On Bounded Second Variation

Transcription

1 Advces Pure Mthemtcs, 0,, -6 htt://dxdoor/0436/m0005 Pulshed Ole Jury 0 (htt://wwwscrpor/ourl/m) O Bouded Secod Vrto José Gméez, Lore Lóez, N Meretes, J L Sáchez Dertmeto de Mtemátcs, Uversdd de los Ades, Mérd, Veezuel Dertmeto de Mtemátcs, Uversdd Cetrl de Veezuel, Crcs, Veezuel Eml:{meez, lomloez}@ulve, merucv@mlcom, oseschez@cesucvve Receved Setemer, 0; revsed Octoer, 0; cceted Novemer, 0 ABSTRACT I ths er, we dscuss vrous sects of the rolem of sce-vrce, uder comostos, of cert suclsses of the sce of ll cotuously dfferetle fuctos o tervl, We reset result out terlty of roducts of the form f f f uder sutle mld codtos d, flly, we rove tht Nemyts oertor S ms BV,, dstushed susce of the sce of ll fuctos of secod ouded vrto, to tself f, d oly f BV loc A smlr result s oted for the sce of ll fuctos of ouded, -vrto, A, Keywords: Fucto of Bouded Vrto; Lschtz Cotuous Fucto; Asolutely Cotuous Fucto; Nemyts Oertor Itroducto Throuhout ths er we use the follow ottos: f, f re ve fuctos, the exresso f stds for the comoste fucto f t, wheever t s welldefed;, deotes comct tervl (the feld of ll rel umers) d deotes the Leesue mesure o As usul, the set of ll turl umers wll e deoted y Recll tht fucto f :, s sd to e of ouded vrto o, f the (totl) vrto of f o, ;, : su I V f V f f I :, Ideotes (fte) rtto of,, d f I : f f o-overl tervls, s deoted s, theorem ([]) sttes tht fucto f :, ouded vrto o, to The clss of ll fuctos of ouded vrto o BV The reowed Jord s s of f, d oly f, t s the dferece of two mootoe fuctos I rtculr, every fucto BV, hs left lmt f x t every ot x, d rht lmt f x + t every ot x, ; lso, y the celerted Leesue s Theorem (see e [, Theorem 8]) every fucto BV, s - e dferetle It s well ow tht the comosto of two fuctos of ouded vrto, sy d f, eerl, eed ot e of ouded vrto; fct, ot eve f we choose the er fucto well-eouh ehved urtees tht the comosto f s of ouded vrto For stce, f 0 f x 0; 3 x x d f x: 3 3 () x s f x 0 x the f C 0,, BV f 0 ut f s ot of ouded vrto However, the multlcto of f y dervtve of f, whch s of ouded vrto, mroves the roertes of tht comoste fucto Ideed, roof of the follow theorem c e foud [3, Theorem 5] Theorem ([3]) Suose tht f hs dervtve f of order every o, If f BV, d f BV c, d, c: m, f, d : mx, f, the the fucto f f s of ouded vrto o, ; moreover,,,, f f f, () s the orm o,, f : su f x V f;,, x, BV defed s Coyrht 0 ScRes

2 J GIMÉNEZ ET AL 3, Let e susce of Gve fucto :, the utoomous Nemyts (or Suerosto, see [4]) oertor S :, eerted y, s defed s : t, t S f t f, (3), Gve two ler sces, d fucto :, rmry oectve of reserch s to vestte uder wht codtos o the eert fucto the ssocted Nemyts oertor ms to Ths rolem s ow s the Suerosto Oertor Prolem Recetly (see e [5]), the Suerosto Oertor Prolem hve ee studed extesvely vrous sces of dfferetle fuctos relted to the sces BV, d AC, I ths er, we dscuss vrous sects of the the Suerosto Oertor Prolem whe the sces, C, d re somehow relted to the sce BV, We rove verso of Theorem out the terlty of roducts of the form f f f whe s terle fucto d f s cotuous, d ot estmto of the orm f f f L, Flly, we rove two results whch we ve ecessry d suffcet codtos for the utoomous Nemyts Oertor to m the sce of ll fuctos of secod ouded vrto to tself d the clss of fuctos of ouded,-vrto to tself Some Fucto Sces I ths secto we recll some deftos d stte some results whch wll e eeded for the further develomet of ths wor We wll use the otto BV M ;, to deote the sce of ll ouded fuctos f such tht f, c e exressed s uo of M sutervls of,, for ll, cd,, c : m f, d : mx f,, M Josehy roved [6] tht for ll M the clss of ll ouded fuctos BV M ;, s coted BV, A fucto F :, s sd to e Lschtz co- tuous o, ff y F x F LF: su : x, y,, x y x y The clss of ll Lschtz cotuous fuctos, L, d the fuctol s deoted s f : mx f, L f defes orm o t Recll tht fucto f :, s sd to e solutely cotu ous o, f, ve 0, there exsts some 0 such tht wheever I, su f I : I, s fte collecto of mutully dsot sutervls of, wth The clss of ll solutely cotuous fuctos o,, whch s ctully ler, s deoted s AC, Defto (Luz N roerty) A rel-vlued fucto defed o comct tervl I s sd to stsfy the Luz N roerty (or smly, roerty N) f t crres sets of - mesure zero to sets of - mesure zero It s esy to see tht the comosto of two fuctos tht hve roerty N lso hs roerty N The clss of ll cotuous fuctos tht stsfy roerty N o tervl, wll e deoted s N, The follow chrcterzto of solutely cotuous fuctos s well ow (cf [, Chter 7]) Proosto The follow sttemets o fucto f :, re equvlet: ) f s solutely cotuous, ) f BV, C, d stsfes roerty N, 3) f exsts - e, s ter le o, d x f ftd t f x The equvlece () () s ow s the Bch- Zrecĭ theorem The fuctol L f : f f AC defes orm o AC, ; fct, f V f;, L Remr 3 The sme fuctos ve () show tht the clss of ll solutely cotuous fuctos s ot closed uder comostos I the yer 908, de L Vllée Pouss ([7]), troduced the oto of ouded secod vrto The clss of ll fuctos of ouded secod vrto o tervl, s deoted y BV, d s chrcterzed y the follow result due to F Resz ([8]): Proosto 4 A rel vlued fucto f s the clss BV, f, d oly f, there s fucto f BV, such tht x f x f f xd x () Coyrht 0 ScRes

3 4 J GIMÉNEZ ET AL I ths cse, the relto f : f f V f;, BV, defes orm o BV, () Defto 5 Us the otto of () we defe, : BV, : f BV BV f C,, Clerly, BV, s ler susce, BV Notce lso tht y the Fudmetl Theorem of Clculus f f BV, the f s dfferetle o, d f f I fct, BV, f, d oly f, f C, d f BV, I 997, N Meretes, [9], troduced the oto of fucto of ouded,-vrto, for, The clss of ll fuct os of ouded, -vrto s deoted y A, d ts chrcterzed s follows:, Proosto 6 ([9]) A fucto f s the clss A, f, d oly f, f AC, d f L, I ths cse the relto f f : f f A, L, defes orm o A, Clerly cotuously dfferetle fucto s Lschtz cotuous d y Lschtz cotuous fucto s solutely cotuous I fct, >, the follow ch of strct clusos holds (see e, [5,0]):,,, L, AC, BV, A BV C 3 M Results (3) We e ths secto y stt some fudmetls fcts cocer comostos of fuctos o BV d AC I these cses the trsc roertes of the er fucto ( the comosto) wll show to ly lso mortt role We recll tht f D d E re ve sets, X s ler E susce of d φ s m from D to E, the ler D comosto oertor C : X s defed y C f : f Remr 3 ) Althouh oth cl sses BV d AC re ot closed uder comos to, they do stsfy weer roerty tht resect More recsely, t redly follows from result ve y M Josehy [7, Theorem 3] tht f :, cd, the, the oertor C ms BV c, d to BV, f, d oly f, BV M ;, for some M From ths, t redly follows tht f BV M ;, AC, the C ms, AC cd to AC, The coverse of ths roosto s lso true (see [3]) ) By the fudmetl Theorem of Aler d Rolle s Theorem, f f s olyoml of deree M, the for ll, f BV M ;, AC, ; lso, every, mootoe fucto s BV M ;, for some M I wht follows we wll oserve more stces of very remrle heomeo tht of te occurs oler fuctol lyss: s the cse whch ve two fuctos, sy d f, the multlcto of f y cotuous dervtve f, of f mroves the roertes of the comosto The follow roosto s corollry of Theorem The result follows from the fct tht the sce N, s ler wth resect to otwse multlcto (see [3]), d the Bch-Zrec Theorem Proosto 3 (Bureov) If f hs solutely th cotuous -dervtve f o, d f AC, the the fucto f f s lso solutely cotuous o, d equlty () holds If L, smlr cosdertos s those dscussed ove ly wth resect to the terlty of roducts of the form f f or eve f f Now, ot eve the fct tht the fucto s terle d the fucto f s solutely cotuous urtees tht the roduct f f s terle; for stce (see [3]), let 0 : f 0 : 0 d, for x 0, let x : x d f x : x 6 s x 3, the f AC0,, s terle f 0,, ut f f s ot terle 0, I tht resect, the follow ro osto s well ow (see, for stce, [, Theorem 354]): Proosto 33 [Che of Vrles] Let : c, d e terle fucto d let f :, cd, e fucto dfferetle -e, The f f s terle d f t dt f fxd holds fo r ll,, G f AC0,, z Gz: td t, z c, d f x x f, d oly f, the fucto c (3) Notce tht G s solutely cotuous fucto, whch r us c to the sme stutos cosdered ove It turs out tht, f s terle fucto, multlcto y cotuous dervtve of f mroves the (terlty) roertes of the roduct f f By loy wth useful oto orted from the theory of rtl dfferetl equtos, we mht cll ths dervtve tert fctor The follow lemm rovdes verso of Theorem whe the outer fucto the comosto s terle fucto The roosto mht e of some terest tself Lemm 34 Suose tht L loc d tht f C, The f f f L, ; Coyrht 0 ScRes

4 J GIMÉNEZ ET AL 5 moreover, f f f f, L, L f Proof The cotuty of f mles tht the oe set S : x, : f x 0 c e exresse d s coutle uo of comoet oe tervls, sy N S,, N or N Now, sce f 0 o S, to ech,, N corresods oetve teer m such tht, c e decomosed to t most m dsot o-deeerted m m tervls,,,,,,,, o whch f s mootoe Now, e f cotuous o,, the Fudmetl Theorem of Clculus mles tht f AC, ; lewse, the def te terl fucto G, defed y (3), s solutely cotuous, thus y Remr 3, the mootocty of f o, mles tht G f AC, ; cosequetly, f f s terle o ths tervl Hece, sce f does ot che s o,, we must hve d f, f f t f t t x dx the ot to,, f or, otherwse Now, sce f s cotuous, the (eerlzed) me vlue theorem for terls mles tht, o ech,, there s o t c such tht stds for f t f t f dt, f f c x d x f Notce tht the roduct f f f s mesurle o, Thus fucto f t f t f N m N m N t, f, f, f, dt f c x dx f f x dx f f x dx f f L f The roof s comlete The Autoomous Nemyts Oertor o the Sces, A, BV d For coveece we stte the ext result s sle roosto The roof of t s sed three serte results of M Josehy [6] (see lso []), N Meretes [3] d N Meretes d S Rvs [4] Proosto 35 Suose, : BV,, AC, or RBV, ([4]) The S ms, to tself f, d oly f, L loc Now we reset result tht ves ecessry d suffcet codto for the Nemyts oertor to m the sce BV, to tself Theorem 36 S ms BV, to tself f, d oly f, BV loc Moreover, ths cse S s utomtclly ouded Proof Suose tht BV loc The, y Proo- sto 35, for ll f BV,, fl, AC, (sce oth d f re Lschtz cotuous); thus, for -e x, f f f, d, y Theorem, wth d, we et f f BV, d, sce t s clerly cotuous o,, t follows tht f BV, Coversely, ssume S ms BV, to tself For y ve r of rel umers α, β wth α < β deote y f the ler dfeomorhsm f :,, defed s f x : m x, m : The, ech f BV, d therefore, for ll : S f BV, Thus, y the frst rt of the roof we hve S f S f f f BV, Hece, BV loc d the roof s comlete The cocluso out utomtc cotuty follows t o ce from () d () we reset smlr result for the sce A Now, At ths ot, let us recll the follow roosto (see, for stce [, Theorem 344]): Suose, f re fuctos defed o tervls d tht f s well defed If, f d f re - e df- feretle fuctos d stsfes the roerty N the, s terreted to e zero whe- f x f x f x for - f x f x ever f x 0 Theorem 37 Let S ms A, self f, d oly f, A e x, utomtclly ouded loc to t- I ths cse S s Coyrht 0 ScRes

5 6 J GIMÉNEZ ET AL By Proos- (s ce Proof Suose frst tht A loc to 35, for ll f A,, f AC, s Lschtz cotuous o, ) Thus f x f x fx or ll x, d sce AC, roerty N), for - e x, f, (d rtculr t stsfes f x f x f x f x f x (3) Now, Sce f s cotuous, the secod summd the rht hd sde of (3) s L,, d O the other hd, f f f f L (33) L, f x f x f x f x f x Hece, y Lemm 34 f f f (34) L L f, f, From (33) d (34) t follows tht s A, to tself d tht, ths cse, S ms ouded sets o ouded sets The roof of the coverse s sm lr to the oe ve for the the ecessty of the codto the roof of Th eorem 36 Acowledemets 4 S m Ths reserch hs ee rtly suorted y the Cetrl B of Veezuel We wt to ve ths to the lrry stff of BCV for coml the refereces We would le to th the oymous referee d the edtors for ther vlule commets d suestos REFERENCES [] C Jord, Sur l Sere de Fourer, Comtes Redus des Séces de l Acdéme des Sceces, Vol, 88, 8-30 [] R K d C K Krueer, Advced Alyss o the Rel Le, Srer, New Yor, 996 [3] V I Bureov, O Iterto y Prts d Prolem o Comosto of Asolutely Cotuous Fuctos Whch Arses Ths Coecto, Theory of Fuctos d Its Alctos, Proceeds of the Stelov Isttute of Mthemtcs, Vol 34, 975, [4] J Aell d P P Zreo, Noler Suerosto Oertor, Cmrde Uversty Press, New Yor, 990 [5] J Aell, Z Jesús d O Meí, Some Remrs o Noler Comosto Oertors Sces of Dfferetle Fuctos, Bolletto Dell Uoe Mtemtc Itl Sere IX, Vol 4, No 3, 0, [6] M Josehy, Comos Fuctos of Bouded Vrto, Proceeds of the AMS Amerc Mthemtcl Socety, Vol 83, No, 98, do:0090/s [7] Ch J de l Vlle Pouss, Sur L terle de Leesue, Trsctos of the AMS Amerc Mthemtcl Socety, Vol 6, 95, [8] F Resz, Sur Certs Systmes Sulers d Qutos Itrles, Ales de l Ecole Normle, Sureure, Vol 8, No 3, 9, 33-6 [9] N Meretes, O Fuctos of Bouded (; )-Vrto, Collecte Mthemtc, Vol 43, No, 99, 7-3 [0] J Aell, Some Couterexmles for Your Clculus Course, Alyss, Vol 3, No, 00, 00-0 [] G Leo, A Frst Course Soolev Sces, Amerc Mthemtcl Socety, Vol 05, Rode Isld, 009 [] J Aell d P P Zreo, Remrs o the Suerosto Oertor Prolem Vrous Fucto Sces, Comlex Vrles d Elltc Equtos, Vol 55, No 8, 00, do:0080/ [3] N Meretes, O the Comosto Oertor AC[,], Collecte Mthemtc, Vol 4, No, 99, - 7 [4] N Meretes d S Rvs, El Oerdor de Comosc e Escos de Fucoes co Alú To de Vrcó Acotd, IX Escuel Veezol de Mtemátcs, Fcultd de Cecs-ULA, Mérd, 996 Coyrht 0 ScRes