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 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 ,, y 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 Riesz’s 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 Poussin’s 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 ,, x 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 , , , . y 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 REFERENCES [1] Ch. 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