 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 ggS maps ,,BVa blocgBV 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,.pAab 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 gf stands for the composite function , gft 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  : ,nnI;, :supVf VfabfI where nIdenotes a (finite) partition of ,ab into non-overlapping intervals ,nabn and  nnn:fIfbfa The class of all functions of bounded variation on ,ab is denoted as ,BVa b. The renowned Jordan’s theorem () 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 fx at every point ,xab and right limit +fx at every point ,xab; 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 gf is of bounded variation. For instance, if 13330if 0;and :1sinif 0.xgxxf xxxx (1.1) then 10, 1fC, 0.1gBVf but gf is not of bounded variation. gHowever, 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. (). Suppose that f has a derivative f of order k everywhere on ,ab,k. If BVa b and if f,BVc d,:min ,abcf, where ,:max ,abdf·kthen the function ff is of bounded variation on ,ab; moreover, ,,,·1, kkabab abff kf (1.2) ,BVa b defined as where ,ab is the norm on  ,,:sup ;,ab xabffxVf ab Copyright © 2012 SciRes. APM J. GIMÉNEZ ET AL. 23Let be a subspace of . Given a function , the autonomous Nemytskij (or Superposi-tion, see ) opera,ab generated by g, is defined a,ab:g:gStor s : ,gSftgf, .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. ), the Superposition Operator Problem have been studied extensively in various spaces of differentiable functions related to the spaces ,BVa b and ,ACab. 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·kBV . We prove a version of Theorem 1.1 about the integrability of products of the form ·gff g is an integrable function and kf whenf is continuous, and obtain an estimation of the norm 1,··kLab we give necessary2. 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 deresults which will be needed for the further development of this work.  We will use the notation ;,Ma b to denote functiBVthe 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 abcf,,:abd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 :,,, yLFxyab xyxyThe class of all Lipschitz continuous functions in . Fx F,ab is denoted as ,Lipab and the functional :max ,ffL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 :,nnfII  whenever I,nnnab is a finite con of mu- tually disjoint subintervals of llectio,ab with 1.innnba The class of all absolutely conus functions on tinuo,ab , which is actually an algebra, is denoted as ,ACab. inition 2.1. (Luzin N property). A real-valued deDeffunction 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 hao fve 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 btisfies property N, fCa b and sale on 3)  exists -a.e., is integrab,ab and  d.axfx The equivalence (a) fa ftt (b) is known as the Banach- Zareckiĭ theorem. The functional 1L:ACffa f Adefines a norm on ,Cab; in fact, ;, .f1LVfab e same fven in (1.1) show that the class of all alutely continuous functions is not cled second variation. The Remark 2.3. Thunctions gibsoosed under compositions.  In the year 1908, de La Vallée Poussin (), intro-duced the notion of boundclass 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 (): Proposition 2.4. A real valued function f is in the class 2,BVa b if, and only if, there is a function ,fBVa b such that x d.afxfa fxx (2.1) Copyright © 2012 SciRes. APM J. GIMÉNEZ ET AL. 24 In this case, the relation  ;,2,:BVa bffa 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 alsolus if mental Theore,fBVa b then fle on is differentiab,ab and .ff In fact, ,BVa b if, and only if, 1,fCab and ,fBVa b.  In 1tes, in , introduced the notif fuof bounded iation, for 1997, N. Merenon onctio,2p.pn -var The clllons of bounded ass of ain functi ,ab,2p-variation is denoted by 2,pAab and its characterized as follows: Proposition 2.6. (). A function ,abf is in the class 2,pAab if, and only if, ,fbAC a and ,.pfLab  In this case the relation  2,pAa Labpf f o,:bfafa  n defines a norm2,pAab . 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,,,,,,.pAabBVabCaip abACa 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 compositthese 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 :DCX is defined by :Cff. Remark 31) 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 retionat respect. Mosult 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 CBV M maps ,ACcd into ,ACab. The converse of also true (see ). rem of Algebra an Theo s a polynomial of degree M, then for all this proposition is 2) By the fundamental Theod Rolle’srem, if f i,ab  ;, ,fBVMa 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 morenomenoten occurs infunctional analysis: is the case in which given two functions, say g and f, the multiplication of gf by a continuous derivative ,kfk 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- cacontinuotion (see ), and the Banach-Zareckii Theorem. Proposition 3.2. (Burenkov). If f has an absolutely us -thkderivative kf on ,ab and if  AC , then the function kff is also abso-lutely continuous on ,ab and inequality (1.2) holds. If 1,gLintegrability of products of the formab similar considerations as those dis-cussed above apply with respect to the gff or even pgff . Now, not even the fact that the function g is integrable and the function f is absolutely continuous guarantees that the product gff grable; for ie ), let is intenstance (se0:0:0gf and, for 0x, let :gx 1xand 63:sin12fx xx, then 0, 1fAC, g is integrable in, but 0,1fgff is not inte-grable in 0,1 . In that reswipositionis well known (see, for instance, [11m 3.54]): [Change of Variables] Lepect, the follong pro , TheoreProposition 3.3. t :,gcd  be an integrable function and let :, ,fab cd be a function differentiable-a.e. in ,ab . Then ·gff is integrable and dff dgtt gfxr all fx x holds fo ,,ab if, and only if, the function 0,1Gf AC, where :zcGz gNotice that Gbsolutely continuous function, which bring us back to the same situatioabove. bility) properties of the product ·d .,,ttzcd (3.1) is an ans considered It turns out that, if g is an integrable function, multi-plication by a continuous derivative of f improves the (integragff. By an lemmaalogy 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 1locgL and that ,kfbCa k. Then 1·;,· kgf fbfLa Copyright © 2012 SciRes. APM J. GIMÉNEZ ET AL. 25moreover, 11,·· .abaLfLgf fg Proof of k,bkkf kf. The continuity f implies that the open : 0kfx can be expresse component open intervals, say Sa or0set Sxcountable unio:,abn ofd as a1,Ni , where N N. Now, since kfiib on S, to each ,N corresponds a nonnegative integer imk such that 1,i,iiab can be decom nt no, ,iimma bposed into at most disjoin-degenerated 112 2,,,, ,ii i iiiab ab   on which f is monotone. ng im intervals,Now, beif continuoon us ,ab , the Fundamen-tal Theorem of Calculus implies that ,;fACab likewise, the indefi, defined by elnite integral function Gby (3.1), is absoluty continuous, thus Remark 3.1, the monotonicity of f on ,jjiiab implies that ,;ACbGaf consequently, ·gff is integrable on this interval. Hence, since f does not change sign on ,jjiiab, we must have  ,ijjiiafa fbddjijbgftf ttgxx whertion e the nota, stands for , if  or , otNow, sinceherwise. kf is continuous, (generalized) for iies ththe mean value theoremntegrals implat, on each ,jjiiba there is a poin,t jic such that ,d.jjiiifa fbdjijibkakjgftf tftfcgxx Notice th·· kat the product gff f is easurable n on a mfunctio,ab . Thus 111 ,11 ,1,, iijjiidddd.jjiijjiikmNkjiij fa fbmNkij fa fbNkifa fbkLfabf tbagftftfcgxxfgx xkfgx xktfg   The proof is complete. The Autonomous Nemytskij Operator on the Spaces  ,BVab and 2,pAab For convenience we state the next result as a single proposition. The proof of it is based in three separate results of M. Josephy  (see also ), N. Merentes  and N. Merentes and S. Rivas . Proposition 3.5. Suppose :,,,BabVba ,RBVpab (). Then Sg maps ,ACab or ,ab into itself if, and only if,  .locgLip 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, .locgBV Moreover, in this case Sg is automatically bounded. po-, for all Proof. Suppose that .locgBV Then, by Prosition 3.5,,fBV a b ,,gfLipabACab (since both g and f are Lipschitz continuous); thus, for -a.e. ,xab ·,gfgff and, by Theorem 1.1, with g and 1,k we get ·,gff 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 abf the linear difeomorphism :,,ab afb defined as :,ab abfxmxa  where:abmba. Then, each ,b and therefore, for all ab:fBVa,ab. Thus, be gabfVSBy the first part of the proof we hav 1gab aba abg b,.gSf ff  S f BVHence, locgBV the proof is comple abouand 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,pAab. At this p, let us recall the following prorem 3.44]): oint oposition (see, for instance [11, TheSuppose g, f are functions defined on intervals and that f is well defined. If g, f and gf are -ga.e if-ferentiable functions and g satisfies the property N then,  . dfxgfxfx for -a.e. x, gwhere gfxf x is interpreted to be zero when-ever 0fx. Theorem 3.7. Let 1.p Sg maps 2,pAab into it-self if, and only if, 2gA In this case Sg is ically boundeplocautomatd. Copyright © 2012 SciRes. APM J. GIMÉNEZ ET AL. Copyright © 2012 SciRes. APM 26 Proof. Suppose first that 2plocgA By Proposi-tion 3.5, for all 2,,pfAab ,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 orand since ,gbAC a (and in particular it satisfies property N), for -a.e. ,xab  2.gfxgfxfx g xffxince (3.2) Now, Sgf is s in continuous, the second summand in the right hand side of (3.2) i,Lab, and p,·.ppLLabgff f (3.3) gfOn the other hand, 2pp222.pgfxfxfx   Lemma 3.4 gfx fx Hence, by,.pfabpfabpLLf g From (3.3) and (3.4) it follows that ps ,212ppgff   (3.4) Sg ma2pA that,ilar to the one givenfor the the necessity of the condition in thTh 4. Acknowledgements This research has been partly supported by the Central Bank of Venezuela. 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