x

gxxf xxx

x

(1.1)

then

10, 1fC

,

0.1gBVf

but

g

f is

not of bounded variation.

g

However, the multiplication of f

k

by a derivative

of f, which is of bounded variation, improves the proper-

ties of that composite function. Indeed, a proof of the

following theorem can be found in [3, Theorem 5].

Theorem 1.1. ([3]). Suppose that f has a derivative

f

of order k everywhere on ,ab

,

k

. If

BVa b

and if

f

,BVc d

,

:min ,

ab

cf

, where

,

:max ,

ab

df

·k

then the function

f

f

is of bounded variation on

,ab

; moreover,

,

,,

·1,

kk

ab

ab ab

ff kf

(1.2)

,BVa b defined as where

,ab is the norm on

,,

:sup ;,

ab xab

f

fxVf ab

C

opyright © 2012 SciRes. APM

J. GIMÉNEZ ET AL. 23

Let be a subspace of . Given a function

, the autonomous Nemytskij (or Superposi-

tion, see [4]) opera

,ab

generated by g, is

defined a

,ab

:g

:

g

S

tor

s

: ,

g

Sftgf, .ttab

,

,ab

:g

(1.3)

Given two linear spaces and a function

, a primary objective of research is to investi-

gate under what conditions on the generating function the

associated Nemytskij operator maps into . This

problem is known as the Superposition Operator Prob-

lem.

Recently (see e.g. [5]), the Superposition Operator

Problem have been studied extensively in various spaces

of differentiable functions related to the spaces

,BVa b

and

,

A

Cab

.

In this paper, we discuss various aspects of the the

Superposition Operator Problem when the spaces

1

,,Cab

and are somehow related to the space

,a b

·k

BV . We prove a version of Theorem 1.1 about the

integrability of products of the form ·

g

ff

g is an integrable function and

k

f

when

f

is continuous, and

obtain an estimation of the norm

1,

··k

L

ab

we give necessary

2. Some Function Spaces

finitions and state some

gfff

.

Finally, we prove two results in which

and sufficient conditions for the autonomous Nemytskij

Operator to map the space of all functions of second

bounded variation into itself and the class of functions of

bounded

,2p-variation into itself.

In this section we recall some de

results which will be needed for the further development

of this work.

We will use the notation

;,Ma b to denote

functi

BV

the space of all boundedons f such that

1,f

can be expressed as a union of M subin-

tervals of

,ab , for all

,,,cd

where

in,max f.

:m ab

cf

,,

:ab

d

M. Josephy proved in [6 all M

] that for

the

class of all bounded functions in

BV M is

contained in

;,ab

,BVa b .

A function

:,abe Lipschitz coF bis said ton-

tinuous on

,ab iff

:sup :,,,

y

LFxyab xy

xy

The class of all Lipschitz continuous functions in

.

Fx F

,ab

is denoted as

,Lipab and the functional

:m

ax ,

f

fL

defines a norm on it.

Recall that a function

f

:,fab is said to

ous on

be ab solutely continu,ab if, given 0,

there exists some 0

such that

sup :,

nn

fII

whenever

I

,

nnn

ab is a finite con of mu-

tually disjoint subintervals of

llectio

,ab with

1

.

i

n

nn

ba

The class of all absolutely conus functions on tinuo

,ab , which is actually an algebra, is denoted as

,

A

Cab.

inition 2.1. (Luzin N property). A real-valued

de

Def

function fined on a compact interval

I

is said to

satisfy the Luzin N property (or simply, property N) if it

carries sets of -

measure zero into set-s of

measure

zero.

It is easy to see that the composition of twunctions

that ha

o f

ve property N also has property N. The class of all

continuous functions that satisfy property N on an inter-

val

,ab will be denoted as

,Nab.

The following characterization of absolutely continu-

ous funons is well known (capctif. [2, Chter 7]).

Proposition 2.2. The following statements on a func-

tion

:,fab are equivalent:

1) f is absolutely continuous,

2)

f

,,BVa btisfies property N,

f

Ca b and sa

le on

3)

exists -

a.e., is integrab,ab and

d.

a

x

f

x

The equivalence (a)

fa ftt

(b) is known as the Banach-

Zareckiĭ theorem. The functional

1

L

:

AC

f

fa f

A

defines a norm on ,Cab; in fact,

;, .

f

1

LVfab

e same fven in (1.1) show

that the class of all alutely continuous functions is not

cl

ed second variation. The

Remark 2.3. Thunctions gi

bso

osed under compositions.

In the year 1908, de La Vallée Poussin ([7]), intro-

duced the notion of bound

class of all functions of bounded second variation on

an interval

,ab is denoted by

2,BVa b and is

characterized by the following result due to F. Riesz

([8]):

Proposition 2.4. A real valued function f is in the class

2,BVa b if, and only if, there is a function

,

f

BVa b such that

x

d.

a

f

xfa fxx

(2.1)

Copyright © 2012 SciRes. APM

J. GIMÉNEZ ET AL.

24

In this case, the relation

;,

2,:

BVa b

f

fa fa Vf ab

(2.2)

defines a norm on

2,BVa b.

Using the notation of (2.1) we

, ,abCab.

Definiti on 2.5.define

,:BV abf

2,:BVabfBV

Clearly,

,BVa b

is a linear subspace

2,BVa b.

that by the Fundam of Calcu-

Notice also

lus if

mental Theore

,

f

BVa b

then fle on is differentiab

,ab

and .

f

f

In fact,

,BVa b

if, and only i

f,

1,

f

Cab and

,

f

BVa b

.

In 1tes, in [9], introduced the notif

fuof bounded iation, for 1

997, N. Merenon o

nctio

,2p.p

n

-var

The clllons of bounded ass of ain functi

,ab

,2p-variation is denoted by

2,

p

A

ab and its

characterized as follows:

Proposition 2.6. ([9]). A function

,ab

f is in the

class

2,

p

A

ab if, and only if,

,

f

bAC a

and

,.

p

f

Lab

In this case the relation

2,

p

Aa Lab

p

f f

o

,

:

bfafa

n

defines a norm

2,

p

A

ab .

Clearly a continuously differentiable functionis Lips-

chitz continuous and any Lipschitz continuous function is

> 1,p the following

ch

absolutely continuous. In fact,

ain of strict inclusions holds (see e.g., [5,10]):

21

,,,,

,,.

p

A

abBVabCaip ab

ACa bBVa b

(2.3)

b L

3. Main Results

We begin this section by stating some fundamentals facts

ions of functions on BV and AC. In

sic properties of the inner function

concerning composit

these cases the intrin

(in the composition) will show to play also an important

role. We recall that if D and E are given sets, X is a linear

subspace of

E

and φ is a map from D to E, the linear

composition operator :

D

CX

is defined by

:Cff

.

Remark 3

1) Although both cl AC are not clo

.1.

asses BV and sed

under composi, they do satisfy a weaker property in

thre precisely, it readily follows from a

re

tion

at respect. Mo

sult given by M. Josephy in [7, Theorem 3] that if

,:,ab cd

then, the operator C

maps

,BVc d

into

,BVa b if, and only if,

;,BVMab

for

some M. From this, it readily follows that if

;,ab,ACa b then C

BV M

maps

,

A

Ccd

into

,

A

Cab. The converse of also

true (see [3]).

rem of Algebra an

Theo s a polynomial of degree M, then for all

this proposition is

2) By the fundamental Theod Rolle’s

rem, if f i

,ab

;, ,

f

BVMa bACa b

; also, every

monotone function

,

ab

is in

;,BVMa b. for

some M

.

instances of a

very remarkable phen that of non-

linear

In what follows we will observe more

nomenoten occurs in

functional analysis: is the case in which given two

functions, say g and f, the multiplication of

g

f by a

continuous derivative

,

k

fk of f improves the

properties of the composition.

The following proposition is a corollary oeorem

1.1. The result followsfact that the space

f Th

from the

,Nab is an algebra with respect to pointwise multipli-

ca

continuo

tion (see [3]), and the Banach-Zareckii Theorem.

Proposition 3.2. (Burenkov). If f has an absolutely

us -

th

kderivative

k

f on

,ab and if

AC , then the function

k

ff

is also abso-

lutely continuous on

,ab and inequality (1.2) holds.

If

1,

g

L

integrability of

products of the form

ab similar considerations as those dis-

cussed above apply with respect to the

g

ff

or even p

g

ff

.

Now, not even the fact that the function g is integrable

and the function f is absolutely continuous guarantees

that the product

g

ff

grable; for ie

[3]), let

is intenstance (se

0:0:0gf

and, for 0x, let

:gx

1

x

and

63

:sin12fx xx

, then

0, 1fAC,

g is integrable in

, but

0,1f

g

ff

is not inte-

grable in

0,1 . In that reswiposition

is well known (see, for instance, [11m 3.54]):

[Change of Variables] Le

pect, the follong pro

, Theore

Proposition 3.3. t

:,gcd

be an integrable function and let

:, ,

f

ab cd

be a function differentiable-a.e.

in

,ab . Then

·

g

ff

is integrable and

d

f

f

d

g

tt gfx

r all

fx x

holds fo

,,ab

if, and only if, the function

0,1Gf AC, where

:

z

c

Gz g

Notice that Gbsolutely continuous function,

which bring us back to the same situatio

above.

bility) properties of the product ·

d .,,ttzcd

(3.1)

is an a

ns considered

It turns out that, if g is an integrable function, multi-

plication by a continuous derivative of f improves the

(integra

g

ff

. By

an

lemma

alogy with an useful notion originated from the theory

of partial differential equations, we might call this de-

rivative an integrating factor. The following pro-

vides a version of Theorem 1.1 when the outer function

in the composition is an integrable function. The propo-

sition might be of some interest in itself.

Lemma 3.4. Suppose that

1loc

gL and that

,

k

fbCa

k. Then

1

·;,· k

gf fbfLa

Copyright © 2012 SciRes. APM

J. GIMÉNEZ ET AL. 25

moreover,

1

1,

·· .

ab

aLf

L

gf fg

Proof of

k

,b

kk

f kf

. The continuity

f

implies that the open

: 0

k

fx can be expresse

component open intervals, say

Sa or

0

set Sx

countable unio

:,ab

n of

d as a

1,

N

i

, where N N. Now, since

k

f

i

i

b

on S

, to each

,N corresponds a

nonnegative integer i

mk such that

1,i

,

ii

ab can be

decom nt no

, ,

ii

mm

a b

posed into at most disjoin-degenerated

112 2

,,,, ,

ii i iii

ab ab

on which f is

monotone.

ng

i

m

intervals,

Now, bei

f

continuoon

us

,ab , the Fundamen-

tal Theorem of Calculus implies that

,;f

A

Cab

likewise, the indefi, defined by

el

nite integral function G

by (3.1), is absoluty continuous, thus Remark 3.1, the

monotonicity of f on ,

j

j

ii

ab

implies that

,;

A

CbGaf consequently, ·

g

ff

is integrable

on this interval. Hence, since

f

does not change sign

on ,

j

j

ii

ab

, we mus

t have

,

ijj

ii

a

fa fb

dd

j

i

j

b

g

ftf ttgxx

whertion e the nota,

stands for

,

if

or

,

ot

Now, since

herwise.

k

f

is continuous, (generalized)

for iies th

the

mean value theoremntegrals implat, on each

,

j

j

ii

ba

there is a poin,t

j

i

c such that

,

d.

jj

ii

i

fa fb

d

j

i

j

i

bk

a

kj

g

ftf tft

f

cgxx

Notice th

·· k

at the product

g

ff f

is easurable

n on

a m

functio

,ab . Thus

1

11 ,

11 ,

1,

,

i

i

jj

ii

d

d

d

d

.

jj

ii

jj

ii

k

m

Nkj

i

ij fa fb

m

Nk

ij fa fb

Nk

ifa fb

k

Lfab

f t

b

agftft

f

cgxx

f

gx x

kfgx x

k

t

fg

The proof is complete.

The Autonomous Nemytskij Operator on the

Spaces

,BVab and

2,

p

Aab

For convenience we state the next result as a single

proposition. The proof of it is based in three separate

results of M. Josephy [6] (see also [12]), N. Merentes [13]

and N. Merentes and S. Rivas [14].

Proposition 3.5. Suppose

:,,,BabVba

,RBVpab ([14]). Then Sg maps ,ACab or ,ab

into itself if, and only if,

.

loc

gLip

Now we present a result that gives a necessary and

sufficient condition for the Nemytskij operator to map

the space

,BVa b

into itself.

Theorem 3.6. Sg maps

,BVa b

into itself if, and

only if,

.

loc

gBV

Moreover, in this case Sg is

automatically bounded.

po-

, for all

Proof. Suppose that

.

loc

gBV

Then, by Pro

sition 3.5

,,

f

BV a

b

,,

g

fLipabACab (since both g and f are

Lipschitz continuous); thus, for -a.e.

,

x

ab

·,

g

fgff

and, by Theorem 1.1, with

g

and 1,k

we get

·,

g

ff BVab

and, since it is

clearly continuous on

,ab , it follows that

,.f BabV

Conversely, assume Sg maps

g

,BVa b

into itself.

For any given pair of rβeal numbers α, with α < β de-

note by ab

f

the linear difeomorphism

:,,

ab afb

defined as

:,

ab ab

fxmxa

where:

ab

mba

.

Then, each

,b

and therefore, for all

ab

:

fBVa

,ab. Thus, b

e

gab

fVSB

y the first part of

the proof we hav

1

ga

b aba

ab

g b

,.

g

Sf ff

S f

BV

Hence,

loc

gBV

the proof is comple

abou

and te.

The conclusiont automatic continuity follows at

on ce from (2.2) and (1.2).

Now we present a similar result for the space

2,

p

Aab

.

At this p, let us recall the following pr

orem 3.44]):

oint oposition (see,

for instance [11, The

Suppose g, f are functions defined on intervals and that

f is well defined. If g, f and

g

f are -

g

a.e if-

ferentiable functions and g satisfies the property N then,

. d

fxgfx

fx

for -

a.e. x,

g

where

g

fxf x

is interpreted to be zero when-

ever

0fx

.

Theorem 3.7. Let 1.p S

g maps

2,

p

Aab

into it-

self if, and only if,

2

gA In this case Sg is

ically bounde

ploc

automatd.

Copyright © 2012 SciRes. APM

J. GIMÉNEZ ET AL.

Copyright © 2012 SciRes. APM

26

Proof. Suppose first that

2

ploc

gA By Proposi-

tion 3.5, for all

2,,

p

fAab

,gf AbCa (sin ce

is Lipschitz continuous on

,ab ). Thus

gf xgfx fx

Séances de l’Académie des Sciences, Vol. 2, 1881, pp.

228-230.

g

f,,allbx a or

and since

,gbAC a

(and in particular it satisfies

property N), for -

a.e.

,xab

2

.

g

fxgfxfx

g xffx

ince

(3.2)

Now, S

g

f

is

s in

continuous, the second summand

in the right hand side of (3.2) i

,Lab, and

p

,

·.

pp

L

Lab

gff f

(3.3) gf

On the other hand,

2p

p22

2.

p

gf

xfxfx

Lemma 3.4

gfx fx

Hence, by

,.

pfab

pfab

p

L

L

f g

From (3.3) and (3.4) it follows that ps

,

212pp

gff

(3.4)

Sg ma

2

p

A

that,

ilar to the one given

for the the necessity of the condition in th

Th

4. Acknowledgements

This research has been partly supported by the Central

Bank of Venezuela. We want to give thanks to the

library staff of B.C.V. for compiling the references.

he anonymous referee an

editors for their valuable comments and suggestions.

[2]

[3] V. I. Burenkov, “On Integration by Parts and a Problem

on Composititinuous Functions

Which Arises in This Connection, Theory of Functions

and P. P. Zabrejko, “Nonlinear Superposition

Remarks on

ematical

,ab

R. Kannan and C. K. Krueger, “Advanced Analysis on

the Real Line,” Springer, New York, 1996.

on of Absolutely Con

and Its Applications,” Proceedings of the Steklov Institute

of Mathematics, Vol. 134, 1975, pp. 38-46.

[4] J. Appell

Operator,” Cambridge University Press, New York, 1990.

[5] J. Appell, Z. Jesús and O. Mejía, “Some

Nonlinear Composition Operators in Spaces of Differen-

tiable Functions,” Bollettino Della Unione Matematica

Italiana—Serie IX, Vol. 4, No. 3, 2011, pp. 321-336.

[6] M. Josephy, “Composing Functions of Bounded Varia-

tion,” Proceedings of the AMS—American Math

Society, Vol. 83, No. 2, 1981, pp. 354-356.

doi:10.1090/S0002-9939-1981-0624930-9

[7] Ch. J. de la Valle Poussin, “Sur L’ntegrale de Lebesgue,”

Transactions of the AMS—American Mathematical Society,

Vol. 16, 1915, pp. 435-501.

[8] F. Riesz, “Sur Certains Systmes Singuliers d’Quations

ariation,”

1992, pp. 117-

o. 1, 2010, pp. 1001-1012.

, “Remarks on the Superposi-

pp. 727-737. doi:10.1080/17476930903568332

Intgrales, Annales de l’Ecole Normale,” Suprieure, Vol.

28, No. 3, 1911, pp. 33-62.

[9] N. Merentes, “On Functions of Bounded (p; 2)-V

Collectanea Mathematica, Vol. 43, No. 2,

123.

[10] J. Appell, “Some Counterexamples for Your Calculus

Course,” Analysis, Vol. 31, N

into itself and in this case, Sg maps bounded sets on

bounded sets.

The proof of the converse is sim [11] G. Leoni, “A First Course in Sobolev Spaces,” American

Mathematical Society, Vol. 105, Rode Island, 2009.

[12] J. Appell and P. P. Zabrejko

e proof of

eorem 3.6.

tion Operator Problem in Various Function Spaces,”

Complex Variables and Elliptic Equations, Vol. 55, No. 8,

2010,

icin

[13] N. Merentes, “On the Composition Operator in AC[a,b],”

Collectanea Mathematica, Vol. 42, No. 1, 1991, pp. 121-

127.

[14] N. Merentes and S. Rivas, “El Operador de Compos

We would like to thank td the

en Espacios de Funciones con Algún Tipo de Variación

Acotada,” IX Escuela Venezolana de Matemáticas, Facultad

de Ciencias-ULA, Mérida, 1996.

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