Advances in Pure Mathematics, 2012, 2, 45-58
http://dx.doi.org/10.4236/apm.2012.21011 Published Online January 2012 (http://www.SciRP.org/journal/apm)
On Second Riesz
-Variation of Normed Space
Valued Maps
Mireya Bracamonte1, José Giménez2, N. Merentes3, J. L. Sánchez3
1Departamento de Matemáticas, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela
2Departamento de Matemáticas, Universidad de los Andes, Mérida, Venezuela
3Departamento de Matemáticas, Universidad Central de Venezuela, Caracas, Venezuela
Email: mireyabracamonte@ucla.edu.ve, jgimenez@ula.ve, nmerucv@gmail.com, jose.sanchez@ciens.ucv.ve
Received September 21, 2011; revised November 11, 2011; accepted November 20, 2011
ABSTRACT
In this article we present a Riesz-type generalization of the concept of second variation of normed space valued func-
tions defined on an interval
,ab [,]ab
fX. We show that a function , where X is a reflexive Banach space, is of
bounded second -variation, in the sense of Riesz, if and only if it can be expressed as the (Bochner) integral of a
function of bounded (first) $\Phi$-variation. We provide also a Riesz lemma type inequality to estimate the total second
Riesz- -variation introduced.
Keywords: Young Function;
-Variation; Second
-Variation of a Function
,2p-variation. 1. Introduction
Functions of bounded variation where first introduced in
1881 by Camille Jordan who established the relation be-
tween these functions and the monotonic ones. Thus, the
Dirichlet Criterion for the convergence of the Fourier
series applies to the class of functions of bounded varia-
tion. This, in turn, has motivated the study of solutions of
nonlinear equations that describe concrete physical phe-
nomena in which, often, functions of bounded variation
intervene.
The interest generated by this notion has lead to some
generalizations of the concept, mainly, intended to the
search of a bigger class of functions whose elements
have point wise convergent Fourier series. As in the clas-
sical case, these generalizations have found many appli-
cations in the study of certain differential and integral
equations. Ch. J. de la Vallée Poussin, introduced in 1908
([1]) the notion of second variation of a function. A few
years later, F. Riesz ([2]) proved that a function f is of
bounded second variation on
,ab

$1 p
if, and only if, it is
the definite Lebesgue integral of a function F of bounded
variation.
More recently, in 1983, A. M. Russell and C. J. F. Upton
[4] obtained a similar result for functions of bounded sec-
ond variation , in the sense of Wiener. In
1992 the third author introduced the notion of
,2p
-
variation, in the sense of Riesz ([4]), presenting, also, a
result that generalizes the renowned Riesz lemma for the
class that he called

2,
p
BVa b, or class of functions of
bounded Riesz
In this article we define the notion of function of
bounded second
-variation in the sense of Riesz. We
show that a function F, with values in a reflexive Banach
space, is of bounded second -variation, in the sense of
Riesz, if and only if it is the integral (in the sense of Bo-
chner) of a function of bounded -variation. In addi-
tion, from the main results presented it is deduced an
inequality that generalizes Riesz’s lemma.
2. Vector Value Functions of Bounded
Variation
We begin this section by recalling some known spaces
and results.
We will also assume that all partitions of an interval
,ab considered, contain at least one point
,tab;
the set of all such partitions will be denoted as
3
The notion of bounded second variation in the sense
De La Vallée Poussin is defined as follows: A function
π,ab .
:,uab





is of bounded second variation if and only
if
3
2
2
12 1
π,0
;, :sup,,
m
ii ii
ab i
Vuabut tutt
 



where
21
12
21
,: ii
ii ii
ut ut
ut ttt




02
π,, ,.
m
tt t and
(2.1)
C
opyright © 2012 SciRes. APM
M. BRACAMONTE ET AL.
46
The class of all the functions of bounded second varia-
tion (on
,ab
), in the sense of De La Vallée Poussin, is
denoted by

2,BVa b
.
The following are known properties of functions in

2,BVa b
([3,5,6]).
Proposition 2. 1. Let

2,uBV ab
.
1) If

2,vBV ab and
is any real constant,
then




2
;,
;, .
ab
Vvab
acdb




22
22
;,
;, ;,
Vuab Vu
Vuvab Vuab
y


2) (Monotonicity) If then


;,Vuab
22
;,Vucd.
3) (Semi-additive) , then
acb
2,uBV ac,

2,cbuBV and



2
;,Vucb
22
;, ;,Vuab Vuac.
4)
01
,uy y
is bounded for all
01
,,
yab
u
.
5) is Lipschitz and therefore absolutely continuous
on
,ab
.
6)
2,uBV ab12
uuu
1
u u if and only if , where
, are convex functions.
2
7) A necessary and sufficient condition for a function
F to be the integral of a function

,fBVab
is that

2,
F
BVa bu
. This result is known as Rieszs lemma.
8) If u is twice differentiable with integrable on
,ab then
,ab
2
V

uB and


2;,Vuab d
b
autt


1p
.p
.
In 1910, F. Riesz introduced the concept of function of
bounded p-variation as follows:
Definiti on 2.2. Let1 A function
:,fab
is said to be of bounded p-variation,
in the sense of Riesz, if




11 .
i i
t t


3
2
([ , ])0
;,
:sup ,
RR
pp
mp
ii
ab i
VfVfab
ut t

The class of all functions of bounded p-variation on
,ab , in the sense of Riesz, is denoted by
,.
p
RBVa b
It readily follows from the definitions that
,,bCa b
p
RBV a and that for all
,,
p
f
gRBVa b
  
;
Rp R
ppp
Vfgf Vg gVf


Rp
in fact, the relation
 

1
:,
R
p
p
aVf
p
ff defines
a norm in
,RBVa b
p respect to which it becomes a
Banach algebra (see e.g., [7]).
Notice that
1
RBVab,,BVab
1p
; on the other hand,
it is well known that, for a function belongs
to ,RBVa b
p if, and only if, it is absolutely continuous
and its derivative (which exists
-a.e. in
,ab) bes
to
long
,
p
Lab; in this case


p
R
p;,
L
p
Vfab f
(this is
renownea, [2]). In particular, thed Riesz’s lemm
,,.
p
Lipa bRBVa b
Further generalizations consider the so call
-func-
tions. As it is customary, we shall denote by
the set
of all continuous convex functions
:0, 0,

such that
0

if and only if 0
and
0
0
lim

. Likewise, t
he notation
shall be use
d
to denote the set of all functions , for which the
Orlicz condition holds:

lim

[8]
 . Following
functions in
shall be called -functions. Any func-
tion
strictly increasing, and the function is
is no decreasing for 0.
One says that a function satisfies a condition 
2
, and writes 2
 , ifare constants 2K
0
t there
and
such that
t for all 0.tt (2.1)
For instance, if
2tK
:,1,xtp
p
 one may chooses
th 2.
p
e optimal constant K
that is a In the sequel we will assume
-function
and X is a normed space with norm ·Xim (or sply ·).
The integral of a normed space va function means
the Bochner integral ([9]).
The following generaliza
lued
tion of the notion of function
of
The
bounded variation is due to V. V. Chistyakov ([11],
see also [12]).
Definition 2.3.
-variation in sense of Riesz of a
map
:,
f
ab X isefined as d
 
1
1
11
,,
:sup
nii
ii
iii
abVf
ft fttt
tt



 


where the supremun is taking over the set of all parti-
i
,,,
RR
Vf abX
tions

0
n
i
t
of the interval
,.ab
Somproperties of theonale known functi
R
V
are the
following
1)
R
V
is no decreasing, that is,
, ,,
RR
b Vfcd

if
,ab,Vfa ,;cd
2)
R
is semi additive
V

, ,,Vfa Vfab
for all
;acb
, ,,
RR
cVfcb

R
and
3)
R
is sequentially lower semi continuous.
V
The set of all functions [,]ab
fX for which
,,
Rfab
V
is not necessarily a linear space, but
Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL.
Copyright ©iRes. APM
47
it is a conve
2012 Sc
x su
bset of [,]ab
X
and

·, ,
R
Vab
is a
convex functional on it.
The class
We now state Theorem 3.3 of [11].



R
Vf


is a linear se, called tnctions of bound
w
[,]
:X
he class of fu
,,
R
BabX:
pac
0,
ab
f

ed
-variation, in the sense of Riesz. It can be equipped
h the norm: it
 
:,
f
fa f
 where
 
1.
:inf 0:fV



f

Definition 2.4. ([11])
A mapping
:,
f
ab Xis
called absolutely continuous if there exists a function

:0, 0,
 
such that for any 0, n
and
of points any finite collection
1
,,
n
ii
abb
h that
1122nn
abab ab  
 

a suc
tion

ii
ba
i
, the condi

implies
1i
n
1
.
ii
i
A proof of the following result can be found in [12]. e.
Th
nfb fa
Proposition 2.5. Let X be a reflexive Banach spac
en every X-valued absolutely continuous function f,
defined on
,ab , is a.e. strongly differentiable and can
be represented as

d,, ,
 
t
a
f
tfa f

sstab
where
f
denotes the strong derivative of f.
Theorem 2.6. Let
,X be a reflexive Banach
space. Suppose that
and that
,;
f
BVa bX

Then f admits a strong derivative
. tX
f
for al-
most all
,b

ta which is strongly measurable and


d,,.
b
f
attVfab

,
Moreover, if
then f absolutely continuous and the following integral
formula for the variation holds:




,, d.
b
R
a
Vfabft t

f
(2.4)
Now we introduce a notion that generalizes the clas-
sical de la Vallée Poussins concept of second bounded
variation.
Definiti on 2.7. Given :, ,ab X
f
3
0
π,
m
jj
x
ab
 and
. We shall use the fol-
lowing notation:



;, :ij
ji ij
f
xfx
Ufxx xx
and define the second variation of a function f on ,ab ,
in the sense of Riesz, as

3
2, 2
π[,]
;,,: sup;,,
R
ab
Vf abXf ab



where

212 1
2
2
02
;, ;,
;,, :.
mjj jj
j
j
jjj
Ufx xUfxx
f
abx x
xx
 
 


 


If

,bX we will say that the function
second
2,
bo;,
R
Vfa
unded
01
;,Ufyy is bounded for all
y

-f ifs ovariation, in the sense of Ri-
esz, and write
R
2, ,, .
f
BVa bX
Lemma 2.8. Let
,, .
2, R
f
BVa b
Xen Th
1) For
x
yz
in
,ab there is a constant 0K
su

ch that
;, ;Ufz f
2) 01
,,,yab
solutely
3) f is ab continuous on
,ab ,
4) If
00
,,
yab with 00
x
y then
0
;,Ufxx
is continuous at
x
y0.
Proof. 1) Conhr
x
yz in sider tee points ,ab ,
an
,y UyxK, d note that if 1zx
.










11
112,
;, ;,
;, ;,;, ;,
;, ;,;,,.
R
Ufzy Ufyx
Ufzy UfyxUfzy Ufyxzxzx
Ufzy UfyxzxV fabX
zx



 








 





stead


If in 1zx

 



 







1
1
12,12,
1
1
;,, ;,
;, ;,;,;,
;, ;,
;,
,.
;,
RR
zx zx
zx zx
zx zx
zx zx
VfabXzxVfabXba
z
Ufzy UfyxUfzyUfyx
Ufzy Ufyx
Ufzy U
x
fyx



 



 









 







 



M. BRACAMONTE ET AL.
48

Then

;, )Ufzy ;,UfyxK where





, ,
, .


12,
12,
:max; ,
;,
R
R
K
Vfa
Vfa

bXba
bX
2) Let 01
,,yab and let

,cab. The proof
depends on the location of 01
,yy with respect to ,ab
and c.
Case 1: 01
.aycyb  In this case, for
21
,,
y
yb we have



0101 12121
1
;,;, ;,;,;,
, ;,
3;,.
UfyyUfyy UfyyUfyyUfcy
y Ufacac
KUfac



where K is given by 1).
Case
;,Ufc ;Uf
2:
_
0_1.aycy b Here, for
21
,
y
cy





01 00
02 121
0 2
2
2
;_0,_1; ,;,;,; ,
;, ;,;, ;,
;,;,;,;, .
UfyyUfyy UfayUfayUfyy
Ufyy UfyyUfyy Ufcy
UfcyUfacUfacKUfac


 

any oceeds in
a
bou
2
where K is given by 1). Inther case one pro
similar fashion.
Since c is arbitrary but fixed

01
;,Ufyymust be
nded.
3) By 2) there exists 0M such that

01
;,yyMfor all
Uf
01
,,.
y
yab Therefore f is
0
Lipschitz, and hence, absolutely continuous.
4) It is immediate.
Ejemplo 2.9. Let C be the subspace of
of all
h the natural norm inherited null sequences equipped wit
from
. Let
0,C
 be defined by :,fab
():
n
t
ft n



. If a0, then for every partition


3
0π,
m
ij
x
ab
 :



21 1
1
2
21 1
2
2
2
jjjj
j
jj
nn n
jj jj
m
jj
jj
xfxfx fx
xx xx
x
xx xx
xx
xx

 
 




 




 








 

221
mjj j
xx


02
jjj
xx

2
xx x
 
11
jjjj
nnnn


 
 
0
j
n
f


2
2
02
00,
mnjj
jjj
xx
xx






which means

2, ,,
R
f
BVa bX
and

2, ;,,0
R
VfabX
. A similar estimation holds if
0
ab.
On the other hand, consider the same function

:
n
t
ft n



on the interval 22
,
33



and let
3
211222
,,, π,
333333


 




. Th
en:
Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL. 49
 
 

 

 
 

13 131323
13 131323
13 23
23 131313
2313131323
23( 13)
13 2323
13 13
nn
fff f
ffff
nn


 

 

















 


 
 

 




13 23
13
13
nn
nn



 









1
221.
n n
n n
 


 



 

 
So that, in this case
11
n
n


2, ;23,232 10.
R
Vf
Lemma 2.10. If

2,
,,,
R
gBVabX,
f
is a com-
plex constant with 1
and
,0,1

such that
1
1)
, then


;,,;,,$,
RR
abXVfabX

2,
Vf
2,
2,

2)
R
V
is convex in the function argument; that is




2, 2,
;,, ;,,
;,,.
RR
VfgabXVfabX
VgabX
 

We omit the proof, which follows immediately from
the convexity of
.
Lemma 2.11. Let

2, ,,
R
fBV abX
and
,cab,
then

2, 2,
,,
RR
fBVab BVcb

 and

2, 2,2,
;,;,;,,.
RR R
VfacVfcbVfabX


(2.5)
Proof. Let
2, R

,cab and let


3
0π,
m
ij
x
ac
 .
Then,
3
π,.bab
 Hence







 
21 1
221 1
2
2
21 11
21 11
2
2 1
;,,
jjjj
mjj jj
jj
jj
jjjj mmm
jj jjmmm
jj
jj m
fxfxfx fx
xx xx
facx x
xx
fxfxfx fxfb fxfxfx
xx xxbxxx
xx
xx bx

 
 

 
 






 





 












1m
bx




2
0j
2
0
m
j



2, ;,,
R
Vf
abX
.
all pa

0
m
i
Since this holds for rtitions
j
x
of
,ac,
it fws that ollo

2, ;,,
R
f
BVfa cX
and


2, 2,
,,;,, .
RR
XVfabX

Similarly, one ;Vfac
gets

2, 2,
;, ,;, ,.
RR
VfcbXVfabX

On the other
hand, if

3
0
π,
m
jj
x
ac



3
0
π,,
n
j
j
x
cb

then
3
π,.ab

 Put

0
mn
jj
y
si 0
:si
j
jjm
xj
y
:,
where
.
m
x
mj


Then
nm
Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL.
50





 


 
21 1
221 1
2
0
j

2
2
10 02 11
1
1002 11
11
1
1
1
;,, ;,,
jjj
mj jj
jjj
jjjj
m
jj jj
m
mj
m
xfx fx
xx
fa
c fcbx
xx
fx fxfxfxfxfxfxfx
xxxxxxx x
xx
xx x

 

 






  



 






2
j
jj
xxx
fx f








22
2
0
21 1
2211
2
02
22,
;,,;,,.
n
j
j
j
j
jjjj
mn jj jj
jj
jjj
R
x
x
x
fyfyfyfy
yyyyyy
yy
fabV fabX


  





















Th
erefore


2, ;,;,
R R
Vfac a

2, 2,
;,, .
R
VfcbVfbX

uality as can be readily verified by con
.9) with, for instance,
Re e replaced by
an eqsidering the
mark 2.12. Inequality (2 .5) cannot b
example (2 23,a 23,b
0.c In this case, g the partition by considerin

,230., one 23,13,13c
 shows that





2, 2,
;23,0;0,230
3,23, .
RR
Vf Vf
X
 
ain Results
Now, we are ready to characterize the class of functions
in [,]ab
2,
21
;2
R
Vf



3. M
X
of second bounded -variation in terms of
the class of functions of (first) bounded -n.
Theorem 3.1. Suppose
variatio

,,
R
f
BVa bX
and

:d
x
a

F
xftt. Then

2, ,
R
BVab
and
F

2, ;,,;,,.
RR
VFabXVfabX

(3.1)
Proof. For a given partition
3
π,,ab
putting
1
21
:j
jj
tx
u
x
x
,
1
:
j
j
j
tx
v
x
x
and making use of Jen-
sen inequality, we obtain








11
22111
00
2
2
02
1
21 11
0
2
2
221 1
1
00
dd
;,,
d
mjj jjjj
2
0
j
j
jjj
jj jjjj
jj
jj
mjjj
j
fuxxxufvxxxv
m
j
F
abx x
xx
fuxxxfuxxxuxx
xx
fux xxfu
 
 
 
 

 
 
 

 




 

 







 




1
2
2
d
jjj
jj
jj
xxx xxu
xx








As
0, 1u,
1 12j jjjjj21jj
x
x
xuxxxxx
 



.
Thus, do 21 11
j
jj jjj
uxxxuxx x
 



K
 , we
must
Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL. 51







221 1
1
2
2
1 1
2
2
221 1
1
00
;,,
mjj
jj
j
mjj jj
j
fuxx
Fab
fuxxK
1
2d
jjj
jj
xx u
x
00
22
1
00
jjj
mjj j
j
x
x1d
jjj
j
fuxx x
j
xfuxx x
x
x
K x
fuxxxfux
 
 













u
x









 





1
2
2
1
0;, d;, .
jj
jj
jj
xx Kd
x
x
K x
VfabuVfab










Since this inequality holds for any partition
u
x
3
π,ab
,
we conclude that


2;,, ,;
F
ab ab which
impli
Th
V f

es (3.1).
us, replacing

b by ,,
R
Vfa


d
b
a
f
tt
in
(3.1) n [1e following.
CSu
(cf. Propositio
orollary 3.2.
1]), one obtains th
ppose

,,
R
f
BV a
bX and let
x
a
 
:d.
F
xf

:d.
x
a
tt
Then

F
xftt
and




2, ;,,d.
b
R
a
VFabXftt

(3.2)
The following example sh ows th a t it is possible to ha ve
strict inequality in (3.2).
Example 3.3. Let
X
with the norm given by the
absolute value. Let
2
:
F
xx defined on
0,1. Sup-
pose


3
0π0,1
m
ii
x
 . Then


 




2222
21 1
22
21 1211
2
22
00
22
222
221
000
;,,
11
1
jjjj
mm
j jjjj jjj

1
1
j
jjj
jj
jj jj
mmm
jjjjjj
jjj
xxxx
xx xxxxxx
mm
f
abx xx x
xx xx
xx xxx
xxx













 

 









 







11 0.xxx 
x

101 1m
Taking now the supremum over all partitions
3
π0,1
, one obtains







3
3
21 1
121 1
2
π,02
11
π,
sup
sup111 2.
jjjj
mjj jj
j
j
ab jjj
m
ab
fxfxfx fx
xx xx
x
x
xx
xx

 











Clearly, x
 
0d
F
xftt2
(where

f
,ab , and let :
g
EX be a map, where X is a Ba-
nach space, and let K be a positive constant such that
tt
) is ab-
tinuous,

solutely con
20,1ft L
 , and

11
1
mjj
jj
gx gx
11
[0,1] 00
2.
L
ff



Now if, for instance


() d2d
 
 
:
p
tt, for
x
1p
, we
have
 

2,
122 ,
R
[0,1] 22 2,.
p
L
f
Vf
So, in all of these cases, inequality (3.2) is
Let 
ab

 
strict.
Lemma 3.4.
01
jjj
,xK
xx




1mm
xx b

where 01
ax x
(3.3)
, E be a dense subset of
 and j
x
E
for
jm
0x exists for all 0,1, 2,,.
Then, E
g
,\
x
ab E, where

0 .
E
hE
0
lim
h
x
g
x

An analogo us asserti o n h ol ds for

0
E
gx,
gxh
 
Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL.
52

,ab which is similarly defined.
roof. It suat

0
E
gx exists for
x
Pffices to show th
all
,\.
x
ab E uch is not the case; i.e.,
suppose t
Suppose that s
hat there is a
,\tab

im
E such that
hst
sE

l
0
thE
lim
g
th th gs
do


es not exist.
Then by Cauchy’s criterion (in the complete metric
space Xe is an 0 and an increasing sequence:
that n
yt and


), ther
yE

nn

,at such
1

1
gy
jj
gy for all j.
Since ytind N so that
n we can f
1.
n
nNy t 
2
Thus, if nN we must
11nnn
yyy

 1.
n
tyt
Now consider the subsequence

1
nn
x defined by
:
nnN
x
y. Notice that in this case we have 11
jj
xx
for all j. Thus for all 0N



1
1
11
1
11
,
Njj
j
j
jjj
N
jj
jjj
gx gx
x
x
xx
x
x
xx






 



which contradicts (3.3) since .

Theorem 3.5. If
2, ;,,
R
VfabX
 where
, then f has bounde d seco n d-variation.
Proof. Let

0
n
ii
t
be a partition of
,ab , since
satisfies the condition 1
, then give 0r there is
a 00x such that if 0
x, then

.
x
x
rx
We write
0
x
12 1
0
2
;, ;,
2 :,
jj jj
jj
Ufx xUfxx
n x
xx
 


 




0, ,ej
the

n




 
00
0
22
1
12 1
00
2
1
2
2
212 1
;, ;
;, ;,
;,;, .
xx
x
mm
j j
jj jjj
jj
jj
j
0
0
x
21
2
,
jj
j
12
;, ;,
j jj
j
j
je jj
jjj jj
je
Ufx xfx
Ufx xUfxxxx
U x
je
j
je
Ux x x
fx Ufxx
x
x
xx
xxxUfxxUfxx














 


(3.4)
 
 

Now, for each 0
x
je must be


1
2
;, ;,
;, ;,.
jj jj12
12 1
2
1
j
jjjjj
jj
fx xUfxx
Ufx xUfxxxx
rxx
 
 


Substituting t) we get
U


his in (3.4
  




00
212 1
1 22
2
12 1
2
02
;, ;,
1
;, ;,
;, ;,
.
xx
mjj jj
12 0
0
0
1
j
jjj
j

jjjj
jeje
jj
jj jj
jj
jjj
Ufx xUfxx
UfxxUfxxxx xx x
rxx
Ufx xUfxxxx
x
 
 

 


 







Since this inequality holds for every partition
n
xbarx


of
,b we obtain

a


0
1;
R
xba fa
22,
;,,,,.abXV bX
r

Corollary 3.6. Suppose X is a reflexive Banach space
and let . If

2, ;,,
R
VfabX
 then f is
absolutely continuous on
,ab . Moreover, by Theorem
2.6,
Vf
is also absolutely continuous.
f
Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL. 53
Theorem 3.7. Let X be a reflexive Banach space. If

,,
2,R
F
BVabX and

then the-re is a func
tion

,
R
f
BVa b
t suchhat ,
F
f
Lebesgue-a.e.,
and


;,,.fabX
Proof. We use the fact that F is absolutely continuous
and consequently it has strong derivative almost every-
wheresgue) on
2, ;,,
RR
VFabXV

e (Leb
,ab . Thus (see [12])
 

d
d.
x
a
x
a
 

F
xFa Ftt
F
aftt


Therefore, by Theorem 3.1, the function
 
:
x
Fx Fa is also of bounded second varia-
the sense of Riesz, and
tion, in



;,,
.
FaabX
Theorem 3.8. Let X be a reflexive Banach space and
2
 0 and 0
t as in (2.2). If


2, 2,
;,,
,,
RR
R
VFabXVF
Vfab


, with constants
K
,
R

2, R,,
F
BV
a bX then there is a function
f
BVa b
such that ,f
Leband esgue-a.e.,
F




;,,;,, 2.
R
f
abFabX ba


(3.5)
ce F is ably contm

2,R
K
X t

0
2
VV
soluteinuous (Lema 2.8)
an flexive Banach space, Theorem 2.6 ensures
that f is strongly differentiable a.e., with derivative
strongly measurable. Let E be a set of zero Lebesgue
measure such that
Proof. Sin
d X is a re
F
exists at every point of
:, .ab E Consider points

0
m
j
x
,ab in
such that 01 ,
m
ax xxb
  and let 01
,, ,
m
hh h
be positive real numbers such that
_
__1xjhjx j
 for0, ,1jm and
11mmmm
all
x
hxh

.
Next, define a partition

21
0
:,
m
jj
y
of
,ab as
/2
11
22
If021 iseven,
If121 isodd,
If2 ,
If2 1.
j
jj
j
mm
m
xjm
xh jm
yxh jm
xjm

 



Then

 

 

 
1
1
1
01
11 1
1
1
1
1 1
0
jjjj
mjj
jj
jjj
m m
mm
mm
mm
jjj j
jj
FxFxFx hFx
hh
xx
xx
h Fx
hh
xx
xx
FxFx hFx
xh
 










 




 


11jj
h


2

11jj
h

 1
11
j
jj
xFx
hx

2m

1jj
xx
mm
m m
Fx Fx hFx



j
F

   
11mm
mmm
xx

1,
mm
xx
f win,
1
1
1
111111 1
11 1
1
jjjj j
jj
j j
jj
jj
mmm mmmmmmmmmm
mmmmm
mm
Fx hFxhFx
xxhh xx
xx
FxFxhFx hFxhFx hFxhFxhFx
hxhx hxhx hh

 


 
 


 
  
 

1mm
xx
m
using the convexity oe obta
Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL.
54




  
11 11
211
1
1
1
21
01
11
12
2
12
2
12
2
jj jj j
mjjjj
jj
jj
jjj
mjj
jjj
m mmmm
mm
FxhFxFxFxh
hxxh
xx
xx
FxFx hFx
xxh
xx
Fx hFxh
hx
 



 




 











j
0j

1
jjj
j
jj
h Fx
hxx

j
mm
Fx Fxh


 

1
1
11 11 1
11 1
1
1
12.
2
mm m
mm
mmmmm mm
mmm mmmm
mm
hx h
xx
Fx hFxhFxhFx
xhx hhxx
xx
  
 








  

 

 




Let
11mm
xx

11 11
11
1
1
0,1,,2:
jj jj
jj
jj
Fx hFxFxFx
hx
Aj mxx
 




0
2
jj
jj
h
xh
t









1
1
1
20,1,...,2:
jjjjj
jjjj
jj
FxFxhFx h
xxh h
jm x
Ax



n, from the above inequality we obtain
0.
2
j
Fx
t









The

 

1
1
1
j
jj
mm
11 1
21
01
11
1
1
jj jjj
mjj
jjj
mmm mmm
mm
mm
FxhFxFx hFx
hh
x
x
xx
















xx
Fx Fx hFxhFx
hh
xx
 

 
 

Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL. 55





1
2
11 11
11
1 1
1
1
1
2 1
1
12
2
12
2
12
2
jj jj jj
jjjj
jj
jA jj
jjjjjj
jjjj
jj
jA jj
mm
j
j
FxhFxFxFxh
hxxh
Ax
xx
FxFxhFx hFx
xxh h
Ax
xx
FxFx h
 


 










 


















x
x
 

 

 
11
11
1
1
1111 1
11 1
1
1
11 11
11
12
2
2
mmmmm
mmmmm
mm
mm
mmmmm mm
mmm mmmm
mm
jj jj jj
jjjj
j
Fx hFxh
hxhxh
xx
xx
Fx hFxhFxhFx
xhx hhxx
xx
FxhFxFxFxh
hxxh
K
x


 
 
 




 






 

 

 




 









  
2
1
01
1
22
1
01
00 1
11
211
01
0
1
22
1
22
m
jj
jj
jjjjjj
mm
jjjj
jj jj
jj jj
mmm mmmm
mmmmmm
jj
j
xx
x
FxFxhFxhFx
xxh h
K
tx xx x
xx
xFxhFx hFxh
hxhxh
K
tx xx













 




 




 

 

1
F


 

1
1
0101
1111 1
11 1
1
1
11
.
22
2
mm
mm
mm mm
mm mmmmm
mmm mmmm
mm
xx
x
txx txx
Fx hFxhFxhFx
xhx hh
Kxx
xx

 
 





 

 

 

 





Copyright © 2012 SciRes. APM
M. BRACAMONTE ET AL.
CopyrighciRes. APM
56
Thus,




11 1
212,
01
;,,
2
jjjjjj
mjj R
jj
jjj
FxhFxFx hFx
hh K


10
2
x
xVFabX
xx
 


 








tba

0,1, 2,,jm, one gets Making 0
j
h,




112,
01
;,,
2
mjj R
jj
jjj
Fx FxK


10
2
x
xVFabX
xx







tba

therefore, by
Lemma 3.4

0Fx
and
0Fx
t © 2012 S
exists for all
,\xab

Fx
. Defiunction
,
,\,
and.
x ab
a a
Clearly,
ne now the f
x
 

when
:0when
0if
fxF x
Fa x


F
f
almost everywhere. We need just to
that f sa

0
:m
j
verify tisfies (3.5). Let
j
x
be a parti-
tion of ,ab . Suppose that there is exactly a point
k
ax
such that k
x
. In this case, we can
choose k
x such that

$_ 1__$
x
kxkxk
 .
Put
01 11
,, ,,,, ,
kkkm
x
xxxxx

. Observe that
 

lim lim lim
0.
kk kk kk
kk
kk k
xx xx xx
xx
kk
f
xfxFx
Fx fx








Therefore


  





 

211
11
011
21
11
1
11
1
1
01
kjjkk
jj kk
jjj kk
mjj
kk
kk jj
jk
kk jj
jj
jj
jjj
fx fxfx fx
xx xx
xx xx
fx fx
fxfxxx xx
xx xx
Fx Fx
xx
xx




















 

 





 
1
2
+
kFx

 

 





21
1
1
11
2,
0
;,, 2
2
+
mjj
kk
kk
jk
kk jj
R
Fx Fx
Fx Fx
1
1
1
11
kk
kk
kk
jj
Fx xx
xx
x
xxx









xx xx
KVFabX tba












and taking the limit as kk
x
x
we have


  








211
011
21
1
1
11
2,
0
+
;,,2.
2
kjjkk
jj
jjjkk
mjj
kk
kk
jk
kk jj
R
fx fxfx fx
xx
xx xx
fx fx
fx fx
11
11
kk
jj
xx
x
xxx


xx xx
KVFabX tba























M. BRACAMONTE ET AL. 57
thhand, 0

00 00
000
limlim0.
xx xx
x
a is thnt of e only poiIf, on e other
not in ay consider a collection
,
, then we m
1
,,,
kk

, ,
m01
,,
f
1k
x
xxxx
e 01
x
. Then
x
wher
 ax
xFxFxfa





Thus, proceeding as above, we get








1
10 2,
10 1 0
1
10 1
;,,2.
2
jj R
jj
jjj
fx fxK
2m
fx fx
x
xxxVFabXtba
xxxx




 






and taking the limit as 00
x
x
:





21
10 2,
101 ;,,2
2
mjj R
jj
fx fx
fx fxK


0
1
10 1
.
jjj
x
xxxV
xxx x





 FabXtba






To complete the proof that nded (f
tion (in sense o



3
0
:π,
m
jj
f is of bouirst) varia-
f Riesz), let
x
ab

arbitrary, and set

01 11
,,,, ,,,
km
kk
x
xxxx x

 


where:
j
j
x
x if j
x; if j
ax
,
. Then,
1'
j
jj
x
xx

where as if a
take 01
ax x
.
Then, from the above estimations we obtain
 




1
1
2,
0
;,,2.
2
jj
jj
R
xx
KVF
abX tba




Hence, by taking the supreme over all partitions
2
0
m
j
1
fx fx
xx
jj



of
,ab :





2,
0
,, ;,,
2
2.
RR
K
VfabVF abX
tba


(3.8)
xive Banach space and
onstant 0K. If

Corollary 3.9. Let X be a refle
let 2
 globally, with 2
-c

2, ,,
R
F
BVabX
then there is a function

,
R
f
BVa b such that
,
F
f
Lebesgue-a.e., and





2, ;,
Rab
2, ;,,d
, .
2
b
R
a
VFabXFtt
KVFX


(3.9)
. Thnsures the existence of a function
Proof eorem 3.8 e

,
R
f
BVa
sense
,
b (in theof Riesz) such that
F
f
and



2,
,,
;,,.
2
R
abX
KVFabX
w, by Theorem 2.6 f must be strongly differentiable
2, ;,, ;
RR
VFabXVf

No
a.e. on
,ab with derivative
f
strongly measurable
anr integrable od Bochnen
,ab,
and



, d.
a
abt Thus ,b
R
Vf ft






2,
2,
;,,
dd
;,,
R
bb
aa
R
VFabX
f
2
tt Ftt
KVF
abX

 

That Theorem 5.8 (Corollary 3.9) generalizes Riesz’s
lemma is brought forward by considering, for 1,p the
function

p
:
ptt, which has sharp 2
-constant
2
p
K
. Indeed, in this case we have:
Corollary 3.10. Let X be a reflexive Banach space. If
1,p and
2, ,,
p
R
F
BVabX
then there is a func-
tion
,
p
R
f
BVab
such that ,
F
f
Lebesgue-a.e.,
and





2,
12,
;,,
2;,,.
p
p
bp
R
a
pR
VFabXFtt
VFabX

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