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Advances in Pure Mathematics, 2012, 2, 22-26 http://dx.doi.org/10.4236/apm.2012.21005 Published Online January 2012 (http://www.SciRP.org/journal/apm) On Bounded Second Variation José Giménez1, Lorena López1, N. Merentes2, J. L. Sánchez2 1Departmento de Matemáticas, Universidad de los Andes, Mérida, Venezuela 2Departmento de Matemáticas, Universidad Central de Venezuela, Caracas, Venezuela Email:{jgimenez, lomalopez}@ula.ve, nmerucv@gmail.com, jose.sanchez@ciens.ucv.ve Received September 21, 2011; revised October 22, 2011; accepted November 2, 2011 ABSTRACT In this paper, we discuss various aspects of the problem of space-invariance, under compositions, of certain subclasses of the space of all continuously differentiable functions on an interval ,ab ··k . We present a result about integrability of products of the form ff f under suitable mild conditions and, finally, we prove that a Nemytskij operator g g S maps ,,BVa b loc gBV a distinguished subspace of the space of all functions of second bounded variation, into itself if, and only if . A similar result is obtained for the space of all functions of bounded -variation ,2 11,p p 2,. p A ab Keywords: Function of Bounded Variation; Lipschitz Continuous Function; Absolutely Continuous Function; Nemytskij Operator 1. Introduction Throughout this paper we use the following notations: if g, f are given functions, the expression g f stands for the composite function , g ft whenever it is well- defined; ,ab denotes a compact interval in (the field of all real numbers) and denotes the Lebesgue measure on . As usual, the set of all natural numbers will be denoted by . Recall that a function :,fab is said to be of bounded variation on ,ab if the (total) variation of f on ,ab : , nn I ;, :supVf VfabfI where n I denotes a (finite) partition of ,ab into non-overlapping intervals , n ab n and nnn : f Ifbfa The class of all functions of bounded variation on ,ab is denoted as ,BVa b. The renowned Jordan’s theorem ([1]) states that a function :,fab is of bounded variation on ,ab if, and only if, it is the diference of two monotone functions. In particular, every function in ,a bBV has left limit f x at every point , x ab and right limit + f x at every point , x ab; also, by the celebrated Lebesgue’s Theorem (see e.g. [2, Theorem 1.2.8]) every function in ,BVab is -..ae diferentiable. It is well known that the composition of two functions of bounded variation, say g and f, in general, need not be of bounded variation; in fact, not even if we choose the inner function well-enough behaved guarantees that the composition g f is of bounded variation. For instance, if 1 33 3 0if 0; and :1 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 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. REFERENCES [1] C. Jordan, “Sur la Serie de Fourier,” Comptes Rendus des |