>sinif 0.
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 Rolles
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 lAcadémie 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
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