Note on Matuzsewska-Orlich indices and Zygmund inequalities

Transcription

1 ARMENIAN JOURNAL OF MATHEMATICS Volume 3, Number 1, 21, Noe on Mauzewka-Orlic indice and Zygmund inequaliie N. G. Samko Univeridade do Algarve, Campu de Gambela, Faro,85 139, Porugal Received by e edior February 1, 21; acceped for publicaion Marc 31, 21. Abrac In i noe we call aenion o e fac a ere exi ome relaion beween e Mauzewka-Orlicz indice m(φ) and M(φ) of e funcion φ, and poible value of e conan in Zygmund ype inequaliie. Key Word: Mauzewka-Orlicz indice, Zygmund ype inequaliie, almo monoonic funcion Maemaic Subjec Claificaion 2: 46E3, 26A48, 26D7 1 Inroducion Te main goal of i noe i o call aenion o e fac a ere exi ome relaion beween e Mauzewka-Orlicz indice m(φ) and M(φ) of e funcion φ, and poible value of e conan c φ and C φ in e inequaliie d 1 φ(), (1) c φ d 1 φ(), (2) C φ were < l <, φ i a non-negaive funcion, ee Teorem 3.1 and

2 NOTE ON MATUSZEWSKA-ORLICH INDICES AND ZYGMUND INEQUALITIES 23 Inequaliie (1) and (2) are known a Zygmund ype inequaliie, we refer for inance o [1], were under ome monooniciy condiion on φ ere wa own in paricular a Zygmund inequaliie are equivalen o e o called Lozinky and Bary-Seckin condiion. In [2], [7] i wa own a monooniciy condiion on φ may be replaced by a of almo monooniciy, or more generally, by e condiion φ W, ee Definiion 2.1; recall a a non-negaive funcion φ i called almo increaing if ere exi a conan c 1 uc a φ(x) φ(y) for all x y. Noe a we prefer o wrie conan on e rig-and ide of (1)-(2) a 1 and 1 by c C reaon wic become clear in e equel, ee for inance Lemma 3.1 and inequaliy (5). 2 Preliminarie Te Mauzewka-Orlicz indice known in e eory of Orlicz pace (ee [5], [3] and [4], were ey were udied mainly for Young funcion φ), are defined a [ ] [ ] φ() φ() ln lim ln lim φ() φ() m(φ) = up = lim >1 ln ln [ ln ln M(φ) = inf = lim >1 ln ln e definiion being applicable o any non-negaive-funcion φ, and in i cae. lim φ() φ() Noe a for φ γ () = γ we ave m(φ) M(φ) + ] [ lim φ() φ() m(φ γ ) = γ + m(φ) and M(φ γ ) = γ + M(φ). ] (1), (2) Definiion 2.1. By W = W ([, l]) we denoe e cla of non-negaive almo increaing funcion on [, l], poiive on (, l) and by W = W ([, l]) we denoe e cla of funcion on [, l], uc a ere exi an a R 1 uc a e funcion x a φ(x) W. In e cae φ W, one a < m(φ) M(φ) +. Variou properie of e indice m(φ) and M(φ) were obained in [3] and [4], and in [2], [7], [8], [9], [1], [11], [12] in connecion wi udy of variou operaor in generalized Hölder pace, were in paricular i wa own a e validiy of e Zygmund inequaliie for a funcion may be caracerized in erm of e indice m(φ), M(φ). 23

3 24 N. G. SAMKO In paricular, e following propery i known (for e proof ee [2], Teorem 3.1 and 3.2 for φ W, a aed in Teorem 2.1, and [3], Tm 6.4 or [4], Tm 11.8 under a differen definiion of e indice and oer aumpion on φ) Teorem 2.1. Le φ W. Ten d cφ() γ < m(φ), (3) 1+γ γ d cφ() ν > M(φ). (4) 1+ν ν 3 A relaion beween e index m(φ) and e conan c φ Given a non-negaive funcion φ, le I (φ) = γ R1 : ere exi c = c(φ, γ) uc a 1+γ d 1 c φ() γ. Obviouly, if γ I (φ), en γ a I (φ) for any a >, o a I (φ) may be only an infinie inerval aring from. For funcion φ W i i known a e e I (φ) i an open inerval wi e exacly calculaed upper bound: wic follow from (3). I (φ) = (, m(φ)), (1) In Lemma 3.1 we ow a e fac ielf a i inerval i open, i valid for an arbirary non-negaive funcion φ, wiou any aumpion on almo monooniciy of φ, and find a relaion beween e conan c(φ, γ) and c(φ, γ + ε). Lemma 3.1. Le be a non-negaive funcion on [, l] uc a e inegral d exi for every (, l). If ere old inequaliy (1) wi ome c φ >, en for any ε (, c φ ) ere alo old e inequaliy were c i e ame a in (1). 1 φ() d (2) 1+ε c φ ε ε Proof. Le Φ() = d. 24

4 NOTE ON MATUSZEWSKA-ORLICH INDICES AND ZYGMUND INEQUALITIES 25 Te formula i valid Indeed, wic yield (3). = ε ε Φ() d = + ε 1+ε ε Φ() d = ε 1+ε d d = 1+ε d 1+ε Since Φ() 1 c φ φ() by (1), from (3) we obain from wic (2) follow. φ() ε d + 1+ε c φ ε c φ Φ() d. (3) 1+ε d ( 1 1 ) d ε ε d, 1+ε Corollary 3.1. Le φ be a non-negaive funcion on [, l] uc a Ten for any ε < c γ. 1+γ d 1 c γ φ() γ = d 1 1+γ+ε c γ ε φ() γ+ε 1+γ d exi, γ R 1. in (2) o a power of e logarimic func- Remark 3.1. In cae we pa from e facor 1 ε ion, e correponding aemen become d 1 φ() = c φ ( ln ) n d 1 φ(), (4) c n+1 φ n! were n = 1, 2, 3,... wic may be obained by e ucceive applicaion of e given inequaliy: φ() c φ d c 2 φ d d = c 2 φ ln d ec Teorem 3.1. Le φ W. If ere old inequaliy (1) wi ome conan c φ >, en c φ m(φ). (5) 25

5 26 N. G. SAMKO Proof. Suppoe o e conrary a m(φ) < c φ. By Lemma 3.1, inequaliy (2) old wi every ε (, c φ ), in paricular, wi every ε (λ, c φ ), λ = max{m(φ), }, wic i impoible, becaue for φ W, inequaliy (2) implie m(φ) > ε by 3. Corollary 3.2. For e index m(φ) of a funcion φ W e eimae old m(φ) inf > Φ() = inf Φ () > Φ(), (6) were Φ() = d. Φ() Proof. Le A = up >. Le fir A =. Ten e rig-and ide of (6) i zero and φ() alo m(φ) =. Indeed, we ave m(φ) for φ W and in cae m(φ) > ere old (1) wi a finie conan c φ, wic would mean a A <. Terefore, (6) rivially old in e cae A =. Le A <. Ten (1) obviouly old wi c φ = 1. Ten 1 m(φ) by Lemma 3.1, A A wic i inequaliy (6). Remark 3.2. In cae of power funcion = λ we ave m(φ) = M(φ) = inf > Φ() = inf Φ () > Φ() = λ, bu in e general cae i may be a m(φ) > inf > Φ(). 4 A relaion beween e index M(φ) and e conan C φ Similarly o e previou ecion we reveal a relaion beween e upper index M(φ) and e conan C φ in e Zygmund inequaliy (2). Le I + (φ) = γ R1 : ere exi C = C(φ, γ) uc a For funcion φ W i i known a I + = (M(φ), + ), 1+γ d 1 C φ() γ. ee (4). Te following lemma exacifie e aemen on e openne of e inerval (M(φ), + ) for an arbirary non-negaive funcion. 26

6 NOTE ON MATUSZEWSKA-ORLICH INDICES AND ZYGMUND INEQUALITIES 27 Lemma 4.1. Le be a non-negaive funcion on [, l] uc a e inegral d exi for every (, l). If ere old inequaliy (2) wi ome C φ >, en for any ε (, C φ ) ere alo old e inequaliy were C φ i e ame a in (2). d 1 1 ε C φ ε ε φ() (1) Proof. Lemma 4.1 wa proved in [6]. We give e proof ere for e compleene of preenaion. Le Φ 1 () = d. Similarly o (3) we ave l d = 1 ε ε Φ 1 () + ε by direc verificaion. Since Φ 1 () 1 C φ φ() by (2), from (2) we obain from wic (1) follow. C φ l d 1 ε ε φ() + ε Φ 1 () d (2) 1 ε d, 1 ε Lemma 4.2. Le φ W. If ere old inequaliy (2) wi ome conan C φ >, en M(φ) C φ. Proof. Suppoe o e conrary a M(φ) > C φ. By Lemma 4.1, inequaliy (1) old wi every ε (, C φ ), in paricular, wi every ε (μ, C φ ), μ = max{ M(φ), }, wic i impoible, becaue for φ W, inequaliy (1) implie M(φ) < ε by (4). Teorem 4.1. If a funcion φ W admi eimae (2) wi ome conan C φ >, en for e index M(φ) e eimae old M(φ) inf < l Φ 1 () = up Φ 1() < l Φ 1 (), (3) were Φ 1 () = d. Φ() Proof. Le A 1 = up. Inequaliy (2) obviouly old wi C φ = 1 A 1. Ten 1 A 1 <<l M(φ) by Lemma 4.2, wic i inequaliy (3). 27

7 28 N. G. SAMKO Remark 4.1. Te indice p(φ) = xφ (x) inf <x l φ(x), xφ (x) q(φ) = up <x l φ(x) (4) are known a Simonenko indice, ee [13], and i i known a p(φ) m(φ) M(φ) q(φ), (5) ee [4], Teorem In ee erm, inequaliie (6) and (3), in cae φ W, mean a p(φ) m(φ) M(φ) q(φ 1 ). (6) Oberve a aloug we can wrie, for inance, p(φ) m(φ) M(Φ) q(φ), o derive e lef-and ide inequaliy p(φ) m(φ) in (6) from ere, we would like o ave e propery m(φ) = m(φ), wic i rue in e cae < m(φ) M(φ) < becaue Φ φ in i cae and en e funcion Φ and φ ave coinciding indice, ee [4], Teorem Similarly one a M(Φ 1 ) = M(φ) wen < m(φ) M(φ) <. 5 A generalizaion of Lemma 3.1 and 4.1 Baed on e paage from (1) o (2) and e example given in (4), we now conider a poibiliy o race a imilar paage wen one deal wi e cale of funcion more fine an ju e cale of power (or power-logarimic) funcion. In e equel e noaion AC(, l) and for e e of funcion on (, l) aboluely coninuou on every cloed ubinerval of (, l). Lemma 5.1. Suppoe a for ome c >. Ten a imilar inequaliy d 1 φ() (1) c ν() d 1 φ() c δ ν() (2) old, were ν() i any non-negaive funcion on [, l] uc a 1 ν AC(, l), and ν () δ =: up [,l] ν() < c. (3) 28

8 NOTE ON MATUSZEWSKA-ORLICH INDICES AND ZYGMUND INEQUALITIES 29 Proof. Inegraion by par yield d ν() = Φ() ν() + ν () Φ()d (4) ν 2 () ince lim Φ() ν() =. To ceck e laer, in view of (1) i uffice o ow a lim φ() ν() =, for wic i i ufficien o verify a m ( φ ν ) >. Since m ( φ ν ) m(φ) + m ( 1 ν ) = m(φ) M(ν), we en may only ceck a M(ν) < m(φ). Te laer follow from condiion (3), wic implie a M(ν) q(ν) < c ( m(φ)). or From (4), by aumpion (1) we obain d 1 φ() ν() c ν() + By aumpion (3) we ave 1 1 c ν () ν() Lemma 5.2. Suppoe a for ome C >. Ten a imilar inequaliy ν () ν 2 () d (5) ( 1 1 ) ν () c ν() ν() d 1 φ() c ν(). (6) 1 δ c wic yield (2). d φ() (7) C λ() d λ()φ() C δ old, were λ() i any non-negaive funcion in AC(, l), and Proof. Inegraing by par, we obain or λ () δ =: up [,l] λ() (8) < C. (9) λ() d = λ()φ 1 () + λ ()Φ 1 ()d, Φ 1 () = d. (1) By aumpion (7) we en ave λ() d 1 λ()φ() + λ () d (11) C wic yield (8) by (9). ( 1 1 ) λ () λ() d 1 λ()φ(), (12) C λ() C 29

9 3 N. G. SAMKO Reference [1] N.K. Bary and S.B. Seckin. Be approximaion and differenial properie of wo conjugae funcion (in Ruian). Proceeding of Mocow Ma. Soc., 5: , [2] N.K. Karapeian and N.G. Samko. Weiged eorem on fracional inegral in e generalized Hölder pace H ω (ρ) via e indice m ω and M ω. Frac. Calc. Appl. Anal., 7(4): , 24. [3] Lec Maligranda. Indice and inerpolaion. Dieraione Ma. (Rozprawy Ma.), 234:49, [4] Lec Maligranda. Orlicz pace and inerpolaion. Deparameno de Maemáica, Univeridade Eadual de Campina, Campina SP Brazil. [5] W. Mauzewka and W. Orlicz. On ome clae of funcion wi regard o eir order of grow. Sudia Ma., 26:11 24, [6] E. Nakai. Hardy-Lilewood maximal operaor, ingular inegral operaor and e Riez poenial on generalized Morrey pace. Ma. Nacr., 166:95 13, [7] N.G. Samko. Singular inegral operaor in weiged pace wi generalized Hölder condiion. Proc. A. Razmadze Ma. In, 12:17 134, [8] N.G. Samko. Crierion of Fredolmne of ingular operaor wi piece-wie coninuou coefficien in e generalized Hölder pace wi weig. In Operaor Teory: Advance and Applicaion, volume 142, page Proceeding of IWOTA 2, Seembro 12-15, Faro, Porugal, Birkäuer, 22. [9] N.G. Samko. On compacne of Inegral Operaor wi a Generalized Weak Singulariy in Weiged Space of Coninuou Funcion wi a Given Coninuiy Modulu. Proc. A. Razmadze Ma. In, 136:91, 24. [1] N.G. Samko. On non-equilibraed almo monoonic funcion of e Zygmund- Bary-Seckin cla. Real Anal. Exc., 24. [11] N.G. Samko. Singular inegral operaor in weiged pace of coninuou funcion wi an ocillaing coninuiy modulu and ocillaing weig. Operaor Teory: Advance and Applicaion, Birkäuer, Proc. of e conference IWOTA, Newcale, July 24, 171: , 26. [12] N.G. Samko. Singular inegral operaor in weiged pace of coninuou funcion wi non-equilibraed coninuiy modulu. Maem. Nacricen, 279(12): , 26. 3

10 NOTE ON MATUSZEWSKA-ORLICH INDICES AND ZYGMUND INEQUALITIES 31 [13] I.B. Simonenko. Inerpolaion and exrapolaion of linear operaor in Orlicz pace. Ma. Sb. (N.S.),