∈ N. Let Π be any partitionwith ‖Π‖ ≤ δ. Then σk(Π) = ∅ and Sk(Π) = 0 for k < ν. Thus, from (5.3.4)


and (5.3.5)

vp(f ; Π) ≤

(∞∑k=ν

2(k+1)θqω1−1/q(f ; 2−k)q

)1/q

≤ 4

(∫ δ

0


t

)1/q

.


Now, we obtain the following embedding theorem for the classes V ωq .

Theorem 5.6. Let 1 < p < q <∞, θ = 1/p− 1/q, and ω ∈ Ω1/q′. Then theembedding

V ωq ⊂ Vp (5.3.6)

holds if and only if ∫ 1

0

(t−θω(t))qdt

t<∞. (5.3.7)

Proof. The sufficiency of (5.3.7) for embedding (5.3.6) follows immediatelyfrom Theorem 5.5. To prove the necessity, we assume that the integral on theleft-hand side of (5.3.7) diverges. Then ω satisfies (2.3.6). Let the sequencenk = nk(ω) be defined by (2.3.7). By Lemma 5.4(i),

∞∑k=1

2nkθqωqnk =∞.

Thus, there exists a strictly increasing sequence of natural numbers νj suchthat ν1 = 1 and

σj ≡

νj+1−1∑k=νj

2nkθqωqnk

1/q

> 2j (5.3.8)

for all j ∈ N. For each j ∈ N, we apply Lemma 5.2 with ν = νj, µ = νj+1,and γ = 2−j. Then we have σ = σj; we will write α∗j instead of αµ (µ = νj+1).By (5.2.3),

∞∑j=1

α∗j ≤1

8

(∞∑j=1

2−j +∞∑k=1

2−nk

)≤ 1

2. (5.3.9)

5.3. Embeddings of the space V ωq 83

By fj we denote the function f defined in Lemma 5.2. Set

β1 = 0, βj =

j−1∑m=1

α∗m for j ≥ 2,

and define the functions

ϕj(x) = fj(x+ βj) (j ∈ N),

and

ϕ(x) =∞∑j=1

ϕj(x).

Observe that by (5.3.9), (suppϕj)∩[0, 1] = [βj, βj+1] ⊂ [0, 1/2], and suppϕj (j ∈N) have disjoint interiors.

Assume that h ∈ (0, 1] and estimate ω1−1/q(ϕ;h). By (5.2.4), we have

ω1−1/q(ϕ;h) ≤∞∑j=1

ω1−1/q(ϕj;h)

≤ 16∞∑j=1

νj+1−1∑k=νj

min(1, (2nkh)1−1/q)ωnk

= 16∞∑k=1

min(1, (2nkh)1−1/q)ωnk .

Let k(h) be the greatest natural number k such that 2nkh ≤ 1. Then h >2−nk(h)+1 . Applying (2.3.8), we have

ω1−1/q(ϕ;h) ≤ 16

h1−1/q

k(h)∑k=1

ωnk +∞∑

k=k(h)+1

ωnk

≤ 32(h1−1/qωnk(h)

+ ωnk(h)+1) ≤ 64ω(h).

Thus, ϕ ∈ V ωq . On the other hand, by (5.2.6) and (5.3.8), we have that

vp(ϕ) ≥ vp(ϕj) > 2j(1−θ) for any j ∈ N.

Thus, ϕ /∈ Vp.


Remark 5.7. Let p = 1 and 1 < q <∞. Then the integral on the left-handside of (5.3.7) diverges for any non-trivial ω ∈ Ω1/q′ . It is easy to show thatembedding

V ωq ⊂ V1 ≡ V (5.3.10)

holds if and only if

ω(t) = O(t1−1/q). (5.3.11)

Indeed, if (5.3.11) is true, then V ωq ⊂ W 1

q (see Chapter 2) and thus we have(5.3.10). On the other hand, if (5.3.11) does not hold, then the function ψdefined in Lemma 5.3 (for p = 1) belongs to V ω

q , but does not belong to V.

5.4 Sharpness of the main estimate

In this section we complete the proof of the main result of this chapter.

Theorem 5.8. Let 1 < p < q < ∞, θ = 1/p − 1/q. There exists a positiveconstant c(p, q) such that for any ω ∈ Ω1/q′ there is a function f for which

ω1−1/q(f ; δ) ≤ ω(δ), δ ∈ [0, 1], (5.4.1)

and

ω1−1/p(f ; δ) ≥ c(p, q)

(∫ δ

0

(t−θω(t))qdt

t

)1/q

, δ ∈ [0, 1]. (5.4.2)

Proof. If the integral on the left-hand side of (5.3.7) diverges, then the state-ment follows from Theorem 5.6. We suppose that (5.3.7) holds. First weassume that ω satisfies condition (2.3.6). Let nk = nk(ω) (see (2.3.7)). Asabove, we set

ρ(ν) =

(∞∑k=ν

2nkθqωqnk

)1/q

(ν ∈ N).

Let ν1 = 1 and

νj+1 = minν ∈ N : ρ(ν) ≤ α0ρ(νj) (j ∈ N),

where α0 = 2−6. Then

ρ(νj+1) ≤ α0ρ(νj) (5.4.3)

5.4. Sharpness of the main estimate 85

andρ(νj+1 − 1) > α0ρ(νj) (j ∈ N). (5.4.4)

For each j ∈ N we apply Lemma 5.2 with ν = νj, µ = νj+1, and γ = 1. Wedenote by fj the function f defined in this lemma. Further, by (5.4.3), wehave

σ ≡ σj =

νj+1−1∑k=νj

2nkθqωqnk

1/q

= [ρ(νj)q − ρ(νj+1)q]1/q ≥ (1− αq0)1/qρ(νj).

Thus,σj ≤ ρ(νj) ≤ 21/qσj. (5.4.5)

Observe also that by (5.2.3), (supp fj) ∩ [0, 1] ⊂ [0, 1/2]. Set

ϕ(x) =∞∑j=1

fj(x).

Then(suppϕ) ∩ [0, 1] ⊂ [0, 1/2]. (5.4.6)

Let h ∈ (0, 1]. By (5.2.4),

ω1−1/q(ϕ;h) ≤∞∑j=1

ω1−1/q(fj;h)

≤ 16∞∑j=1

νj+1−1∑k=νj

min(1, (2nkh)1−1/q)ωnk

= 16∞∑k=1

min(1, (2nkh)1−1/q)ωnk .

Estimating the last sum exactly as in the proof of Theorem 5.6, we obtain

ω1−1/q(ϕ;h) ≤ 64ω(h). (5.4.7)

Now we estimate the modulus of p-continuity of the function ϕ frombelow. We shall denote

nk = lj if k = νj and nk = l∗j if k = νj+1 − 1. (5.4.8)


By (5.2.5),ω1−1/p(fj;h) ≤ 16 min(σj, h

1−1/pωl∗j ) (5.4.9)

for all h ∈ (0, 1] and any j ∈ N. Besides, by (5.2.6), (5.2.7), and (5.4.5), wehave

ω1−1/p(fj; 2−lj) ≥ ρ(νj), (5.4.10)

and

ω1−1/p(fj; 2−l∗j ) ≥ ρ(νj)

(2l∗j θωl∗jρ(νj)

)q/p

(5.4.11)

for any j ∈ N. Let h ∈ (0, 1]. By (5.4.9), (5.4.5), and (5.4.3),

∞∑j=m+1

ω1−1/p(fj;h) ≤ 16∞∑

j=m+1

ρ(νj) ≤ 32ρ(νm+1) (5.4.12)

for any m ∈ N. Also, by (5.4.9) and (2.3.8),

m−1∑j=1

ω1−1/p(fj;h) ≤ 16h1−1/p

m−1∑j=1

ωl∗j

≤ 32h1−1/pωl∗m−1≤ 8h1−1/pωlm (5.4.13)

for any m ≥ 2.Fix now s ∈ N, s ≥ 2. First we assume that

ρ(νs) ≥ β0ρ(νs − 1) (β0 = 2−12q). (5.4.14)

By (5.4.12) and (5.4.13), for any h ∈ (0, 1]

ω1−1/p(ϕ;h) ≥ ω1−1/p(fs;h)−∑j 6=s

ω1−1/p(fj;h)

≥ ω1−1/p(fs;h)− 32ρ(νs+1)− 8h1−1/pωls .

Taking here h = 2−ls and applying (5.4.10) and (5.4.3), we obtain

ω1−1/p(ϕ; 2−ls) ≥ 1

2ρ(νs)− 2lsθ+3ωls . (5.4.15)

By (5.4.14) and (5.4.4), for any νs−1 ≤ k < νs,

ρ(νs) ≥ β0ρ(νs − 1) ≥ α0β0ρ(νs−1) ≥ α0β0ρ(k). (5.4.16)


In what follows, we denote

λ(k) = 2nkθωnk . (5.4.17)


ω1−1/p(ϕ; 2−nk) ≥ c1ρ(k)− 8λ(νs) (c1 > 0) (5.4.18)

for any νs−1 ≤ k < νs, provided that (5.4.14) holds.Now we assume that

ρ(νs) < β0ρ(νs − 1). (5.4.19)

Then (see notation (5.4.17))

λ(νs − 1) ≥ ρ(νs − 1)− ρ(νs) ≥ (1− β0)ρ(νs − 1) ≥ 1

2ρ(νs − 1).

By (5.4.4), ρ(νs − 1) > α0ρ(νs−1). Whence, we obtain

λ(νs − 1) ≥ α0

2ρ(νs−1). (5.4.20)

Set hs = 2−l∗s−1 . We have l∗s−1 = nνs−1 (see (5.4.8)). Thus, it follows from

(5.4.11) and (5.4.20) that

ω1−1/p(fs−1;hs) ≥ ρ(νs−1)

(λ(νs − 1)

ρ(νs−1)

)q/p≥(α0

2

)q/pρ(νs−1). (5.4.21)

We observe also that by (5.4.19)

ρ(νs) < β0ρ(νs−1). (5.4.22)

Now we apply (5.4.21), (5.4.12) and (5.4.13) (in the case s ≥ 3) for m = s−1.We obtain

ω1−1/p(ϕ;hs) ≥ ω1−1/p(fs−1;hs)−∑j 6=s−1

ω1−1/p(fj;hs)

≥(α0

2

)q/pρ(νs−1)− 32ρ(νs)− 8h1−1/p

s ωls−1 .

Taking into account (5.4.22), we have

ω1−1/p(ϕ;hs) ≥ c2ρ(νs−1)− 8h1−1/ps ωls−1 ,


where c2 = 2−7q/p − 25−12q > 0. Let νs−1 ≤ k < νs. Then hs ≤ 2−nk andωls−1 ≤ ωnk . Thus,

ω1−1/p(ϕ; 2−nk) ≥ c2ρ(k)− 8λ(k) (5.4.23)

for νs−1 ≤ k < νs, provided that (5.4.19) holds.Let now ψ be the function defined by Lemma 5.3. Set

F (x) = ϕ(x) + ψ(x+ 1/2).

By (5.4.7) and (5.2.11),

ω1−1/q(F ; δ) ≤ cω(δ), 0 ≤ δ ≤ 1. (5.4.24)

On the other hand, taking into account (5.4.6) and (5.2.10), and applying(2.2.6), we have that

ω1−1/p(F ; δ)p = ω1−1/p(ϕ; δ)p + ω1−1/p(ψ; δ)p. (5.4.25)

Let νs−1 ≤ k < νs. Then, by (5.2.12)

ω1−1/p(ψ; 2−nk) ≥ cλ(k),

andω1−1/p(ψ; 2−nk) ≥ ω1−1/p(ψ; 2−nνs ) ≥ cλ(νs)

(c > 0). Moreover, at least one of the inequalities (5.4.18) or (5.4.23) is true.Thus, taking into account (5.4.25), we obtain that

ω1−1/p(F ; 2−nk) ≥ cρ(k), (5.4.26)

for any k ∈ N. Finally, let nk ≤ n < nk+1. By (5.4.25) and (5.2.12),

ω1−1/p(F ; 2−n) ≥ ω1−1/p(ψ; 2−n) ≥ c2nθωn.

On the other hand, by (5.4.26),

ω1−1/p(F ; 2−n) ≥ ω1−1/p(F ; 2−nk+1) ≥ cρ(k + 1).

Thus,ω1−1/p(F ; 2−n) ≥ c(ρ(k + 1) + 2nθωn) (c > 0).


Applying Lemma 5.4, we have that

ω1−1/p(F ; 2−n) ≥ c′

(∞∑m=n

2mθqωqm

)1/q

(c′ > 0)

for any integer n ≥ 0. This estimate and (5.4.24) yield that the theorem istrue provided that (2.3.6) holds.

Now we assume that (2.3.6) does not hold. Then ω(t) = O(t1−1/q). Take1 − 1/p < γ < 1 − 1/q and set ωn(t) = ω(t) + tγ/n (n ∈ N). Clearly, ωnsatisfies (2.3.6) and (5.3.7). As we have proved, there exists a constant c > 0and a sequence of continuous 1-periodic functions fn such that fn(0) = 0,

ω1−1/q(fn; δ) ≤ ωn(δ) ≤ ω1(δ) for all δ ∈ [0, 1], n ∈ N, (5.4.27)

and

ω1−1/p(fn; δ) ≥ c

(∫ δ

0

(t−θω(t))qdt

t

)1/q

(5.4.28)

for all δ ∈ [0, 1], n ∈ N. By (5.4.27) and Theorem 5.5,

ω1−1/p(fn; δ) ≤ 4

(∫ δ

0

(t−θω1(t))qdt

t

)1/q

, δ ∈ [0, 1], (5.4.29)

for all n ∈ N. By the compactness criterion in Cp (see [24]), there exist asubsequence fnk and a function f ∈ Vp such that f(0) = 0 and

vp(f − fnk)→ 0 as k →∞. (5.4.30)

Since f(0) = fnk(0) = 0, it follows that fnk converges uniformly to f .Thus, by (5.4.27),

ω1−1/q(f ; δ) ≤ limn→∞

ω1−1/q(fn; δ) ≤ ω(δ), δ ∈ [0, 1].

Thus, f satisfies (5.4.1). Besides, (5.4.28) and (5.4.30) imply that f satisfies(5.4.2).

Remark 5.9. Recall that for 1 < p < q <∞ and ω ∈ Ω1/q′ , we denote

ρp,q,ω(δ) =

(∫ δ

0

(t−θω(t))qdt

t

)1/q

, θ = 1/p− 1/q.


It follows from Theorems 5.5 and 5.8 that we have

c(p, q) ≤ supf∈V ωq

inf0<δ≤1

ω1−1/p(f ; δ)

ρp,q,ω(δ)≤ sup

f∈V ωq

sup0<δ≤1

ω1−1/p(f ; δ)

ρp,q,ω(δ)≤ 4, (5.4.31)

where c(p, q) > 0 depends only on p and q.At the same time, for the classes H

ω

r the term corresponding to the firstof the inequalities (5.4.31) is weaker; that is, we only have that

inf0<δ≤1

supf∈Hω

r

ω(f ; δ)sµr,s,ω(δ)

≥ c(r, s) > 0

(see (5.1.3)), where infimum and supremum cannot be interchanged.

Chapter 6

On functions of boundedΛ-variation

In this chapter, we study some properties of the functions of bounded Λ-variation. We shall first recall the definition of this class of functions.

Let f be a 1-periodic function on the real line. For any interval I = [a, b],we set f(I) = f(b) − f(a). Denote by S the collection of all positive andnondecreasing sequences Λ = λn such that λn →∞ and

∞∑n=1

1

λn=∞.

Let Λ = λn ∈ S, a function f is said to be of bounded Λ-variation if

vΛ(f) = supI

∞∑n=1

|f(In)|λn

<∞,

where the supremum is taken over all sequences I = In of nonoverlappingintervals contained in a period. The class of functions of bounded Λ-variationis denoted ΛBV .

Recall also that for 1 ≤ p <∞ and 0 < α ≤ 1, we denote

Lip(α; p) = f ∈ Lp([0, 1]) : ω(f ; δ)p = O(δα).

The main objective of this chapter is to obtain the necessary and sufficientcondition for the embedding

Lip(α; p) ⊂ ΛBV (1 1/p in (6.0.1) is essential; for α ≤ 1/p, theclass Lip(α; p) contains unbounded functions and (6.0.1) cannot hold.

We also show that Vp (p ≥ 1) can be expressed in terms of spaces ΛBV .For p = 1, this result is due to Perlman [58].


For f ∈ Lip(α; p) (1 ≤ p <∞, 0 < α ≤ 1), we denote

‖f‖Lip(α;p) = supδ>0

ω(f ; δ)pδα

. (6.1.1)

Let 1 0 such that for the modified function f ,

cp,α supδ>0

ω1−1/p(f ; δ)

δα−1/p≤ ‖f‖Lip(α;p) ≤ sup

δ>0

ω1−1/p(f ; δ)

δα−1/p. (6.1.2)

These statements follow from (2.2.9) and (1.0.10). Thus, if p > 1, 1/p < α ≤1 and f is a continuous 1-periodic function, then

ω(f ; δ)p = O(δα) if and only if ω1−1/p(f ; δ) = O(δα−1/p). (6.1.3)

The following is a slight generalization of the construction given by Defi-nition 5.1.

Definition 6.1. Let I = [a, b] ⊂ [0, 1] be an interval, N ∈ N and H =(H0, H1, ..., HN−1) ∈ RN be a vector with Hj ≥ 0 for 0 ≤ j ≤ N − 1. Seth = (b − a)/N , ξj = a + jh (j = 0, 1, ..., N) and ξ∗j = a + (j + 1/2)h (j =0, 1, ..., N − 1). The function F (x) = F (I,N,H;x) is defined to be thecontinuous 1-periodic function such that F (x) = 0 for x ∈ [0, 1] \ I, F (ξj) =0 (j = 0, 1, ..., N), F (ξ∗j ) = Hj (j = 0, 1, ..., N − 1), and F is linear on eachof the intervals [ξj, ξ

∗j ] and [ξ∗j , ξj+1] (j = 0, 1, ..., N − 1).

Thus, the graph of F consists of N isosceles triangles of heights Hj (j =0, ..., N − 1) and bases h. Using (2.2.1) and Lemma 2.6, we have

vp(F ) = 21/p

(N−1∑j=0

Hpj

)1/p

(1 ≤ p <∞). (6.1.4)


It is also easy to see that

‖F ′‖p = 2h−1/p′

(N−1∑j=0

Hpj

)1/p

(1 ≤ p <∞). (6.1.5)

The next lemma is of a known type (cf. [60]). In particular, it can beproved in the same way as Lemma 2.4 in [35].

Lemma 6.2. Let αk ∈ l1 be a sequence of non-negative numbers and letθ > 1 and γ > 0. There exists a sequence βk of positive numbers such that

αk ≤ βk, k ∈ N,∞∑k=1

βk ≤θ1+γ

(θ − 1)(θγ − 1)

∞∑k=1

αk,

and

θ−γ ≤ βk+1

βk≤ θ, k ∈ N.

We shall also use the following Hardy-type inequality (see [39]).

Lemma 6.3. Let β > 0 and 1 < r < ∞ be fixed. Let ak be a sequence ofnonnegative real numbers, and νn an increasing sequence of positive realnumbers with ν0 = 1. Then there exists a constant cβ,r > 0 such that

∞∑n=0

2−nβ

( ∑1≤k≤νn

ak

)1/r

≤ cβ,r

∞∑n=1

2−nβ

∑νn−1≤k≤νn

ak

1/r

. (6.1.6)

Finally, we formulate the next well-known result (see, e.g., [17, Ch.6]).

Lemma 6.4. Let 1 < p <∞. Then xn ∈ lp if and only if

∞∑n=1

αnxn <∞,

for all αn ∈ lp′. Moreover,

sup‖αn‖p′≤1

∞∑n=1

αnxn = ‖xn‖p.


6.2 Embedding of Lipschitz classes

We shall now prove the main results of this chapter. Recall that ‖f‖Lip(α;p)

is given by (6.1.1).

Theorem 6.5. Let Λ ∈ S be given and 1 0 depending only on α and p such that for anyf ∈ Lip(α; p),

vΛ(f) ≤ cp,α‖f‖Lip(α;p)

∞∑n=0

(2n+1∑k=2n

(1

kα−1/pλk

)p′)r′/p′1/r′

. (6.2.2)

Proof. In light of (6.1.2), we may without loss of generality assume that

supδ>0

ω1−1/p(f ; δ)

δα−1/p= 1. (6.2.3)

Take an arbitrary sequence I = Ij of nonoverlapping intervals containedin a period. Denote

σk(I) = j : 2−k−1 < |Ij| ≤ 2−k (k ≥ 0).

Then we have

V =∞∑j=1

|f(Ij)|λj

=∞∑k=0

∑j∈σk(I)

|f(Ij)|λj

.

We shall estimate V . By Holder’s inequality, we have

V ≤∞∑k=0

∑j∈σk(I)

|f(Ij)|p1/p ∑

j∈σk(I)

(1

λj

)p′1/p′

≤∞∑k=0

ω1−1/p(f ; 2−k)

∑j∈σk(I)

(1

λj

)p′1/p′

. (6.2.4)

6.2. Embedding of Lipschitz classes 95

Thus, by (6.2.3) and (6.2.4)

V ≤∞∑k=0

2−k(α−1/p)

∑j∈σk(I)

(1

λj

)p′1/p′

. (6.2.5)

Let the sequence δn be defined by

card

(n⋃k=0

σk(I)

)= 2nδn,

where, card(A) denotes the number of elements of the finite set A. Set alsoδ−1 = 0. There exists an n0 ≥ 0 such that δn > 0 for all n ≥ n0, and we mayassume n0 = 0. We observe that ‖δn‖l1 ≤ 4. Indeed, first note that

∞∑k=0

2−kcard(σk(I)) ≤ 2∞∑j=0

|Ij| ≤ 2.

On the other hand, for n ≥ 0, we have

2−ncard(σn(I)) = δn − δn−1/2.

Whence, for any N ∈ N, we have

N+1∑n=0

(δn − δn−1/2) = δN+1 +1

2

N∑n=0

δn ≥1

2

N∑n=0

δn,

and consequently, ‖δn‖l1 ≤ 4.Applying Lemma 6.2 with θ = 2 and γ = 1/2 to δk yields a sequence

βk such that δk ≤ βk,

2−1/2 ≤ βk+1

βk≤ 2 (k ∈ N) and ‖βk‖l1 ≤ 64. (6.2.6)

Set νk = 2kβk. By the first relation of (6.2.6), we have

2νk ≤ νk+2 ≤ 16νk (k ∈ N). (6.2.7)

Since card(σk(I)) ≤ 2kδk = νk, and λj is increasing, we have by (6.2.5)

V ≤∞∑k=0

2−k(α−1/p)

( ∑1≤j≤νk

(1

λj

)p′)1/p′

.


Applying (6.1.6) to the right-hand side of the previous inequality, we get

V ≤ cp,α

∞∑k=1

2−k(α−1/p)

∑νk−1≤j≤νk

(1

λj

)p′1/p′

,

for some constant cp,α > 0. Since 2−k = βk/νk, we have

V ≤ cp,α

∞∑k=1

(βkνk

)α−1/p ∑νk−1≤j≤νk

(1

λj

)p′1/p′

. (6.2.8)

By using Holder’s inequality with exponents r and r′, and the second in-equality of (6.2.6), we estimate the right-hand side of (6.2.8)

V ≤ cp,α‖β1/rk ‖lr

∞∑k=1

ν−r′(α−1/p)k


(1

λj

)p′r′/p′

1/r′

≤ 64cp,α

∞∑k=1


(1

jα−1/pλj

)p′r′/p′

1/r′

. (6.2.9)

By collecting the terms of the sum at the right-hand side of (6.2.9) in pairs,and using that aq + bq ≤ 2(a+ b)q for any q ≥ 0 and a, b ≥ 0, we get

V ≤ 128cp,α

∞∑k=0

∑ν2k≤j≤ν2k+2

(1

jα−1/pλj

)p′r′/p′

1/r′

. (6.2.10)

For k ≥ 0, we define mk ≥ 0 as the greatest integer m such that

2m < ν2k.

By (6.2.7), we have 2ν2k ≤ ν2k+2, and thus,

2mk+1 < ν2k+2.

Consequently,mk+1 ≥ mk + 1 (k ≥ 0). (6.2.11)


Further, by (6.2.7), we have ν2k+2 ≤ 16ν2k. Therefore, for all k ≥ 0,

[ν2k, ν2k+2] ⊂ [2mk , 2mk+5].

Whence, ∑ν2k≤j≤ν2k+2

(1

jα−1/pλj

)p′≤

2mk+5∑j=2mk

(1

jα−1/pλj

)p′.

Since the terms of the previous sum decrease, it follows that

2mk+5∑j=2mk

(1

jα−1/pλj

)p′≤ 40

2mk+1∑j=2mk

(1

jα−1/pλj

)p′,

Consequently, by the previous inequality and (6.2.10),

V ≤ c′p,α

∞∑k=0

2mk+1∑j=2mk

(1

jα−1/pλj

)p′r′/p′

1/r′

,

for some c′p,α > 0. By (6.2.11), for each k ≥ 0, the intersection of [2mk , 2mk+1]and [2mk+1 , 2mk+1+1] consists of at most one point. Hence,

V ≤ c′p,α

∞∑n=0

(2n+1∑j=2n

(1

jα−1/pλj

)p′)r′/p′1/r′

.

This proves (6.2.2).

The estimate (6.2.2) is sharp in a sense. Namely, we have the followingresult.

Theorem 6.6. Let Λ ∈ S be given, 1 0

depending only on α and p such that

ω1−1/p(g; δ) ≤ c′p,αδα−1/p (0 < δ ≤ 1), (6.2.12)

and

vΛ(g) ≥ c′′p,α

∞∑n=1

(2n+1∑k=2n

(1

kα−1/pλk

)p′)r′/p′1/r′

. (6.2.13)


Proof. Let δn ∈ l1 be a fixed but arbitrary positive sequence with ‖δn‖l1 ≤1. Applying Lemma 6.2 with γ = 1 and θ = 3/2 (the value of γ does notmatter, it is only important that 1 < θ < 2) to the sequence δn, we obtaina positive sequence βn such that δn ≤ βn (n ∈ N),

2

3<βn+1

βn≤ 3

2(n ∈ N) and L = ‖βn‖l1 ≤ 9. (6.2.14)

Subdivide the interval [0, 1] into non-overlapping intervals Jn (n ∈ N) with|Jn| = βn/L. For n ∈ N, denote

Sn =

(2n+1−1∑k=2n

(1

λk

)p′)1/p′

,

and

H(n)k = (2−nβn)α−1/pλ

−1/(p−1)k S−p

′/pn for 2n ≤ k ≤ 2n+1 − 1.

Let also Hn = (H(n)2n , H

(n)2n+1, ..., H

(n)

2n+1−1) ∈ R2n . Put Fn(x) = F (Jn, 2n,Hn;x)

(see Definition 6.1), and

g(x) =∞∑n=1

Fn(x).

It is clear that

vΛ(g) ≥ 2∞∑n=1

2n+1−1∑k=2n

H(n)k

λk. (6.2.15)

On the other hand,

2n+1−1∑k=2n

H(n)k

λk= (2−nβn)α−1/pSn.

Thus, since δk ≤ βk for k ∈ N, we have

∞∑n=1

2n+1−1∑k=2n

H(n)k

λk=

∞∑n=1

βα−1/pn

(2−np

′(α−1/p)

2n+1−1∑k=2n

(1

λk

)p′)1/p′

≥ 2−1/p+α

∞∑n=1

δα−1/pn

(2n+1−1∑k=2n

(1

kα−1/pλk

)p′)1/p′

.


By the previous inequality and (6.2.15),

vΛ(g) ≥ 2−1/p+α

∞∑n=1

δα−1/pn

(2n+1∑k=2n

(1

kα−1/pλk

)p′)1/p′

. (6.2.16)

We proceed to estimate ω1−1/p(g; δ). By the first relation of (6.2.14), wehave

1

3≤ 2−n−1βn+1

2−nβn≤ 3

4< 1.

In particular, the sequence 2−nβn is strictly decreasing and 2−nβn → 0 asn→∞. Fix 0 < δ ≤ 1. If δ > 2−1β1, then we set m = 0. Otherwise, definem ∈ N to be the unique natural number such that

2−m−1βm+1 < δ ≤ 2−mβm.

By (2.2.8), we have

ω1−1/p(g; δ) ≤ δ1/p′m∑n=1

‖F ′n‖p +∞∑

n=m+1

vp(Fn). (6.2.17)

(The first sum is taken as zero if m = 0). We shall estimate the terms at the

right-hand side of (6.2.17). It follows from (6.1.4) and the definition of H(n)k

that

vp(Fn) = 21/p

(2n+1−1∑k=2n

(H(n)k )p

)1/p

= 21/p(2−nβn)α−1/p. (6.2.18)

Further, by (6.1.5),

‖F ′n‖p = 2

(βn/L

2n

)−1/p′(

2n+1−1∑k=2n

(H(n)k )p

)1/p

= 2L1/p′(2−nβn)α−1. (6.2.19)

By the estimate L ≤ 9, (6.2.17), (6.2.18) and (6.2.19),

ω1−1/p(g; δ) ≤

≤ 18δ1/p′m∑n=1

(2−nβn)α−1 + 21/p

∞∑n=m+1

(2−nβn)α−1/p. (6.2.20)


Since(2−n+1βn−1

2−nβn

)α−1

=

(2βn−1

βn

)α−1

=

(βn

2βn−1

)1−α

≤(

3

4

)1−α

< 1,

we get

m∑n=1

(2−nβn)α−1 ≤ (2−mβm)α−1

∞∑n=0

(3

4

)n(1−α)

= cα(2−mβm)α−1

≤ cαδα−1. (6.2.21)

Similarly,

∞∑n=m+1

(2−nβn)α−1/p ≤ (2−m−1βm+1)α−1/p

∞∑n=0

(3

4

)n(α−1/p)

≤ cp,αδα−1/p. (6.2.22)

Thus, by (6.2.17), (6.2.21) and (6.2.22),

ω1−1/p(g; δ) ≤ c′p,αδα−1/p (0 < δ ≤ 1). (6.2.23)

Denote

Ln =

(2n+1∑k=2n

(1

kα−1/pλk

)p′)1/p′

.

Clearly, δn ∈ l1 is equivalent to δα−1/pn ∈ lr. By Lemma 6.4, we can

choose δn ∈ l1 such that

∞∑n=1

δα−1/pn Ln ≥

1

2

(∞∑n=1

Lr′

n

)1/r′

. (6.2.24)

If Ln /∈ lr′, then we must interpret (6.2.24) in the sense that we may choose

δn ∈ l1 such that the left-hand side of (6.2.24) is infinite. In any case, thefunction g constructed above with this choice of δn satisfies (6.2.12) and(6.2.13), by (6.2.23), (6.2.16) and (6.2.24).

Remark 6.7. As was mentioned in the Introduction, Wang observed thatthe condition

∞∑n=1

(1

λn

)1/(1−α)

<∞. (6.2.25)

6.3. A Perlman-type theorem 101

is necessary for the embedding (6.0.1) to hold, and he then conjectured that(6.2.25) is also sufficient. However, by combining Theorems 6.5 and 6.6, weobtain that the necessary and sufficient condition for (6.0.1)) is

∞∑n=0

(2n+1∑k=2n

(1

kα−1/pλk

)p′)r′/p′

<∞,

where r, r′ are given by (6.2.1). Clearly, this disproves Wang’s conjecture.

Remark 6.8. For 1 ≤ p < ∞, α = 1, we have Lip(1; p) = W 1p . It is easy

to show that the embedding W 1p ⊂ ΛBV holds for all sequences Λ ∈ S.

Remark 6.9. Recall that for 1 ≤ p <∞ and ω ∈ Ω1, we denote

Hωp = f ∈ Lp([0, 1]) : ω(f ; δ)p = O(ω(δ)),

andHω = f ∈ C : ω(f ; δ)C = O(ω(δ)),

where ω(f ; δ)C is the modulus of continuity in C.The problem of finding the necessary and sufficient condition for the em-

bedding Hωp ⊂ ΛBV with general ω ∈ Ω1 and 1 ≤ p < ∞ is still open.

On the other hand, the necessary and sufficient condition for the embeddingHω ⊂ ΛBV was obtained independently by Belov [4] and Medvedeva [47, 48].Later, Leindler [40, 41] generalized these results.

6.3 A Perlman-type theorem

Perlman [58] showed that

V1 =⋂Λ∈S

ΛBV.

We shall prove a similar result for Vp. Let 1 < p < ∞, denote by Sp′ theclass of all sequences Λ = λn ∈ S such that

∞∑n=1

(1

λn

)p′<∞.

Then we have the following theorem.


Theorem 6.10. Let 1 < p <∞. Then

Vp =⋂

Λ∈Sp′

ΛBV.

Proof. Let f be a given function and In an arbitrary sequence of nonover-lapping intervals contained in a period. Applying Holder’s inequality, wehave

∞∑n=1

|f(In)|λn

≤ vp(f)

(∞∑n=1

(1

λn

)p′)1/p′

.

Thus, if Λ ∈ Sp′ , then Vp ⊂ ΛBV . Whence,

Vp ⊂⋂

Λ∈Sp′

ΛBV.

Let now f be a bounded function with f /∈ Vp. Then there exists a sequenceJn of nonoverlapping intervals contained in a period such that

∞∑n=1

|f(Jn)|p =∞.

Since |f(Jn)| /∈ lp, there exists αn ∈ lp′

such that

∞∑n=1

αn|f(Jn)| =∞,

by Lemma 6.4. We may assume that αn > 0 for all n ∈ N and that |f(Jn)|is ordered nonincreasingly. Let α∗n be the nonincreasing rearrangement ofαn, set λn = 1/α∗n and Λ = λn. Since |f(Jn)| is nonincreasing, wehave

∞∑n=1

|f(Jn)|λn

=∞∑n=1

α∗n|f(Jn)| ≥∞∑n=1

αn|f(Jn)| =∞, (6.3.1)

whence f /∈ ΛBV . It remains to show that Λ ∈ Sp′ . Clearly Λ is a positiveand nondecreasing sequence. Moreover, |f(I)| ≤ 2‖f‖∞ for any interval.Therefore,

∞∑n=1

1

λn≥ 1

2‖f‖∞

∑n=1

|f(Jn)|λn

=∞,

6.3. A Perlman-type theorem 103

by (6.3.1). Whence, Λ ∈ S. Furthermore, since αn ∈ lp′,

∞∑n=1

(1

λn

)p′=∞∑n=1

(α∗n)p′=∞∑n=1

αp′

n <∞.

Thus, λn ∈ Sp′ .

In connection to Theorem 6.10, we mention that embeddings betweenΛBV and other spaces of functions of generalized bounded variation werepreviously studied in, e.g., [4, 6, 59, 62].

Remark 6.11. A result similar to Theorem 6.10 can also be proved forclasses VΦ of functions of bounded Φ-variation.

Remark 6.12. We can apply Theorem 6.10 to prove that there is a sequenceΛ ∈ S that satisfies (6.2.25) but still Lip(α; p) 6⊂ ΛBV (thus disprovingWang’s conjecture mentioned above).

Note first that 1 < 1/α < p <∞. By Theorem 5.6, there exists a functionf such that ω1−1/p(f ; δ) = O(δα−1/p), and at the same time f /∈ V1/α. In lightof (6.1.3), this means exactly that there is a function f ∈ Lip(α; p) such thatf /∈ V1/a. Theorem 6.10 states that

V1/α =⋂

Λ∈S1/(1−α)

ΛBV. (6.3.2)

Observe that S1/(1−α) is the collection of all sequences in S that satisfies(6.2.25). Since f /∈ V1/α, (6.3.2) implies that for some Λ ∈ S1/(1−α), we havef /∈ ΛBV . But since f ∈ Lip(α; p), we have shown that there exists a Λ thatsatisfies (6.2.25) while the embedding (6.0.1) does not hold.

Chapter 7

Multidimensional results

The main objectives of this chapter is to study some problems related tobounded p-variation of bivariate functions (i.e., the classes V

(2)p , H

(2)p defined

in the Introduction). In particular, we shall investigate the following:

• sharp estimates of the Hardy-Vitali type p-variation and L∞-norm ofa function in terms of its mixed Lp-modulus of continuity;

• Fubini-type properties of the class H(2)p (p ≥ 1).


Recall that Ω denotes the class of all moduli of continuity (see Chapter 2).Let f ∈ Lp([0, 1]2), as we remarked before, ω(f ; ·)p ∈ Ω. Further, it is

easy to show that for any fixed v ∈ [0, 1], the function ω(f ; ·, v)p ∈ Ω. Thus,by (2.1.2),

ω(f ;u1, v)pu1

≤ 2ω(f ;u2, v)p

u2

, 0 < u2 ≤ u1 ≤ 1. (7.1.1)

Similar relations hold with respect to the second variable v for a fixed u ∈[0, 1].

Let h ∈ R, we shall use the following notations.

∆1(h)f(x, y) = f(x+ h, y)− f(x, y) (7.1.2)

and∆2(h)f(x, y) = f(x, y + h)− f(x, y). (7.1.3)

105

106 Chapter 7. Multidimensional results

The mixed difference (1.0.27) can be written as an iterated difference

∆(s, t)f(x, y) = ∆1(s)∆2(t)f(x, y) = ∆1(s)∆2(t)f(x, y).

From here,‖∆(s, t)(∆1(h)f)‖p = ‖∆1(s)∆1(h)∆2(t)f‖p.

Applying the triangle inequality, we obtain the second estimate of the nextlemma (the first inequality is proved similarly).

Lemma 7.1. Let f ∈ Lp([0, 1]2) (1 ≤ p <∞) and h ∈ R. Then

ω(∆1(h)f ; δ)p ≤ 2 minω(f ; δ)p, ω(f ;h)p, (7.1.4)

andω(∆1(h)f ;u, v)p ≤ 2 minω(f ;u, v)p, ω(f ;h, v)p. (7.1.5)

Similar estimates also hold if we consider ∆2(h)f .

Let f ∈ Lp([0, 1]2) (1 < p <∞). We shall use the following notations

Jp(f) =

∫ 1

0

t−1/pω(f ; t)pdt

t, (7.1.6)

Kp(f) =

∫ 1

0

t−1/p[ω(f ; t, 1)p + ω(f ; 1, t)p]dt

t, (7.1.7)

and

Ip(f) =

∫ 1

0

∫ 1

0

(uv)−1/pω(f ;u, v)pdu

u

dv

v. (7.1.8)

Let f ∈ Lp([0, 1]2) (1 < p <∞), then we have

Kp(f) ≤ 4

p′Ip(f). (7.1.9)

Indeed, by (7.1.1)

Ip(f) =

∫ 1

0

u−1/p−1

(∫ 1

0

v−1/pω(f ;u, v)pv

dv

)du

≥ 1

2

∫ 1

0

u−1/p−1ω(f ;u, 1)pdu

∫ 1

0

v−1/pdv.


Thus, ∫ 1

0

t−1/pω(f ; t, 1)pdt

t≤ 2

p′Ip(f).

Similarly, one shows ∫ 1

0

t−1/pω(f ; 1, t)pdt

t≤ 2

p′Ip(f),

and (7.1.9) follows. In the same way, one demonstrates that

ω(f ; 1, 1)p ≤4

(p′)2Ip(f). (7.1.10)

Denote by Lp0([0, 1]2) the subspace of Lp([0, 1]2) that consists of functionsf such that ∫ 1

0

f(x, t)dt =

∫ 1

0

f(t, y)dt = 0

for a.e. x, y ∈ R. Observe that every function f ∈ Lp([0, 1]2) can be writtenas

f(x, y) = f(x, y) + φ1(x) + φ2(y), a.e. (x, y) ∈ R2, (7.1.11)

where f ∈ Lp0([0, 1]2). Indeed, let

φ1(x) =

∫ 1

0

f(x, t)dt, (7.1.12)

φ2(y) =

∫ 1

0

f(t, y)dt−∫∫

[0,1]2f(s, t)dsdt. (7.1.13)

Then the function

f(x, y) = f(x, y)− φ1(x)− φ2(y)

belongs to Lp0([0, 1]2).It was proved in [65] that if f ∈ Lp0([0, 1]2), then

ω(f ; δ)p ≤ 3[ω(f ; δ, 1)p + ω(f ; 1, δ)p], 0 ≤ δ ≤ 1.

Whence, it follows that if f ∈ Lp0([0, 1]2) (1 < p <∞), then

Jp(f) ≤ 3Kp(f). (7.1.14)


If f(x, y) = g(x)h(y), then for all p ≥ 1, there holds

v(2)p (f) = vp(g)vp(h), (7.1.15)

andω(f ;u, v)p = ω(g;u)pω(h; v)p, u, v ∈ [0, 1]. (7.1.16)

Recall that when defining the class H(2)p (see the Introduction), we require

in addition to (1.0.25) also that the sections fx, fy ∈ Vp for all x, y ∈ R. Infact, it is sufficient to assume that there exists at least two values x0, y0 ∈ Rsuch that f(x0, ·), f(·, y0) ∈ Vp. Indeed, assume that f(x0, ·) ∈ Vp for somex0 ∈ R and let x ∈ R be fixed but arbitrary. Take any partition Π =y0, y1, ..., yn and set

∆f(x0, yj) = f(x, yj+1)− f(x, yj)− f(x0, yj+1) + f(x0, yj),

for 0 ≤ j ≤ n− 1. By the Minkowski inequality, we have

vp(fx; Π) =

(n−1∑j=0

|f(x, yj+1)− f(x, yj)|p)1/p

≤

(n−1∑j=0

|∆f(x0, yj)|p)1/p

+ vp(fx0).

Whence,vp(fx) ≤ v(2)

p (f) + vp(fx0).

A similar inequality holds for vp(fy).The next result is due to Golubov [26].

Lemma 7.2. Assume that f ∈ L10([0, 1]2) and let

F (x, y) =

∫ x

0

∫ y

0

f(s, t)dsdt.

Then

v(2)1 (F ) =

∫ 1

0

∫ 1

0

|f(x, y)|dxdy. (7.1.17)

Remark 7.3. The condition f ∈ L10([0, 1]2) is imposed to assure that F is

1-periodic in both variables.

7.2. Estimates of the L∞-norm 109

We shall also need the following lemma, which is a special case of a Helly-type principle proved in [42].

Lemma 7.4. Let fn be a sequence of functions in H(2)1 . Assume that there

exist x0, y0 ∈ R and M > 0 such that the estimate

v(2)1 (fn) + v1(fn(·, y0)) + v1(fn(x0, ·)) + |fn(x0, y0)| ≤M

holds uniformly in n. Then there exists a subsequence fnj that converges

at every point to a function f ∈ H(2)1 .

7.2 Estimates of the L∞-norm

Recall the notations (7.1.6) and (7.1.8). Potapov [63, 64, 65] obtained es-timates of the L∞-norm of a function in terms of its mixed Lp-modulus ofcontinuity (see also [66]). However, the behaviour of the constant coeffi-cients in these estimates were not investigated. In this section, we study thisproblem.

Observe first that for f ∈ Lp([0, 1]2) (1 < p < ∞), the condition Ip(f) <∞ alone is not sufficient to ensure that f ∈ L∞([0, 1]2). Indeed, if f(x, y) =g(x, y)+φ(x), then Ip(f) = Ip(g), but φ is an arbitrary function (in particular,φ can be unbounded).

Theorem 7.5. Let f ∈ Lp([0, 1]2) (1 < p <∞) and suppose that

Jp(f) <∞ and Ip(f) <∞. (7.2.1)

Then f is equal a.e. to a continuous function and

‖f‖∞ ≤ A

[‖f‖p +

1

pp′Jp(f) +

(1

pp′

)2

Ip(f)

], (7.2.2)


Proof. Assume that (7.2.1) holds, we shall first prove the estimate (7.2.2).For each x ∈ [0, 1], we apply (3.2.2) to the x-section fx. Using also (2.1.5),we have

‖fx‖∞ ≤ A

[‖fx‖p +

1

pp′

∫ 1

0

v−1/p−1‖∆(v)fx‖pdv], (7.2.3)


where ∆(v)fx(y) = f(x, y + v)− f(x, y). Put

α(x) = ‖fx‖p, βv(x) = ‖∆(v)fx‖p, (7.2.4)

and

Φ(x) = α(x) +1

pp′

∫ 1

0

v−1/p−1βv(x)dv. (7.2.5)

By (7.2.3)‖f‖∞ = ess sup

0≤x≤1‖fx‖∞ ≤ A ess sup

0≤x≤1Φ(x). (7.2.6)

We shall estimate ‖Φ‖∞. By (3.2.2) and (2.1.5), we have

‖Φ‖∞ ≤ A

[‖Φ‖p +

1

pp′

∫ 1

0

u−1/p−1‖∆(u)Φ‖pdu], (7.2.7)

where ∆(u)Φ(x) = Φ(x + u) − Φ(x). It follows easily from the definitions(7.2.4) that

‖α‖p = ‖f‖p, ‖∆(u)α‖p ≤ ω(f ;u)p, (7.2.8)

and‖βv‖p ≤ ω(f ; v)p, ‖∆(u)βv‖p ≤ ω(f ;u, v)p. (7.2.9)

We estimate both terms of (7.2.7), starting with ‖Φ‖p. By Minkowski’sinequality and the left inequalities of (7.2.8) and (7.2.9), we get

‖Φ‖p ≤ ‖f‖p +1

pp′

(∫ 1

0

(∫ 1

0

v−1/p−1βv(x)dv

)pdx

)1/p

≤ ‖f‖p +1

pp′

∫ 1

0

v−1/p−1

(∫ 1

0

βv(x)pdx

)1/p

dv

≤ ‖f‖p +1

pp′Jp(f). (7.2.10)

We proceed to estimate ‖∆(u)Φ‖p. Put

I(x) =

∫ 1

0

v−1/p−1βv(x)dv.

Then, by Minkowski’s inequality and the right inequality of (7.2.8)

‖∆(u)Φ‖p ≤ ‖∆(u)α‖p +1

pp′‖∆(u)I‖p

≤ ω(f ;u)p +1

pp′‖∆(u)I‖p. (7.2.11)

7.2. Estimates of the L∞-norm 111

Further, since

|I(x+ u)− I(x)| ≤∫ 1

0

v−1/p−1|βv(x+ u)− βv(x)|dv,

we get after applying Minkowski’s inequality that

‖∆(u)I‖p ≤(∫ 1

0

(∫ 1

0

v−1/p−1|∆(u)βv(x)|dv)p

dx

)1/p

≤∫ 1

0

v−1/p−1‖∆(u)βv‖pdv. (7.2.12)

By (7.2.11), (7.2.12) and the right inequality of (7.2.9), we have

‖∆(u)Φ‖p ≤ ω(f ;u)p +1

pp′

∫ 1

0

v−1/p−1ω(f ;u, v)pdv. (7.2.13)

Now, (7.2.2) follows from (7.2.6), (7.2.7), (7.2.10) and (7.2.13).We now prove that f agrees a.e. with a continuous function. To do this,

it is sufficient to show that ω(f ; δ)∞ → 0 as δ → 0. Fix δ ∈ (0, 1], then

ω(f ; δ)∞ ≤ sup0≤h≤δ

‖∆1(h)f‖∞ + sup0≤h≤δ

‖∆2(h)f‖∞,

where ∆1(h)f,∆2(h)f are defined by (7.1.2) and (7.1.3) respectively. Forh ∈ (0, δ], we have by (7.2.2) that

‖∆1(h)f‖∞ ≤ A

[‖∆1(h)f‖p +

1

pp′Jp(∆1(h)f) +

(1

pp′

)2

Ip(∆1(h)f)

].

By using Lemma 7.1, (2.1.2) and (7.1.1), we get for any 0 < h ≤ δ

Jp(∆1(h)f) ≤ c

∫ δ

0

t−1/pω(f ; t)pdt

t,

and

Ip(∆1(h)f) ≤ c

∫ δ

0

∫ 1

0

(uv)−1/pω(f ;u, v)pdv

v

du

u,

for some constant c that is independent of δ. It follows that

limδ→0

( sup0≤h≤δ

‖∆1(h)f‖∞) = 0.


In exactly the same way, we can show that

limδ→0

( sup0≤h≤δ

‖∆2(h)f‖∞) = 0.

Hence, limδ→0 ω(f ; δ)∞ = 0. This concludes the proof.

Corollary 7.6. Let f ∈ Lp([0, 1]2) (1 < p < ∞) and assume that Ip(f) <∞. Then there exist a continuous function g ∈ Lp0([0, 1]2) and univariatefunctions φ1, φ2 such that

f(x, y) = g(x, y) + φ1(x) + φ2(y),

for a.e. (x, y) ∈ R2.

Proof. By (7.1.11), we have

f(x, y) = f(x, y) + φ1(x) + φ2(y)

for a.e. (x, y) ∈ R2, where f ∈ Lp([0, 1]2). We shall prove that f is equala.e. to a continuous function g. Clearly Ip(f) = Ip(f) < ∞, and sincef ∈ Lp0([0, 1]2), we also have Jp(f) < ∞, by (7.1.14) and (7.1.9). The resultnow follows from Theorem 7.5.

7.3 Estimates of the Vitali type p-variation

In this section we shall consider the relationship between mixed integralsmoothness and the Vitali type p-variation.

In the case p = 1, we have the following theorem.

Theorem 7.7. Assume that f ∈ L1([0, 1]2) and that

ω(f ;u, v)1 = O(uv).

Then there exist a function g ∈ H(2)1 and univariate functions φ1, φ2 such

that for a.e. (x, y) ∈ R2,

f(x, y) = g(x, y) + φ1(x) + φ2(y).

Moreover,

v(2)1 (g) = sup

u,v>0

ω(f ;u, v)1

uv. (7.3.1)

7.3. Estimates of the Vitali type p-variation 113

Proof. We may without loss of generality assume that f ∈ L10([0, 1]2). For

n ∈ N, denote

fn(x, y) = n2

∫ 1/n

0

∫ 1/n

0

f(x+ s, y + t)dsdt.

We shall first prove that

v(2)1 (fn) ≤ sup

u,v>0

ω(f ;u, v)1

uv. (7.3.2)

Observe that

fn(x, y) =

∫ x

0

∫ y

0

D1D2fn(s, t)dsdt− fn(x, 0)− fn(0, y) + fn(0, 0)

= Fn(x, y)− fn(x, 0)− fn(0, y) + fn(0, 0).

Moreover, D1D2fn(s, t) = n2∆(1/n, 1/n)f(s, t). Thus, by (7.1.17),

v(2)1 (fn) = v

(2)1 (Fn) = n2

∫ 1

0

∫ 1

0

|∆(1/n, 1/n)f(x, y)|dxdy

≤ supu,v>0

ω(f ;u, v)1

uv.

This proves (7.3.2).Let E be the set of Lebesgue points of f . Since R2 \ E has Lebesgue

measure 0, there exist (x0, y0) ∈ E such that the sections

E(x0) = y ∈ R : (x0, y) ∈ E and E(y0) = x ∈ R : (x, y0) ∈ E,

have full measure. That is,

mes1(R \ E(x0)) = mes1(R \ E(y0)) = 0, (7.3.3)

where mes1 denotes linear Lebesgue measure. For n ∈ N, define now

gn(x, y) = fn(x, y)− fn(x, y0)− fn(x0, y) + fn(x0, y0).

For each n ∈ N, we have gn(x, y0) = gn(x0, y) = 0 for all x, y ∈ R. Thus, by(7.3.2),

v(2)1 (gn) + v1(gn(·, y0)) + v1(gn(x0, ·)) + |gn(x0, y0)| =

= v(2)1 (gn) = v

(2)1 (fn) ≤ sup

u,v>0

ω(f ;u, v)1

uv. (7.3.4)


By Lemma 7.4, there is a subsequence gnj that converges at all points to a

function g ∈ H(2)1 . On the other hand, by (7.3.3) and Lebesgue’s differentia-

tion theorem, for a.e. (x, y) ∈ R2 there holds

g(x, y) = f(x, y)− f(x, y0)− f(x0, y) + f(x0, y0).

Take φ1(x) = f(x, y0) and φ2(y) = f(x0, y) − f(x0, y0), then f(x, y) =g(x, y) + φ1(x) + φ2(y) for a.e. (x, y) ∈ R2.

We now prove (7.3.1). Since gnj converges to g at all points, it followsfrom (7.3.4) that for any net N ,

v(2)1 (g;N ) = lim

jv

(2)1 (gnj ;N ) ≤ sup

u,v>0

ω(f ;u, v)1

uv.

Thus, v(2)1 (g) ≤ supω(f ;u, v)/uv. On the other hand, since f = g a.e., we

have for any u, v ∈ [0, 1]

ω(f ;u, v)1 = ω(g;u, v)1 ≤ v(2)1 (g)uv,

by (1.0.28). Whence, supω(f ;u, v)1/uv ≤ v(2)1 (g). This proves (7.3.1).

Recall the notations (7.1.7) and (7.1.8).

Theorem 7.8. Let f ∈ Lp([0, 1]2) (1 < p <∞) and assume that Ip(f) <∞.

Then there exists a continuous function g ∈ H(2)p and univariate functions

φ1, φ2 such that for a.e. (x, y) ∈ R2, we have

f(x, y) = g(x, y) + φ1(x) + φ2(y). (7.3.5)

Moreover,

v(2)p (g) ≤ A

[ω(f ; 1, 1)p +

1

pp′Kp(f) +

(1

pp′

)2

Ip(f)

], (7.3.6)

where A is an absolute constant. If f ∈ Lp0([0, 1]2), then we may take φ1 =φ2 = 0 in (7.3.5).

Proof. By Corollary 7.6, there is a continuous function g ∈ Lp0([0, 1]2) suchthat

f(x, y) = g(x, y) + φ1(x) + φ2(y)


for a.e. (x, y) ∈ R2 (if f ∈ Lp0([0, 1]2), then φ1 = φ2 = 0). We shall prove

that g ∈ H(2)p .

Take any net

N = (xi, yj) : 0 ≤ i ≤ m, 0 ≤ j ≤ n,

and setgi(y) = g(xi+1, y)− g(xi, y), 0 ≤ i ≤ m− 1.

Clearly,

v(2)p (g;N ) =

(m−1∑i=0

n−1∑j=0

|∆g(xi, yj)|p)1/p

=

(m−1∑i=0

n−1∑j=0

|gi(yj+1)− gi(yj)|p)1/p

≤

(m−1∑i=0

vp(gi)p

)1/p

. (7.3.7)

By (3.2.3) and (2.1.5), we have for 0 ≤ i ≤ m− 1

vp(gi) ≤ A

(Ωp(gi) +

1

pp′

∫ 1

0

v−1/p−1‖∆(v)gi‖pdv), (7.3.8)

where ∆(v)gi(y) = gi(y + v)− gi(y). Set

Ii =

∫ 1

0

v−1/p−1‖∆(v)gi‖pdv.

By (7.3.8),(m−1∑i=0

vp(gi)p

)1/p

≤ A

(m−1∑i=0

Ωp(gi)p

)1/p

+1

pp′

(m−1∑i=0

Ipi

)1/p . (7.3.9)

Denote gy,v(x) = g(x, y + v)− g(x, y). Since

Ωp(gi)p =

∫ 1

0

∫ 1

0

|gi(y + v)− gi(y)|pdydv,


we have

m−1∑i=0

Ωp(gi)p =

∫ 1

0

∫ 1

0

m−1∑i=0

|gi(y + v)− gi(y)|pdydv

=

∫ 1

0

∫ 1

0

m−1∑i=0

|gy,v(xi+1)− gy,v(xi)|pdydv

≤∫ 1

0

∫ 1

0

vp(gy,v)pdydv.

Further, by (3.2.3) and (2.1.5), we have

vp(gy,v)p ≤ A

[Ωp(gy,v)

p +

(1

pp′

∫ 1

0

t−1/p−1‖∆(t)gy,v‖pdt)p]

.

Thus, (m−1∑i=0

Ωp(gi)p

)1/p

≤ A

[(∫ 1

0

∫ 1

0

Ωp(gy,v)pdydv

)1/p

+

+1

pp′

(∫ 1

0

∫ 1

0

(∫ 1

0

t−1/p−1‖∆(t)gy,v‖pdt)p

dydv

)1/p].

Observe that

Ωp(gy,v)p =

∫ 1

0

∫ 1

0

|∆(h, v)g(x, y)|pdxdh, (7.3.10)

thus (∫ 1

0

∫ 1

0

Ωp(gy,v)pdydv

)1/p

≤ ω(g; 1, 1)p.

Next, by Minkowski’s inequality,(∫ 1

0

∫ 1

0

[∫ 1

0

t−1/p−1

(∫ 1

0

|gy,v(x+ t)− gy,v(x)|pdx)1/p

dt

]pdydv

)1/p

≤∫ 1

0

t−1/p−1

(∫ 1

0

∫ 1

0

∫ 1

0

|gy,v(x+ t)− gy,v(x)|pdxdydv)1/p

dt

≤∫ 1

0

t−1/p−1

(∫ 1

0

ω(g; t, v)ppdv

)1/p

dt ≤∫ 1

0

t−1/p−1ω(g; t, 1)pdt


Thus, we have(m−1∑i=0

Ωp(gi)p

)1/p

≤ A

[ω(g; 1, 1)p +

1

pp′Kp(g)

]. (7.3.11)

Now we estimate the second term of (7.3.9). Applying Minkowski’s in-equality, we obtain(

m−1∑i=0

Ipi

)1/p

≤∫ 1

0

v−1/p−1

(m−1∑i=0

‖∆(v)gi‖pp

)1/p

dv. (7.3.12)

Furthermore,

m−1∑i=0

‖∆(v)gi‖pp =

∫ 1

0

(m−1∑i=0

|gi(y + v)− gi(y)|p)dy

≡∫ 1

0

Sv(y)dy. (7.3.13)

On the other hand,

Sv(y) =m−1∑i=0

|g(xi+1, y + v)− g(xi, y + v)− g(xi+1, y) + g(xi, y)|p

=m−1∑i=0

|gy,v(xi+1)− gy,v(xi)|p,

where gy,v(x) = g(x, y+ v)− g(x, y). Thus, by (3.2.3) and (2.1.5), for a fixedy ∈ [0, 1], we have the following estimate

Sv(y) ≤ A

(Ωp(gy,v) +

1

pp′

∫ 1

0

u−1/p−1‖∆(u)gy,v‖pdu)p

≤ 2pA

[Ωp(gy,v)

p +

(1

pp′

∫ 1

0

u−1/p−1‖∆(u)gy,v‖pdu)p]

. (7.3.14)

Further, by (7.3.10), ∫ 1

0

Ωp(gy,v)pdy ≤ ω(g; 1, v)pp.


This inequality, (7.3.14) and Minkowski’s inequality yield(∫ 1

0

Sv(y)dy

)1/p

≤ A′

[ω(g; 1, v)p+

+1

pp′

∫ 1

0

u−1/p−1

(∫ 1

0

‖∆(u)gy,v‖ppdy)1/p

du

]. (7.3.15)

Since

‖∆(u)gy,v‖pp =

∫ 1

0

|∆(u, v)g(x, y)|pdx,

we obtain from (7.3.13) and (7.3.15)(m−1∑i=0

‖∆(v)gi‖pp

)1/p

≤

≤ A′[ω(g; 1, v)p +

1

pp′

∫ 1

0

u−1/p−1ω(g;u, v)pdu

].

Integrating this inequality with respect to v and taking into account (7.3.12),we have (

m−1∑i=0

Ipi

)1/p

≤ A′[Kp(g) +

1

pp′Ip(g)

].

The above inequality together with (7.3.9) and (7.3.11) yield(m−1∑i=0

vp(gi)p

)1/p

≤

≤ A′

[ω(g; 1, 1)p +

1

pp′Kp(g) +

(1

pp′

)2

Ip(g)

]. (7.3.16)

The estimate (7.3.6) follows now from (7.3.7), (7.3.16), and the fact thatω(g;u, v)p = ω(f ;u, v)p.

To show that g ∈ H(2)p , we also need to demonstrate that there exist

x, y ∈ R such that gx, gy ∈ Vp. By applying (3.2.3) and (2.1.5) to an arbitraryx-section gx, we get

vp(gx) ≤ A

[‖gx‖p +

1

pp′

∫ 1

0

v−1/p−1‖∆(v)gx‖pdv]

= AΦ(x).


It was shown in the proof of Theorem 7.5 that if Jp(g) and Ip(g) are finite,then Φ ∈ L∞([0, 1]). Now, since g ∈ Lp0([0, 1]2), we have Jp(g) ≤ 12Ip(g)/p′,by (7.1.14) and (7.1.9). Thus, for a.e. x ∈ R,

vp(gx) ≤ A‖Φ‖∞ <∞.

In the same way, we have gy ∈ Vp for a.e. y ∈ R. This concludes theproof.

Below we shall demonstrate that the estimate (7.3.6) is sharp in a sense.For this, we use the following results. Let

tn(x) = sin 2πnx,

for n ∈ N. It is easy to show that we have

n1/p ≤ vp(tn) ≤ 2πn1/p (7.3.17)

andω(tn; δ)p ≤ 2πmin(1, nδ). (7.3.18)

Remark 7.9. Let 1 < p ≤ 2, by (7.1.9) and (7.1.10), we have

ω(f ; 1, 1)p +1

p′Kp(f) ≤ 8

(p′)2Ip(f)

Whence, for 1 < p ≤ 2, the estimate (7.3.6) assumes the form

v(2)p (f) ≤ A

(p′)2Ip(f). (7.3.19)

The constant 1/(p′)2 has the optimal order as p → 1. Indeed, let f(x, y) =

t1(x)t1(y), then f ∈ H(2)p for all p ≥ 1. By (7.3.17) and (7.1.15), we have

v(2)p (f) ≥ 1 for all p ≥ 1. On the other hand, by (7.3.18) and (7.1.16), we

easily get that Ip(f) ≤ 4π2(p′)2 for p > 1. This shows that the constantcoefficient 1/(p′)2 at the right-hand side of (7.3.19) cannot be replaced withsome cp such that limp→1(p′)2cp = 0.

Remark 7.10. Let p > 2, then 1 < p′ < 2 and the estimate (7.3.6) takesthe form.

v(2)p (f) ≤ A

[ω(f ; 1, 1)p +

1

pKp(f) +

1

p2Ip(f)

]. (7.3.20)


We shall prove that the first term at the right-hand side of (7.3.20) cannotbe omitted, and that the constant coefficients of the other two terms havethe optimal asymptotic behaviour as p→∞.

Take first f(x, y) = t1(x)t1(y). As above, v(2)p (f) ≥ 1 for all p > 1 and

thus limp→∞ v(2)p (f) ≥ 1. On the other hand, by (7.3.18) and (7.1.16), we

have for all p > 2 the inequalities

1

pKp(f) ≤ 16π2

pand

1

p2Ip(f) ≤ 16π2

p2,

This shows that the term ω(f ; 1, 1)p of (7.3.20) cannot be omitted.We proceed to show the sharpness of the constant coefficients. For fixed

but arbitrary 1 < p <∞, let αp, βp be any coefficients such that

v(2)p (f) ≤ A [ω(f ; 1, 1)p + αpKp(f) + βpIp(f)] , (7.3.21)

holds for some absolute constantA and all (continuous) functions f ∈ Lp([0, 1]2)with Ip(f) < ∞. In light of Theorem 7.8, we may assume that αp ≤ 1/pand βp ≤ 1/p2. We shall prove that these decay rates are optimal, i.e., thatlimp→∞ pαp > 0 and limp→∞ p

2βp > 0.Let f(x, y) = tn(x)t1(y), where n ∈ N is fixed but arbitrary. By (7.3.18)

and (7.1.16), we have

ω(f ;u, v)p ≤ 4π2vmin(nu, 1).

Simple calculations shows that there exists an absolute constant A > 0 suchthat Kp(f) ≤ Apn1/p and Ip(f) ≤ Apn1/p. On the other hand, by (7.3.17)

and (7.1.15), we have v(2)p (f) ≥ n1/p. Putting these estimates into (7.3.21)

and taking into consideration that βp ≤ 1/p2 yield that for all p > 2 and alln ∈ N, we have

n1/p ≤ A

[1 +

(pαp +

1

p

)n1/p

],

where A is an absolute constant. Assume that limp→∞ pαp = 0. Then, givenany ε > 0, we may choose r = r(ε) such that for all n ∈ N, there holds

n1/r ≤ A(1 + εn1/r).

In particular, take ε = 1/(2A) and choose subsequently n ∈ N large enoughto have n1/r > 2A. This gives the contradiction

n1/r ≤ A

(1 +

n1/r

2A

)< n1/r.


Whence, limp→∞ pαp > 0. To show that limp→∞ p2βp > 0, take f(x, y) =

tn(x)tn(y), where n ∈ N is fixed but arbitrary. As above, we have

ω(f ;u, v)p ≤ 4π2 min(nv, 1) min(nu, 1).

Then there exists an absolute constant A > 0 such that Kp(f) ≤ Apn1/p and

Ip(f) ≤ Ap2n2/p. On the other hand, v(2)p (f) ≥ n2/p. Putting these estimates

into (7.3.21) yields that for all n ∈ N and p > 2,

n2/p ≤ A[1 + pαpn1/p + p2βpn

2/p]

where A > 0 is an absolute constant. Dividing by n1/p and taking intoconsideration that pαp ≤ 1, we see that

n1/p ≤ A[2 + p2βpn1/p],

for all p > 2 and all n ∈ N. From here, we can give a proof by contradictionof the inequality limp→∞ p

2βp > 0, as above.

Remark 7.11. We shall consider trigonometric polynomials of two variablesand degree (n,m):

Tn,m(x, y) =n∑j=0

m∑k=0

[aj,k cos 2πjx cos 2πky + bj,k cos 2πjx sin 2πky

+ cj,k sin 2πjx cos 2πky + dj,k sin 2πjx cos 2πky]. (7.3.22)

Oskolkov [54] proved that for any trigonometric polynomial (7.3.22) of degree(n,m) and any 1 ≤ p <∞, there holds

v(2)p (Tn,m) ≤ A(nm)1/p‖Tn,m‖p, (7.3.23)

where A is an absolute constant. We can obtain (7.3.23) directly from (7.3.6).Indeed, take any trigonometric polynomial T of degree (n,m). The estimate

ω(T ;u, v)p ≤ min(uv‖D1D2T‖p, 4‖T‖p), u, v ∈ [0, 1], (7.3.24)

is immediate. By using (7.3.24), we get

Kp(T ) ≤ 2‖D1D2T‖p∫ 1/nm

0

t−1/pdt+ 4‖T‖p∫ 1

1/nm

t−1/p−1dt

≤ 2p′(nm)1/p−1‖D1D2T‖p + 4p(nm)1/p‖T‖p. (7.3.25)


It is a simple consequence of Bernstein’s inequality (see [14, p. 97]) that

‖D1D2T‖p ≤ 4π2nm‖T‖p. (7.3.26)

By (7.3.25) and (7.3.26), we get

Kp(T ) ≤ 12π2pp′(nm)1/p‖T‖p. (7.3.27)

Similarly, by (7.3.24),

Ip(T ) ≤ ‖D1D2T‖p∫ 1/n

0

∫ 1/m

0

u1/pv1/pdvdu

+ 4‖T‖p∫ 1

1/n

∫ 1

1/m

(uv)−1/p−1dvdu

≤ (p′)2(nm)1/p−1‖D1D2T‖p + 4p2(nm)1/p‖T‖p.

By the above estimate and (7.3.26), we have

Ip(T ) ≤ 8π2(pp′)2(nm)1/p‖T‖p. (7.3.28)

Now, (7.3.23) is derived from (7.3.6), the estimate ω(T ; 1, 1)p ≤ 4‖T‖p,(7.3.27) and (7.3.28).

7.4 Fubini-type properties of H(2)p

Recall that for p ≥ 1, the set Vp [Vp ]sym of functions of bounded iteratedp-variation consists of all functions f such that if

ϕ(x) = vp(fx) and ψ(y) = vp(fy),

then ϕ, ψ ∈ Vp. We observe first that Vp [Vp ]sym is not a vector space.

Proposition 7.12. There are two functions f and g such that for any 1 ≤p <∞, we have f, g ∈ Vp [Vp ]sym but (f + g) /∈ Vp [Vp ]sym.

Proof. Let f, g be functions that are 1-periodic in each variable, and definedas follows on [0, 1]2. Let f(x, y) = 1 if y = x and f(x, y) = 0 otherwise.Set g(x, y) = 1 if y = x and x /∈ Q, g(x, y) = −1 if y = x and x ∈ Q and

7.4. Fubini-type properties of H(2)p 123

g(x, y) = 0 otherwise. Then it is easy to see that for any x, y ∈ [0, 1], wehave

vp(fx) = 21/p, vp(fy) = 21/p.

Since vp(fx), vp(fy) are constant functions, they are of bounded p-variation,that is, f ∈ Vp [Vp ]sym. In the same way, we have g ∈ Vp [Vp ]sym. On theother hand,

(f + g)(x, y) =

2 if y = x and x /∈ Q,0 if y = x and x ∈ Q,0 otherwise.

Then vp([f + g]x) = 21+1/p if x /∈ Q and vp([f + g]x) = 0 for x ∈ Q. Clearly,the function x 7→ vp([f + g]x) /∈ Vp.

As was mentioned before, it was shown in [1] that

H(2)1 ⊂ V1 [V1 ]sym. (7.4.1)

The inclusion (7.4.1) is strict. In fact, we have the following result.

Proposition 7.13. Let 1 ≤ p < ∞, then there is a function f ∈ Vp [Vp ]sym

such that f /∈ H(2)p .

Proof. Define f on (0, 1]2

f(x, y) =

1 if 0 < x ≤ y ≤ 10 if 0 < y < x ≤ 1,

and extend to the whole plane by periodicity. It is clear that vp(fx) =vp(fy) = 21/p for all x, y. Thus, f ∈ Vp [Vp ]sym for 1 ≤ p <∞.

On the other hand, fix n ∈ N and let Nn = (xi, yj), where

xi =i

nand yj =

j + 1/2

n, 0 ≤ i, j ≤ n.

Then

|∆f(xi, yi)|p = 1

for 0 ≤ i ≤ n− 1, whence, v(2)p (f ;Nn) ≥ n1/p. Thus, f /∈ H(2)

p .


We will now proceed to consider the embedding H(2)p ⊂ Vp [Vp ]sym for

p > 1.We will use the following function

φ(x) = infk∈Z|x− k|, x ∈ R. (7.4.2)

For each n ∈ N, denote φn(x) = φ(nx). It is easy to see that

vp(φn) = 21/p−1n1/p. (7.4.3)

Definegn(x) = φ(2nx− 1)χ[0,1](2

nx− 1) for x ∈ [0, 1]. (7.4.4)

and extend gn to a 1-periodic function. Restricted to [0, 1], gn is supportedon [2−n, 2−n+1] and the graph of gn is an isosceles triangle with height 1/2.

Lemma 7.14. Let αn be any sequence of real numbers, and define

g(x) =∞∑n=1

αngn(x),

where the functions gn are given by (7.4.4). Then, for 1 ≤ p <∞, we have

vp(g) ≤ 21/p

(∞∑n=1

|αn|p)1/p

. (7.4.5)

Proof. For n ∈ N, set fn(x) = αngn(x). Clearly, the functions fn havepairwise disjoint supports. Moreover, it is easy to see that

vp(fn) = 21/p−1|αn| (n ∈ N). (7.4.6)

Assume first that all αn are nonnegative. Then the functions fn are nonneg-ative, and by Lemma 2.6 and (7.4.6), we have

vp(g) = 21/p−1

(∞∑n=1

αpn

)1/p

. (7.4.7)

When αn changes sign, we set α′n = max(αn, 0) and α′′n = −min(αn, 0).Then α′n, α

′′n ≥ 0 for all n ∈ N, and

g(x) =∞∑n=1

α′ngn(x)−∞∑n=1

α′′ngn(x) = h1(x)− h2(x).


Applying (7.4.7) to h1, h2, we obtain

vp(g) ≤ vp(h1) + vp(h2) = 21/p−1

( ∞∑n=1

(α′n)p

)1/p

+

(∞∑n=1

(α′′n)p

)1/p .

Since α′n, α′′n ≤ |αn|, (7.4.5) follows.

Theorem 7.15. For p > 1, we have

H(2)p 6⊂ Vp [Vp ]sym.

Proof. Let 1 < p <∞ and set

f(x, y) =∞∑k=1

2−k/pgk(x)φ(2ky), (7.4.8)

where φ is given by (7.4.2) and gk (k ∈ N) by (7.4.4). We shall prove that

the function f defined by (7.4.8) belongs to H(2)p \ Vp [Vp ]sym.

First, we show that f ∈ V (2)p . Fix any net

N = (xi, yj) : 0 ≤ i ≤ m, 0 ≤ j ≤ n.

For each j ∈ 0, 1, ..., n− 1, denote

fj(x) = f(x, yj+1)− f(x, yj).

Since∆f(xi, yj) = fj(xi+1)− fj(xi),

we getm−1∑i=0

|∆f(xi, yj)|p =m−1∑i=0

|fj(xi+1)− fj(xi)|p ≤ vp(fj)p.

Thus,

v(2)p (f ;N )p ≤

n−1∑j=0

vp(fj)p. (7.4.9)

On the other hand, we note that

fj(x) =∞∑k=1

2−k/p[φ(2kyj+1)− φ(2kyj)]gk(x).


By Lemma 7.14, we have

vp(fj)p ≤ 2

∞∑k=1

2−k|φ(2kyj+1)− φ(2kyj)|p. (7.4.10)

Thus, by (7.4.9) and (7.4.10),

v(2)p (f ;N )p ≤ 2

n−1∑j=0

∞∑k=1

2−k|φ(2kyj+1)− φ(2kyj)|p. (7.4.11)

Set σl = j : 2−l−1 < yj+1 − yj ≤ 2−l for integers l ≥ 0. Subdividing thesum at the right-hand side of (7.4.11), we have

v(2)p (f ;N )p ≤ 2

∞∑l=0

∑j∈σl

∞∑k=1

2−k|φ(2kyj+1)− φ(2kyj)|p. (7.4.12)

We shall estimate the right-hand side of (7.4.12). Observe that

|φ(2kyj+1)− φ(2kyj)| ≤ min(1, 2k(yj+1 − yj)). (7.4.13)

Indeed, since φ is a nonnegative function, we have

|φ(2kyj+1)− φ(2kyj)| ≤ ‖φ‖∞ = 1/2,

and, at the same time,

|φ(2kyj+1)− φ(2kyj)| ≤ 2k(yj+1 − yj)‖φ′‖∞ = 2k(yj+1 − yj).

Fix l ≥ 0 and let j ∈ σl. Then, yj+1 − yj ≤ 2−l, and by (7.4.13), we have

∞∑k=1

2−k|φ(2kyj+1)− φ(2kyj)|p ≤∞∑k=1

2−k min(1, 2k−l)p

= 2−lpl∑

k=1

2k(p−1) +∞∑

k=l+1

2−k.

Since p > 1, it follows that there is a constant cp > 0 such that

∞∑k=1

2−k|φ(2kyj+1)− φ(2kyj)|p ≤ cp2−l,


for all j ∈ σl. Consequently, for l ≥ 0, there holds∑j∈σl

∞∑k=1

2−k|φ(2kyj+1)− φ(2kyj)|p ≤ cp2−lcard(σl), (7.4.14)

where card(σl) denotes the cardinality of the finite set σl. To sum up, by(7.4.12) and (7.4.14), we have

v(2)p (f ;N )p ≤ cp

∞∑l=0

2−lcard(σl)

≤ 2cp

∞∑l=0

∑j∈σl

(yj+1 − yj) = 2cp.

Thus, f ∈ V (2)p . To prove that f ∈ H(2)

p , it suffices to show the existence ofx0, y0 ∈ R such that fx0 , fy0 ∈ Vp. For all x ∈ R we have f(x, 0) = 0 andthus f(·, 0) ∈ Vp. Similarly, f(1, y) = 0 for all y ∈ R, so f(1, ·) ∈ Vp. Thus,

f ∈ H(2)p .

Now we demonstrate that f /∈ Vp [Vp ]sym. First, we observe that gn(2−k) =0 (n, k ∈ N). Thus, vp(fx) = 0 for x = 2−k (k ∈ N). On the other hand, ifx = (2−k+1 + 2−k)/2 (k ∈ N), then

fx(y) = 2−k/p−1φ(2ky),

and by (7.4.3), we have

vp(fx) = 2−k/p−1vp(φ2k) = 21/p−2.

Clearly, the function x 7→ vp(fx) does not belong to Vp. Thus, f /∈ Vp [Vp ]sym.

It follows from Proposition 7.13 and Theorem 7.15 that Fubini-type prop-erties fail in H

(2)p for p > 1.

Remark 7.16. It is easy to see that for any p ≥ 1, we have

H(2)p ⊂ L∞ [Vp ]sym. (7.4.15)

Moreover, the function constructed to prove Theorem 7.15 shows that forp > 1, the exterior L∞-norm of (7.4.15) cannot be replaced by a strongerVq-norm. That is,

H(2)p 6⊂ Vq [Vp ]sym, for p > 1 and q ≥ 1.

However, for p = 1 we have (7.4.1), which is much stronger than (7.4.15).

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The classical concept of the total variation of a function has been extended in several directions. Such extensions find many applications in different areas of mathematics. Consequently, the study of notions of generalized bounded varia-tion forms an important direction in the field of mathematical analysis.

This thesis is devoted to the investigation of various properties of functions of generalized bounded variation. In particular, we obtain the following results:

• sharp relations between spaces of generalized bounded variation and spaces of functions defined by integral smoothness conditions (e.g., Sobolev and Besov spaces);

• optimal properties of certain scales of function spaces of fractional smooth-ness generated by functionals of variational type;

• sharp embeddings within the scale of spaces of functions of bounded p-variation;

• results concerning bivariate functions of bounded p-variation, in particular sharp estimates of total variation in terms of the mixed L

p-modulus of conti-

nuity, and Fubini-type properties.


ISSN 1403-8099

ISBN 978-91-7063-486-4

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Functions of Generalized Bounded Variation605177/FULLTEXT01.pdf · Such functions are called p-continuous, and the class of all p-continuous func-tions is denoted by C p. For f2Lp([0;1])(1