Inequalities for Some Classes of Hardy Type Operators and Compactness in Weighted Lebesgue Spaces

Transcription

1 DOCTORAL T H E SIS Ineuliies for Some Clsses of Hrdy Tye Oerors nd Comcness in Weighed Lebesgue Sces Abo Abylyev Mhemics

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3 Ineuliies for some clsses of Hrdy ye oerors nd comcness in weighed Lebesgue sces by Abo Muhmediyrovn Abylyev Dermen of Engineering Sciences nd Mhemics Luleå Universiy of Technology Luleå, Sweden & Dermen of Fundmenl Mhemics Fculy of Mechnics nd Mhemics Eursin Nionl Universiy Asn 8, Kzhsn December 26

4 Keywords: Ineuliies, oerors, weighs, weighed Lebesgue sces, Hrdy-ye ineuliy, weighed differenil Hrdy ineuliy, Hrdy oeror, Rimnn-Liouville oeror, Weyl oeror, inegrl oeror wih vrible limis of inegrion, logrihmic singulriies, Oinrov ernels, boundedness, comcness. Prined by Luleå Universiy of Technology, Grhic Producion 26 ISSN ISBN (rin) ISBN (df) Luleå 26

5 To my rens nd fmily

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7 Absrc This PhD hesis is devoed o invesige weighed differenil Hrdy ineuliies nd Hrdy-ye ineuliies wih ernel when he ernel hs n inegrble singulriy, nd lso he ddiiviy of he esime of Hrdy ye oeror wih ernel. The hesis consiss of seven ers (Pers, 2, 3, 4, 5, 6, 7) nd n inroducion where review on he subjec of he hesis is given. In Per weighed differenil Hrdy ye ineuliies re invesiged on he se of comcly suored smooh funcions, where necessry nd sufficien condiions on he weigh funcions re esblished for which his ineuliy nd wo-sided esimes for he bes consn hold. In Pers 2, 3, 4 more generl clss of α - order frcionl inegrion oerors re considered including he well-nown clssicl Weyl, Riemnn-Liouville, Erdelyi-Kober nd Hdmrd oerors. Here <α<. In Pers 2 nd 3 he boundedness nd comcness of wo clsses of such oerors re invesiged nmely of Weyl nd Riemnn-Liouville ye, resecively, in weighed Lebesgue sces for < < nd < < <. As licions some new resuls for he frcionl inegrion oerors of Weyl, Riemnn-Liouville, Erdelyi-Kober nd Hdmrd re given nd discussed. In Per 4 he Riemnn-Liouville ye oeror wih vrible uer limi is considered. The min resuls re roved by using loclizion mehod euied wih he uer limi funcion nd he ernel of he oeror. In Pers 5 nd 6 he Hrdy oeror wih ernel is considered, where he ernel hs logrihmic singulriy. The crieri of he boundedness nd comcness of he oeror in weighed Lebesgue sces re given for < < nd < < <, resecively. In Per 7 we invesiged he weighed ddiive esimes uk ± f C ( ρ f + vh ± f ), f ( ) for inegrl oerors K + nd K defined by K + f () := K(, s) f (s)ds, K f () := K(, s) f (s)ds. I is ssumed h he ernel K = K(, s) of he oeror K ± belongs o he generl Oinrov clss. We derived he crieri for he vlidiy of he ineuliy ( ) when <. v

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9 Prefce This PhD hesis is minly devoed o inroduce nd sudy weighed differenil Hrdy ineuliies nd new Hrdy ye inegrl ineuliies involving Riemnn-Liouville ye oeror nd is conjuge Weyl ye oeror. Furher we invesige boundedness nd comcness of Hrdy ye oerors wih vrible uer limi nd inegrl oerors wih logrihmic singulriy in weighed Lebesgue sces. Moreover, we hve found ddiive esimes of clss of inegrl oerors, which is much wider hn reviously sudied. We lso resen some licions, which cover much wider clsses of inegrl oerors hn sudied before. The hesis consiss of n inroducion nd he following seven ers: [] A.M. Abylyev, A.O. Birysnov nd R. Oinrov, A weighed differenil Hrdy ineuliy on AC(I), Siberin Mh. J. 55 (24), No.3, [2] A.M. Abylyev, Boundedness, comcness for clss of frcionl inegrion oerors of Weyl ye, Eursin Mh. J. 7 (26), No., [3] A.M. Abylyev, R. Oinrov, nd L.-E. Persson, Boundedness nd comcness of clss of Hrdy ye oerors, Reserch reor 26 (submied). [4] A.M. Abylyev, Boundedness nd comcness of he Hrdy ye oeror wih vrible uer limi in weighed Lebesgue sces, Reserch reor 26-4, ISSN: 4-43, Dermen of Engineering Sciences nd Mhemics, Luleå Universiy of Technology, Sweden. Submied o n Inernionl Journl. [5] A.M. Abylyev nd L.-E. Persson, Hrdy ye ineuliies wih logrihmic singulriies, Reserch reor 26-5, ISSN: 4-43, Dermen of Engineering Sciences nd Mhemics, Luleå Universiy of Technology, Sweden. [6] A.M. Abylyev, Comcness of clss of inegrl oerors wih logrihmic singulriies, Reserch reor 26-6, ISSN: 4-43, Dermen of Engineering Sciences nd Mhemics, Luleå Universiy of Technology, Sweden. [7] A.M. Abylyev, A.O. Birysnov, L.-E. Persson nd P. Wll, Addiive weighed L esimes of some clsses of inegrl oerors involving generlized Oinrov ernels, J. Mh. Ineul. (JMI), o er 26. vii

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11 Acnowledgemens Firs of ll, I would lie o eress my dee griude o my suervisors Professor Lrs-Eri Persson (Dermen of Engineering Sciences nd Mhemics, Luleå Universiy of Technology, Sweden) nd Professor Rysul Oinrov (L.N. Gumilyov Eursin Nionl Universiy, Kzhsn) for heir consn suor, hel, ience, undersnding nd encourgemen during my sudies. I lso hn hem for heir wise suggesions nd helful discussions. They devoed mny hours of heir gold ime for dvising me. I hn God h I me such clever, ind, comeen nd wise rofessors in my life. I will forever be hnful o hem. I lso sincere hns my hird suervisor Professor Peer Wll (Dermen of Engineering Sciences nd Mhemics, Luleå Universiy of Technology, Sweden) for suoring nd heling me in vrious wys nd for giving me such mzing ossibiliy o visi nd wor he Dermen of Mhemics of Luleå Universiy of Technology. Everybody of hem re remendous menors. I is gre honour for me o be one of heir sudens. Secondly, my secil hns goes o Professor Lech Mligrnd for his ind dvices nd vluble remrs. I lso would lie o hn Luleå Universiy of Technology for heir gre suor nd for cceing me s PhD suden in heir inernionl PhD rogrm. I m lso greful o L.N. Gumilyov Eursin Nionl Universiy for cceing me s PhD suden in heir inernionl PhD rogrm, which mde my PhD sudies ossible. Furhermore, I would lie o hn everybody he Dermen of Engineering Sciences nd Mhemics Luleå Universiy of Technology, esecilly Professor Nsh Smo nd Elen Miroshniov, for heling me in differen wys nd for lwys being so wrm, suorive nd friendly. I m lso very greful o collegues nd friends he Dermen of Fundmenl Mhemics in L.N. Gumilyov Eursin Nionl Universiy for heling nd suoring me. Moreover, I wn o eress my sincere reciion o my echer of English Professor Krlygsh Zhzibev for siriul suor nd fih in me. Finlly, I give my hery hns o my der rens nd fmily. Esecilly I ronounce my invluble griude o my husbnd PhD docor of mhemics Mdi Murbeov nd my dughers for love nd regulr encourgemen during ll of my sudy. i

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13 Inroducion Inegrl oerors re wide clss of liner oerors h hve licions in vrious fields of science, such s hysics, economics, echnicl sciences nd mny ohers. Therefore he sudy of inegrl oerors e n imorn lce in modern mhemics. In he ls decdes he issues of finding necessry nd sufficien condiions for he weighed ineuliy K f,u C f,v (.) nd wo-sided esimes for he bes consn C in (.) re inensively sudied for vrious inegrl oerors K, where f,v := f () v()d <. In he cse when one of he rmeers nd is eul o or, here is generl resul ([28] Cher XI,.5, Theorem 4, see lso [8], Theorem.) esblishing he ec vlue of he bes consns in (.). However, when <, < in he generl cse his roblem remins oen. Therefore soluion of his roblem for vrious clsses of inegrl oerors is urgen. In 925 G.H.Hrdy [24] obined he ineuliy (.) when = for he Hrdy oeror defined by K f () Hf() := f ()d wih he weighed funcions u() =, v wih he ec vlue C = for he bes consn C in (.), i.e. he ineuliy ( ( ) f ()d) d f ()d, f, (.2) holds which is clled he clssicl Hrdy ineuliy. In 928 G.H.Hrdy [25] roved he firs weigh modificion of ineuliy (.2), nmely he ineuliy ( ( ) f ()d) α d α f () α d, f, (.3)

14 wih he bes consn C = (, α ) when > α> (see [26], Theorem 33). I is nowdys nown h he ineuliies (.2) nd (.3) re in sense euivlen nd lso euivlen o some oher ower weighed vrins of Hrdy s ineuliy, see [56]. Since he middle of he ls cenury he suding of generl weighed form of ineuliy (.) wih he Hrdy oeror H i.e. he ineuliy ( u() ( f ()d d) C f () ) v()d (.4) for = ws iniied (see for insnce [8] by P.R. Beesc, [27] by J. Kdlec nd A. Kufner, [57] by V.R. Pornov, [63] by V.N. Sedov nd [76] by F.A. Sysoev). However, for he cse = he necessry nd sufficien condiion for he vlidiy of ineuliy (.4) ws firs obined, indeendenly, in he wors of G.Tleni [77] nd G.Tomselli [78]. In 972 B.Mucenhou in [42] gve simle ecellen roof of his resul, even in he more generl cse, when u ()d nd v ()d were relced by generl Borel mesures dμ() nd dν(), resecively. A crierion for he ineuliy (.4) o hold when < < ws given indeendenly by J.Brdley [], V.Koilshvili [29] nd B.Mz y [39]. And he cse < < < ws firs described by B.Mz y nd A.Rozin in he le sevenies, see [38] nd [39]. These resuls hve been eended by G. Sinnmon [64] o he vlues of he rmeers < < <, >, nd he cse < < = hs been described by G.Sinnmon nd V.D.Senov [65]. G.Tomselli [78] gve n lernive crierion for he weighed Hrdy ineuliy (.4) o hold when =, which V. Senov nd L.-E. Persson generlized his resul o he cses < < nd < < < in [54]. There re sudies on he descriion of he ineuliies in oher erms [5] nd [32], differen from he bove uhors nd lso for negive vlues of he rmeers, see e.g. [6]. Le us sum u some of he resuls bove in he following Theorem: Theorem A. (i) If <, hen he ineuliy (.4) holds for ll mesurble funcions f () on (, b) if nd only if ( A := su <<b ) ( u()d v ()d 2 ) <

15 or A PS := su > w() v (y)dy d v (y)dy <. (ii) If < < <, hen he ineuliy (.4) holds if nd only if A 2 := ( ) r ( ) r r u()d v ()d v ()d < or B PS := r w() v (y)dy d r v (y)dy v ()d r <, where r =. (iii) If < < < <, hen he ineuliy (.4) holds if nd only if A 3 := ( ) r ( ) r r u()d v ()d u()d <. (iv) If < < =, hen he ineuliy (.4) holds if nd only if A 4 := where v() = ess su <<. v() [ ) v() u()d u()d <, I is nowdys nown h he condiions in (i)-(ii) in fc cn be relced by infinie mny euivlen condiions, even by four differen scles of condiions, see [5] (he cse (i)), [55] (he cse (ii)) nd for even more informion of his ye he review ricle [34]. In connecion wih he invesigion of oerors in Lorenz sces since 99 he Hrdy-ye oerors were cively sudied on he clss of monoone funcions, see for emle [8], [9], [2], [2], [22] nd he references herein. Moreover, oerors including he suremum, hs begn o be invesiged recenly, see for emle [3], [6], [7], [53] nd he references herein. 3

16 The ineuliy (.4) nd is dul ineuliy re euivlen o he differenil ineuliy y,u C y,v (.5) resecively for y() = nd for y( ) =. We remr h P.Gur [23] described he ineuliy (.5) under he condiion y() =, y( ) =. (.6) Hisoricl bcground, review of he reserch, he min resuls nd heir licions re given in he boos [], [2], [26], [3], [33], [4] nd [5]. The ineuliy (.5) wih condiion (.6) ws considered in [5], [3], bu only in [5] n ended version of he wor of P. Gur [23] ws considered nd wo-sided esimes for he bes consn C of (.5) ws sed. The im of his PhD hesis is o comlemen nd eend severl resuls in he re described bove which is ody clled Hrdy ye ineuliies nd reled boundedness nd comcness resuls. Below we give shor descriion nd moivion for hese new conribuions resened in his PhD hesis. In Per, using new mehod, we obined necessry nd sufficien condiions for he vlidiy of he ineuliy (.5) wih condiion (.6) for he cses < < nd < < <, >. We lso derived wo-sided esimes for he bes consn C of (.5), which re beer hn hose in [5]. In 979 O.D.Ayshev nd M.Oelbev [7] considered he ineuliy (.5) for higher order derivive, nmely he ineuliy y,u C y n,v, n > (.7) y (i) () =, i =,,...n. (.8) Bu crierion for he ineuliy (.7) o hold ws obined only under cerin resricions on he weigh funcions. We menion h Cher 4 of he boo [3] is devoed only o such higher order Hrdy ye ineuliies. We remr h he ossible boundry vlues (of ye (.8)) re very crucil o me such invesigions ossible (see [3]). The ineuliy (.7) wih he condiion (.8) is euivlen o he ineuliy (.), when he inegrl oeror K is eul o he Riemnn-Liouville 4

17 oeror I α defined by for α = n, i.e. I α f () := ( y) (α ) f (y)dy, >, (.9) Γ(α) I α f,u C f,v. (.) A sisfcory crierion for he ineuliy (.) o hold for he Riemnn- Liouville oeror when α>ws obined in he ers [67], [7] nd [69] of V.D.Senov. An oher generlizion of (.4) is norm ineuliy of he form (, y) f (y)dy u()d C f (y)v(y)dy for he Hrdy-Volerr inegrl oeror K given by Kf() :=, f, (.) (, y) f (y)dy,, (.2) wih ernel (, y), which is ssumed o be non-negive nd mesurble on he ringle {(, y) : y }. A number of uhors hve sudied in heir wors severl differen clsses of such oerors. In [37] i ws obined chrcerizion of (.) in he cse < < wih he secil ernel (, y) = ϕ(/y), where ϕ :(, ) (, ) is non-incresing nd sisfying h ϕ(b) D(ϕ() + ϕ(b)) for ll <, b <. Moreover, crierion of he L,v L,w boundedness ws given in [7] nd [72] by V.D. Senov for he Volerr convoluion oeror (.2) wih (, y) = ( y) for boh he cses < < nd < < <. An oher clss of sudied oerors of he ye (.2) hs ernels sisfying some ddiionl monooniciy nd coninuiy condiions (see e.g. [9] by S. Bloom nd R. Kermn). In he nineies i ered some imorn wors (see e.g. [45], [46] by R. Oinrov nd [73], [74] by V.D. Senov) devoed o he clss of he oerors (.2) wih so clled Oinrov ernels. A ernel (, y) sisfies he Oinrov condiion if here is consn D indeenden on, y, z such h D (, y) (, z) + (z, y) D(, y), y z. (.3) 5

18 Le he ernel (, y) of he oeror (.2) sisfy he Oinrov condiion (.3). If nd A (α) := su > A (α) := su > B (α) := B (α) := hen i is nown h K (, )u()d u()d K (, )u()d u()d v (y)dy K (, y)v (y)dy v (y)dy ( ) K (, y)v (y)dy ( ),, v ()d u()d K L,v L,u A (α) + A (α), < <, (.4) nd K L,v L,u B (α) + B (α), < < <. (.5) Ler on wo-sided esimes of he yes (.4) nd (.5) were derived for more generl oerors nd sces, see e.g. [37], [35], [75], [3], [4], [2], [3], [47], [48] nd [49]. The clss of Oinrov ernels includes ll bove menioned clsses of ernels ece Riemnn-Liouville ernels for <α<. The Riemnn-Liouville oeror is wely singulr inegrl oeror when <α<nd behves very differenly hn when α>. For ower weigh funcion v() nd u(y) he following clssicl resul [26], Theorem 42, is well nown: If >, <α</, /( α) or α /, < <, hen ( +α) (I α f ) ()d 6 C f. (.6),,

19 The ineuliy (.6) hs been generlized in he following wy in er [6] of K.F.Andersen nd E.T.Swyer: Le <α< nd < < =. Then ( α) ui α (uf) C f if nd only if K <, where K := su h <h<α +h u ()d h h u ()d Moreover, in [59] D.V.Prohorov nd V.D.Senov roved he followings resul: Le <α< nd < < =. Then ( α) ui α f C,v f, (.7) if nd only if v <. When α, = = 2 nd v he ineuliy (.7) hs been 2 chrcerized by S. Newmn nd M. Solomy wihin he secrl heory of seudo-differenil oerors on he hlf-is, see [44] nd lso references herein. A crierion for he ineuliy (.) o hold for < < ws derived by M.Loreni [36]. However, due o imliciness of he condiions he crieri in [36] me hem difficul o verify. Therefore, we se gol o derive elici L,v L,u crieri for he boundedness of he Riemnn- Liouville oeror in subseuen wors. In he cse < <, < <, α> nd v( ) elici crieri for L,v L,u boundedness of he Riemnn-Liouville nd Weyl oerors re obined indeendenly in wors of A.Meshi [4] nd D.V.Prohorov [58], see lso [66]. A generlizion of hese resuls o he cse when he funcion u( ) is no incresing ws climed in he er [3] of S.M.Frsni. In he er [59] of D.V.Prohorov nd V.D.Senov crieri for L,v L,u boundedness nd comcness of he Riemnn-Liouville oeror re given for < < in he following cses: ) < <α nd he funcion v is no decresing; b) < <α nd he funcion u is no incresing. 7.

20 A generlizion of hese crieri for L L boundedness of he Riemnn-Liouville oeror in he cse of convoluion ye oeror K, defined by K f () := v() K( s)u(s) f (s)ds, >, re given in he ers of N.A.Ruin [52] nd R.Oinrov [5]. For he cse when he ernel of he oeror K, defined by (.2) is (, y) = ( y) nd he funcion ( ) hs n inegrble singulriy in zero lie he Riemnn- Liouville oeror he resuls in [52] were generlized by D.V.Prohorov nd V.D.Senov [59] in he cse of ineuliy (.). Moreover, R.Oinrov [5] roved generl resul of he ye climed by S.M.Frsni [3]. In ddiion o he Riemnn-Liouville nd Weyl oerors he Erdey- Kober nd Hdmrd oerors re imorn boh in mhemics nd for severl licions. One of he generlizions nd unificions of hese oerors is he frcionl inegrion oeror I α g defined by: I α gϕ() := Γ(α) ϕ()g ()d, [g() g()] α >, α >, (.8) where g( ) is locl bsolue coninuous nd incresing funcion on I (, ). In [62] he oeror I α g is clled frcionl inegrl of he funcion ϕ wih resec o he funcion g of order α. In riculr, in (.8) when g() =, g() = σ, σ> nd g() = ln, we obin he frcionl inegrl Riemnn- Liouville, Erdelyi-Kober ye nd Hdmrd oeror, resecively. In Pers 2 nd 3 of his PhD hesis we consider he more generl oerors K α,β nd T α,β defined s follows: nd K α,β f () := T α,β f () := u(s)w β (s) f (s)w(s)ds, I, α (W(s) W()) u(s)w β () f (s)v(s)ds, I, α (W(s) W()) 8

21 where <α<, β R, I = (, b), < b nd W( ) is loclly bsoluely coninuous nd monooniclly incresing funcion on I, dw() = d w() nd u( ) - non-negive mesurble funcion in I. In Per 2 when <α<, >, β (β< α, ifw(b) = ) nd α u is non-decresing funcion we obined necessry nd sufficien condiions for he boundedness nd comcness of he oeror K α,β from L,w ino L,v, for he cses < < nd < < <, when b < α nd for he cse < < < when b =. Conseuenly, from hese semens we obin necessry nd sufficien condiions for he boundedness nd comcness of he weighed Weyl oeror Iα, defined by Iα f () := w() u(s)s β f (s)ds, >, <α<, α (s ) from L o L. Noe h from hese resuls i seems h Theorems 3, 4, 7 nd 8 of er [3] re no rue in generl. Similrly, in Per 3 when <α<, >, β nd u is nonincresing funcion we derived necessry nd sufficien condiions for he α boundedness nd comcness of he oeror T α,β from L,w ino L,v, for he cses < < nd < < <, when b < nd for he cse α < < < when b =. Conseuenly, we obined in riculr necessry nd sufficien condiions for he boundedness nd comcness of he weighed Riemnn- Liouville, Erdelyi-Kober nd Hdmrd oerors from L ino L, which generlize he well nown resuls for hese oerors when >. α In Per 4 we considered he roblem of boundedness nd comcness of he oeror K α,ϕ, defined in he following wy K α,ϕ f () := ϕ() f (s)w(s)ds (W() W(s)) α, <α<, from L,w ino L,v, where ϕ() is sricly incresing loclly bsoluely coninuous funcion, which sisfies he following condiions lim ϕ() =, lim + ϕ() = b, nd ϕ(). b 9

22 Obviously, he resuls resened in his er clerly generlizes he resuls in [] nd [4]. In Pers 5 nd 6 we considered he oeror K γ wih logrihmic singulriy defined by K γ f () := v() u(s)s γ ln f (s)ds, >. s When γ =, v( ) u( ) his oeror is clled frcionl inegrion oeror of infiniesiml order nd i hs wide licions in mhemicl biology, see [43]. In Per 5 we ssumed h he funcion u is non-incresing nd derived necessry nd sufficien condiions for he boundedness of he oeror K γ from L ino L, when < < nd < < <, >. Moreover, he comcness of he oeror K γ from L ino L ws roved in Per 6 when < <. We remr h he resuls in ers 5 nd 6 clerly generlizes he min resuls in [5] nd [2], resecively. In Per 7 we considered he weighed ddiive esimes uk ± f C ( ρ f + vh ± f ), f (.9) for he inegrl oerors K + nd K defined by K + f () := K(, s) f (s)ds, K f () := K(, s) f (s)ds, where he secil cses H + nd H re he usul Hrdy oerors defined by H + f () := f (s)ds, H f () := f (s)ds. We ssumed h ernel of he oerors K + nd K belong o he generlised Oinrov clss [48] nd hus found ec crieri for he vlidiy of he ineuliy (.9) when < in much more generl cses hn reviously nown.

23 References [] A.M. Abylyev nd G.Zh. Ismgulov. Crieri such s he limi oeror of frcionl inegrion wih vrible uer limi in weighed Lebesgue sces. Eursin Mh. J. (28), No., 2-2 (in Russin). [2] A.M. Abylyev nd A.O. Birysnov. Comcness crierion for frcionl inegrion oeror of infiniesiml order. Uf Mh. J. 5 (23), No., 3. [3] A.M. Abylyev nd A.O. Birysnov. Hrdy ye ineuliies conining suremum. Mh. J. 4 (24), No.4 (54), 5-7 (in Russin). [4] A.M. Abylyev. Crierion of he boundedness of frcionl inegrion ye oeror wih vrible uer limi in weighed Lebesgue sces. Inernionl Conference on Anlysis nd Alied Mhemics (ICAAM 26) AIP Conf. Proc. 759, Published by AIP Publishing 26. [5] A.M. Abylyev. Weighed esimes for he inegrl oeror wih logrihmic singulriy. Inernionl Conference on Anlysis nd Alied Mhemics (ICAAM 26) AIP Conf. Proc. 759, Published by AIP Publishing 26. [6] K.F. Andersen nd E.T. Swyer. Weighed norm ineuliies for he Riemnn - Liouville nd Weyl frcionl inegrl oerors. Trns. Amer. Mh. Soc. 38 (988), No.2, [7] O.D. Ayshev nd M.O. Oelbyev. On he secrum of clss of differenil oerors nd some embedding heorems. Izv. Ad. Nu SSSR Ser. M. 43 (979), No.4, (in Russin). [8] P.R. Beesc. Hrdy s ineuliy nd is eensions. Pcific J. Mh., (96), [9] S. Bloom nd R. Kermn. Weighed norm ineuliies for oerors of Hrdy ye. Proc. Amer. Mh. Soc. 3 (99), No., [] J.S. Brdley. Hrdy ineuliies wih mied norms. Cnd. Mh. Bull. 2 (978), No., [] D.E. Edmunds nd W.D. Evns. Hrdy oerors, funcion sces nd embeddings. Sringer-Verlg, Berlin, 24. [2] D.E. Edmunds, V. Koilshvili nd A. Meshi. Bounded nd comc inegrl oerors. Kluwer Acdemic Publishers, Dordrech, 22.

24 [3] S.M. Frsni. On he boundedness nd comcness of he frcionl Riemnn-Liouville oerors. Sibirs. M. Zh. 54 (23), No.2, (in Russin); rnslion in Sib. Mh. J. 54 (23), No.2, [4] I. Genebshvili, A. Gogishvili, V. Koilshvili nd M. Krbec. Weigh heory for inegrl rnsforms on sces of homogeneous ye, Longmn, Hrlow, 998. [5] A. Gogishvili, A. Kufner, L.-E. Persson nd A. Wedesig. An euivlence heorem for inegrl condiions reled o Hrdy s ineuliy. Rel Anl. Echnge 29 (23/4), [6] A. Gogishvili, B. Oic nd L. Pic. Weighed ineuliies for Hrdy-ye oerors involving surem. Collec. Mh. 57 (26), No.3, [7] A. Gogishvili nd L. Pic. A reducion heorem for suremum oerors. J. Comu. Al. Mh. 28 : (27), [8] A. Gogishvili nd V.D. Senov. Reducion heorems for weighed inegrl ineuliies on he cone of monoone funcions. Russin Mh. Surveys 68 (23), [9] M. L. Goldmn. Order-shr esimes for Hrdy-ye oerors on cones of usimonoone funcions. Eursin Mh. J. 2:3 (2), [2] M. L. Goldmn. Order-shr esimes for Hrdy-ye oerors on he cones of funcions wih roeries of monooniciy. Eursin Mh. J. 3:2 (22), [2] M. L. Goldmn. Some euivlen crieri for he boundedness of Hrdyye oerors on he cone of usimonoone funcions. Eursin Mh. J. 4:4 (23), [22] M. L. Goldmn. Esimes for resricions monoone oerors on he cone decresing funcions in Orlicz sce. Mh. Zm. : (26), [23] P. Gur. Generlized Hrdy s ineuliy. Csois Pěs. M. 9 (984), [24] G.H. Hrdy. Noes on some oins in he inegrl clculus, LX. An ineuliy beween inegrls, Messenger of Mh. 54 (925), [25] G.H. Hrdy. Remrs on hree recen noes in he journl. J. London Mh. Soc. 3 (928),

25 [26] G.H. Hrdy, J.E. Lilewood, G. Póly. Ineuliies. Cmbridge Univ. Press, 324 (952). [27] J. Kdlec nd A.Kufner. Chrcerizion of funcions wih zero rces by inegrls wih weigh funcions II. Csois Pěs. M. 92 (967), [28] L.V. Knrovich nd G.R. Ailov. Funcionl nlysis. Nu, Moscow, 977 (in Russin). [29] V.M. Koilshvili. On Hrdy s ineuliies in weighed sces. Soobsch. Acd. Nu Gruzin. SSR, 96 (979), No., [3] V. Koilshvili, L-E. Persson nd A. Meshi. Weighed norm ineuliies for inegrl rnsforms wih roduc ernels. Nov Science Publishers, Inc. NY, 29. [3] A. Kufner nd L.-E. Persson. Weighed ineuliies of Hrdy ye. World Scienific, New Jersey-London-Singore-Hong Kong, 23. [32] A. Kufner, L.-E. Persson nd A. Wedesig. A sudy of some consns chrcerizing he weighed Hrdy ineuliy. Bnch Cener Publ. 64, Orlicz Cenenry Volume, Polish Acd. Sci., Wrsw (24), [33] A. Kufner, L. Mligrnd nd L-E. Persson. The Hrdy Ineuliy. Abou is Hisory nd Some Reled Resuls. Vydvelsý Servis, Plzeˆn, 27. [34] A.Kufner, L.E.Persson nd N.Smo. Some new scles of weigh chrcerizions of Hrdy-ye ineuliies, Oeror Theory, Pseudo- Differenil Euions, nd Mhemicl Physics, , Oeror Theory: Adv. Al. 228, Birhöuser/Sringer, Bse/ A.G.,Bsel, 23. [35] E. Lomin nd V.D. Senov. On he Hrdy-ye inegrl oerors in Bnch funcion sces. Publ. M. 42 (998), [36] M. Lorene. A chrcerizion of wo weigh norm ineuliies for one-sided oerors of frcionl ye. Cn. J. Mh. 49 (997), No. 5, -33. [37] P.J. Mrin-Reyes nd E. Swyer. Weighed ineuliies for Riemnn- Liouville frcionl inegrls of order one nd greer. Proc. Amer. Mh. Soc. 6 (989), [38] V.G. Mz y. Einbeungssäze für Sobolewsche Räume. Teil [Imbedding Theorems for Sobolev Sces. Pr ], Teubner -Tee zur Mhemics, Leizig,

26 [39] V.G. Mz y. Sobolev sces. Sringer-Verlg, Berlin, Heidelberg, 985. [4] A. Meshi. Soluion of some weigh roblems for he Riemnn-Liouville nd Weyl oerors. Georgin Mh. J. 5 (998), No.6, [4] D.S. Mirinovic, J.E. Pečrić nd A.M. Fin. Ineuliies involving funcions nd heir inegrls nd derivives. Kluver Acd. Publishers, Dordrech, 99. [42] B. Mucenhou. Hrdy s ineuliies wih weighs. Sudi Mh. 34 (972), No., [43] A.M.Nhushev. Euions of mhemicl biology. M.: Vysshy shol, 995 (in Russin). [44] J. Newmn nd M. Solomy. Two-sided esimes on singulr vlues for clss of inegrl oerors on he semi-is. J. Func. Anl. 5 (997), [45] R. Oinrov. Weighed ineuliies for clss of inegrl oerors. Sovie Mh. Dol. 44 (992), (in Russin). [46] R. Oinrov. Two-sided esimes of cerin clss of inegrl oerors. Proc. Selov Ins. Mh. 24 (993), (in Russin). [47] R. Oinrov. A weighed esime for n inermedie oeror on he cone of nonnegive funcions. Siberin Mh. J. 43 (2), No., [48] R. Oinrov. Boundedness nd comcness of Volerr-ye inegrl oerors, Siberin Mh. J. 48 (27), No. 5, [49] R. Oinrov. Boundedness nd comcness in weighed Lebesgue sces of inegrl oerors wih vrible inegrion limis, Siberin Mh. J. 52 (2), No. 6, [5] R. Oinrov. Boundedness nd comcness of clss of convoluion inegrl oerors of frcionl inegrion ye. Proc. Selov Ins. Mh. 293 (26), No., (in Russin). [5] B. Oic nd A. Kufner. Hrdy-ye Ineuliies. Longmn, Hrlow, 99. [52] N.A. Ruin. On he boundedness of clss of frcionl ye inegrl oerors. Sborni Mhemics. 2 (29), No.2,

27 [53] E. Pernecá nd L. Pic. Comcness of Hrdy oerors involving surem. Boll. Unione M. Il. Ser. 9, 6: (23), [54] L.E. Persson nd V.D.Senov. Weighed inegrl ineuliies wih he geomeric men oeror. J. Ineul. Al. 7 (22), No.5, [55] L.E. Persson, V.D. Senov nd P. Wll. Some scles of euivlen weighed chrcerizion of Hrdy s ineuliy, he cse <. Mh. Ineul. Al. (27), No.2, [56] L.E. Persson nd N. Smo. Wh should hve hened if Hrdy hd discovered his? J. Ineul. Al. 22, 22:29. [57] V.R. Pornov. Two imbedding heorems for sces L,b (Ω R +) nd heir licions. Dol. Ad. Nu SSSR, 55 (964), [58] D.V. Prohorov. On he boundedness nd comcness of clss of inegrl oerors. J. London Mh. Soc. 6 (2), [59] D.V. Prohorov nd V.D.Senov. Weighed esimes for Riemnn- Liouville oerors nd some of heir licions. Proc. Selov Ins. Mh. 243 (23), No.4, [6] D.V. Prohorov. Weighed esimes for he Riemnn-Liouville oerors wih vrible limis. Siberin Mh. J. 44 (23), No.6, [6] D.V. Prohorov. Weighed Hrdy s ineuliies for negive indices. Publiccions M. 48 (24), [62] S.G. Smo, A.A. Kilbs nd O.I. Mrichev. Frcionl order inegrls nd derivives, nd some licions. Mins: Science nd Technology, 987. [63] V.N. Sedov. Weighed sces. The imbedding heorem. Diff. Urvneniy, 8 (972), (in Russin). [64] G. Sinnmon. A weighed grdien ineuliy. Proc. Roy. Soc. Edinburgh Sec. A (989), No. 3-4, [65] G. Sinnmon nd V.D. Senov. The weighed Hrdy ineuliy: new roofs nd he cse =. J. London Mh. Soc. 54 (996), No., 89-. [66] M. Solomy. Esimes for he roimion numbers of he weighed Riemnn-Liouville oeror in he sces L. Oer. Theory: Adv. nd Al. 3 (2),

28 [67] V.D. Senov. Two-weighed esimes for he Riemnn-Liouville inegrls. Czesoslovens Ademie Ved Memicy Usv. 39 (988), 28. [68] V.D.Senov. Weighed ineuliies of Hrdy ye for higher order derivives. Proc. Selov Ins. Mh. 3 (99), [69] V.D. Senov. On weighed ineuliy of Hrdy ye for frcionl Riemnn-Liouville inegrls. Sib. Mh. J. 3 (99), No. 3, (in Russin). [7] V.D. Senov. Two-weighed esimes of Riemnn-Liouville inegrls. Izv. Ad. Nu SSSR Ser. M. 54 (99), No. 3, ; rnslion in Mh. USSR-Izv. 36 (99), No. 3, [7] V.D. Senov. Weighed ineuliies for clss of Volerr convoluion oerors. J. London Mh. Soc. 45 (992), No.2, [72] V.D. Senov. On weighed esimes for clss of inegrl oerors. Siberin Mh. J. 34 (993), [73] V.D. Senov. Weighed norm ineuliies of Hrdy ye for clss of inegrl oerors. J. London Mh. Soc. 5 (994), No.2, 5 2. [74] V.D. Senov. Weighed norm ineuliies for inegrl oerors nd reled oics. Proceedings of he sring school on Nonliner Anlysis, Funcionl sces nd Alicions, Prgue: Promeheus Publishing House 5 (994), [75] V.D. Senov nd E.P. Ushov. On inegrl oerors wih vrible limis of inegrion. Proc. Selov Ins. Mh. 232 (2), [76] F.A. Sysoev. Generlizions of cerin Hrdy ineuliy. Izv. Vysš. Učeb. Zved. Memi 49 (965), No.6, 4 43 (in Russin). [77] G.Tleni. Osservzione sor un clsse di disuguglinze. Rend. Sem. M. Fiz. Milno 39 (969), [78] G.Tomsselli. A clss of ineuliies. Boll. Unione M. Il. 2 (969),

29 Per A weighed differenil Hrdy ineuliy on AC(I) Siberin Mhemicl Journl 55 (24), No.3, Remr: The e is he sme bu he form hs been modified o fi he syle in his PhD hesis.

30

31 Siberin Mhemicl Journl, Vol. 55, No. 3, , 24 Originl Russin Te Coyrigh c 24 Abylyev A.M., Birysnov A.O., nd Oinrov R. A WEIGHTED DIFFERENTIAL HARDY INEQUALITY ON AC(I) A. M. Abylyev, A. O. Birysnov, nd R. Oinrov Absrc: A weighed differenil Hrdy ineuliy is emined on he se of loclly bsoluely coninuous funcions vnishing he endoins of n inervl. Some generlizions of he vilble resuls nd shrer esimes for he bes consn re obined. DOI:.34/S X Keywords: weighed differenil Hrdy ineuliy, Lebesgue sce, loclly bsoluely coninuous funcions. Inroducion Assume h I = (, b), < b,<, <, + =, ρ, υ nd ρ = re nonnegive loclly summble funcions on I nd υ. ρ Le < < nd le L,ρ L,ρ (I) be he sce of mesurble funcions f on I such h he norm f,ρ ρ () f () d is finie. The symbol W,ρ W(ρ, I), >, snds for he collecion of f loclly bsolluely coninuous on I nd hving he norm f W = f,ρ + f ( ) (2),ρ finie, where I is fied oin. Assume h lim + f () f (), lim b f () f (b), nd AC (ρ, I) = { f W,ρ : f () = f (b) = }, AC,l (ρ, I) = { f W,ρ : f () = }, AC,r (ρ, I) = { f W,ρ : f (b) = }. The closures of AC (ρ, I), AC,l (ρ, I) nd AC,r (ρ, I) under (2) re denoed resecively by W (ρ, I), W (ρ, I) nd,l W,r(ρ, I).

32 2 We consider he weighed Hrdy ineuliy in differenil form on AC (ρ, I) [] : υ() f () d C ρ() f () d. (2) Ineuliy (2) nd is generlizions were he subjec of invesigions of mny seciliss in he ls 5 yers, nd so hese re sudied well on AC,l (ρ, I) nd AC,r (ρ, I). The hisory of he roblem nd he resuls cn be found in [, 2, 3]. In he recen yers numerous euivlen crierions, ensuring his ineuliy, re obined (for insnce, see [4, 5]). Bu (2) is AC (ρ, I). Some resuls cn be found in [, 2] no sudied deuely on nd only in he ricle [] wo-sided esimes for he bes consn C > of (2) re given. Vrious licions of (2) in he uliive heory of differenil eu- AC (ρ, I) wih shrer es- ions (see [6, 7, 8, 9]) necessie sudying i on imes for he bes consn. In he resen ricle by mehod differen from h in [] we esblish more genrl resul generlizing hose in he bove ers nd give shrer wo-sided esimes for he bes consn C > in (2). 2. Necessry Noions nd Semens We sudy (2) on AC (ρ, I) in deendence on he behvior of ρ he endoins of I. The weighed funcion ρ my vnish he endoins of I nd hus we hve Theorem A. Le < <. Then ( ) (i) if ρ L (I) hen, for every f W ρ, I, here eis lim + f () f (), lim b f () f (b), nd W (ρ, I) = { f W(ρ, I) : f () = f (b) = } AC (ρ, I); ( ) (ii) if ρ L (, c) nd ρ L (c, b), c I, hen, for every f W ρ, I, here eis f () nd W (ρ, I) = W ( ) {,l ρ, I = f W (ρ, I) : f () = } ( ) AC,l ρ, I ; ( ) (iii) if ρ L (, c) nd ρ L (c, b), c I, hen, for every f W ρ, I, here eis f (b) nd

33 3 W (ρ, I) = W,r ( ) { ρ, I = f W (ρ, I) : f (b) = } ( ) AC,r ρ, I ; (iv) if ρ L (, c) nd ρ L (c, b), c I, hen W (ρ, I) = W ( ),l ρ, I = W ( ),r ρ, I = f W ( ) ρ, I. Generlly seing, he semens of Theorem A re nown nd hey cn be deduced from he resuls in [,, 2]. We resen he roof of (ii). The remining semens re roven by nlogy. Assume ( ) h ρ L (, c) nd ρ L (c, b), c I. Then for f ρ, I we hve W c f () c d ρ b ρ() f () d <. Therefore, f () is ( defined. ) { } ( ) Le f W,l ρ, I. Then here eiss seuence fn AC,l ρ, I such h W f fn sn. Since,ρ f () f n () for < < < b, he Hölder ineuliy yields f () f n () m, Hence, f () =. Le <α < b. In his cse or f (α) α f (α) α f (s) f n(s) ds + f ( ) f n ( ) ρ ρ ρ α α W f f n.,ρ ρ() f d ρ() f d

34 4 Le oin α = α (,α) (,α) sisfy he relion α α ρ = ρ. α Inroduce funcion, < α, ( )( α ) f α () = f (α) ρ ρ, α α, α α f (), α < b. Obviously, f α AC,l ( ρ, I ). We hve f f α W = α ρ f f α α ρ f + f (α) α α ρ ) ( α + 2 ρ f, nd so W ( ) ( ) f fα sα. Hence f W,l ρ, I nd W,l ρ, I = { ( ) } f W ρ, I : f () =. ( ) ( ) ( ) ( ) Demonsre h W ρ, I = W,l ρ, I. Since W ρ, I W,l ρ, I, ( ) ( ) ( ) i suffices o esblish h W ρ, I W,l ρ, I. Le f W,l ρ, I nd <α <β<b. Since β = β ( β, b ) ( β, b ) such h β ρ ds =, for every β I here eiss oin f (β) β β ρ ρ() f () d. β

35 5 Consruc f α,β In his cse AC ( ρ, I ) such h f α (), < β, β β f α,β () = f (β) ρ ρ, β β, β, β < b. f f α,β W,ρ α ) ( α + 2 ρ f f α ρ f ) ( α β β β ρ f ρ f ρ f f α,β f (β) β ρ f + β β. β ρ ρ f Hence, W f fα,β sα nd β b. There fore, f ( ) W ρ, I.,ρ Theorem A is roven. Le α<β b. Pu ( ) A α, β, = ( ) A 2 α, β, = α α ρ β A ( ) α, β, = ρ ρ α β υ() α υ()d α ρ, υ()d, d,

36 6 β A ( ) 2 α, β, = A i ρ β υ() β ρ ( ) ( ) A i α, β = su A i α, β,, α<<β ( ) α, β = su A i α<<β ( γ = min ( ), d ( α, β, ), i =, 2, ( ) ),γ 2 =.,α< <β; The bes consns C in (2) on AC ( ( )) ( ( )) ( ( )) ( ) ρ, α, β (, AC,l ) ρ, α, β ( nd ) AC,r ρ, α, β re denoed by C = J α, β, C = Jl α, β, nd C = Jr α, β, resecively. In view of [3, 3], we cn sy h Theorem B. Le < <. Then m { ( ) ( )} ( ) { ( ) ( )} A α, β, A2 α, β Jl α, β min γ A α, β,γ2 A 2 α, β, (22) m { A ( ) α, β, A ( ) } ( ) { 2 α, β J r α, β min γ A ( ) α, β,γ2 A ( ) } 2 α, β. (23) Assume h B ( α, β ) = B ( α, β ) = β α β α β α υ υ α β ρ ρ ( ) ( ) ρ()d ρ()d Since ρ is loclly summble on I, weby[3, 4] hve (see [4], Remr) Theorem C. Le < < <, >. Then μ B ( α, β ) J l ( α, β ) μ + B ( α, β ), μ B ( α, β ) J r ( α, β ) μ + B ( α, β ), where μ = ( ( ) μ + = ), μ + = ( ) ( for < < < nd μ = ) ( ) for < < < <.,.,

37 7 3. The Min Resuls 3.. The cse of < <. Le ρ (s)ds <. (24) Definiion. A oin c i I, i =, 2, is clled midoin for ( A i, A i) if A i (, c i ) = A i (c i, b) T ci (, b) <, i =, 2. Theorem. Assume h < < nd (24) holds. Then (2) is fulfilled on AC ( ) ρ, I if nd only if here eis midoin ci I for ( A i, Ai) les for one of he numbers i =, 2 nd he bes consn J (, b) in (2) in his cse sisfies he esime 2 m { Tc (, b), T c2 (, b) } J (, b) min { γ T c (, b),γ 2 T c2 (, b) }. (25) Corollry [9]. In he cse of =, we hve m { T c (, b), T c2 (, b) } { ( J (, b) min ) } T c (, b), T c2 (, b). To rove Theorem, we use Lemm. Le < < nd ssume h (24) holds. Then midoin for ( A i, A i), i=,2, eiss if nd only if, for given c I, here eis lim su A i (, c, ) <, lim su A i (c, b, ) <, i =, 2. (26) b Proof of Lemm. Sufficiency: (26) yields lim c A i (, c) <, lim A i (c, b) <, i =, 2. c b Demonsre h lim A i (, c) > lim A i (c, b), i =, 2. (27) c b c b Indeed, if lim A i (, c) lim A i (c, b) < (28) c b c b

38 8 hen (24) imlies h υ()d <, c I. c Hence, lim A i (c, b) =, i =, 2. (29) c b For i =, (29) is obvious nd, for i = 2, i follows from he ineuliy h c ρ c υ() ρ d c ρ b c υ()d Since A i (, c) is nonnegive nondecresing coninuous funcion in c I, from (28) nd (29) i follows h A i (, b), i =, 2. Thus, υ () on I nd he ler conrdics he condiions on υ. Hence, (27) holds. In he sme wy, we jusify he ineuliy lim A c i (c, b) > lim A i (, c), i =, 2. (3) c b In view of (27) nd (3), he coninuiy nd monooniciy of A i (, c) nd A (c, b) in c I imly he eisence of oins c i i I such h A i (, c i ) = A (c i i, b), i =, 2. Necessiy: Le midoin c i I for ( A i, Ai), i =, 2, eis. The definiion of c i yields A i (, c i ) = A i (c i, b) <, i =, 2. If c c hen (24) imlies h su <<c lim su A (, c, ) = lim ρ = A (, c ) + lim c su ρ << υ()d ρ + lim c su ρ << c c υ()d c c. υ()d υ()d = A (, c ) <,

39 9 lim b su A su c <<b (c, b, ) = lim ρ su ρ b <<b c υ()d In he cse of c < c we similrly hve lim b su A A (c, b) + lim c υ()d = A (c, b) <. lim su A (, c, ) A (, c ) <, (c, b, ) = lim su ρ b <<b ρ In he cse of A 2 nd A 2 we hve c c υ()d c υ()d = A (c, b) <. lim su A 2 (, c, ) = lim su ρ << υ() ρ d su <<c 2 ρ for every c I nd similrly υ() ρ d = A 2 (, c 2 ) < lim su A 2 (c, b, ) A 2 (c 2, b) <. b Lemm is roven. Proof of Theorem. Necessiy: Le (2) hold on AC ( ) ρ, I wih he bes consn C = J (, b). Assume h < c < c < c + < b. Pu

40 c ρ ρ, < < c, f () =, c c +, ρ, c + < b. c + ρ The funcion f is loclly bsoluely coninuous on I nd (3) ρ(s) c f (s) ds = ρ(s) c+ f (s) ds + ρ(s) f (s) ds + = c ρ c c c ρρ ( ) + = ρ + ( ) Hence, f W ρ, I nd lim f () f () =, + by consrucion. In his cse f J (, b) The direc clculion yields υ() f () d = c c + c + ρ ρ c + c + lim f () f (b) = b b ρρ ( ) ρ(s) f (s) ds <. (32) AC ( ρ, I ). Insering f in (2), we hve υ() f () c+ d + υ() f () d ρ() f () d υ() f () d + b υ() f () d (33) = c ρ c c υ() ρ d c +

41 c+ + υ()d + c c + ρ By (32) - (34), we obin he ineuliies J (, b) c c ρ υ() c ρ + J (, b) ( ρ c + c + υ() ρ ρ c + ) ρ d υ() c + + c ρ b + c c + υ()d + c c c ρ + υ()d ρ c + d. (34) ρ c + ρ d, (35). (36) Mulilying he numeror nd denominor of he righ-hnd sides in c (35) nd (36) by ρ, we derive J (, b) c ρ + c c υ() ( b ρ J (, b) c ρ c + ρ ) d + c ρ b + c ρ c + b ρ υ() c + ρ c + ρ d (37) ρ c c υ()d + ρ c + υ()d c c c + b ρ. (38) c + ρ Since he lef-hnd sides of (37) nd (38) re indeende of c (, c), ssing o he limi s c on he righ-hnd sides, we infer,

42 2 lim su J (, b) + lim c c ( + lim c ρ b + lim c ) ρ c υ() b ρ ρ c + c b ρ υ() c + ( ρ c + ρ c + ρ ) d ρ d = lim su ρ υ() ρ d = lim su A (, c, ), (39) 2 J (, b) lim su = lim su ( ρ ) + lim ρ c c c c υ()d + lim c b ρ c ρ c + ρ υ()d υ()d = lim su A (, c, ). (4) Mulilying he numeror nd denominor of he righ-hnd sides in (35) nd (36) by nd ssing o he limi s c + b, we obin c + ρ lim c J + (, b) b ρ c + lim c + b c ρ c + c ρ υ() ( c ρ + ρ ) d

43 = lim b su + lim su b ρ lim c J + (, b) b = lim b su lim c + b b ρ c + ρ c + υ() ρ b υ() ρ d c ρ + ρ 3 d = lim b su ( A 2 (c, b, )), (4) c υ()d + lim su b ρ c lim c ρ + c + b ρ c + ρ c c υ()d υ()d = lim b su ( A (c, b, )). (42) Relions (39) - (42) ensure (26). By Lemm, here eis midoins c i I for ( A i, A i), i=,2. Definiion yields he euliy Ai (, c i ) = A i (c i, b) T ci (, b) <, i =, 2. Since A i (, c i, ) nd A i (c i, b, ) re coninuous in on (, c i ] nd [c i, b), resecively, nd A i (, c i ) lim su A i (, c i, ), A i (c i, b) lim b su A i (c i, b, ), he wo ossibiliies re oen: If A i (, c i ) = lim su A i (, c i, ) or A (c i i, b) = lim b su A (c i i, b, ) hen (39) - (42) imly he esime J (, b) T ci (, b), i =, 2, i.e., he lef r of (25) holds. Oherwise, here eis oins c, i ( c+, < i ) c c i i, c i c + < b, such h c c i, c + c, A i (, c i ) = A i, ci, c i, nd A (c ( ) i i, b) = A ci, b, c + i i. To jusify he lef esime in (25), we esime T c (, b) nd T c2 (, b) serely. Firs, we emine T c2 (, b). Le c = c 2 in (35). Esime (37) yields

44 4 J (, b) c 2 c ρ 2 c + 2 υ() ( ) ρ d c ρ 2 + ρ c + 2 ρ + c + 2 ρ c + 2 b c + 2 υ() c ρ 2 d ρ c ρ 2 + ρ ( ) ( ) (we e he eressions for A 2, c2, c 2 nd A 2 c2, b, c + 2 ino ccoun) = ( ( )) b A2, c2, c 2 c + 2 c + 2 ρ + ( A 2 ( )) c c2, b, c c ρ 2 + ρ ρ ( ) (by he definiion of c 2, we hve A 2 (, c 2 ) = A 2, c2, c 2 nd A (, 2 c 2) = ( ) A 2 c2, b, c + 2 ) = (A 2 (, c 2, )) b c + 2 c + 2 ρ + ( A (c 2 2, b) ) c 2 ρ c ρ 2 + ρ

45 5 (since c 2 is midoin for ( A 2, A 2) ) = ( T c2 (, b) ) c + 2 c + 2 ρ + c 2 ρ c ρ 2 + ρ 2 ( T c2 (, b) ). (43) Esime T c (, b). Similrly, uing c = c, c = c, nd c+ = c + 2 view of (38) we obin in (36), in J (, b) = c ρ c + 2 c c υ()d c + ρ c ρ 2 + ρ ( ( )) b A, c, c = c + 2 c + (A (, c, )) b c + 2 ρ + + ( A c + ρ c + 2 c υ()d ρ c + c c ρ 2 + ρ c ( )) c, b, c + c ρ 2 + ρ c + ρ ρ + ( A (c, b) ) c ρ c ρ 2 + ρ ρ = ( T c (, b) ) c + c ρ + c ρ 2 + ρ c ( T c (, b) ). (44)

46 6 The lef esime in (25) resuls from (43) nd (44). The necessiy is roven. Sufficiency: Assume he eisence of midoin c i I for (A i, A ), i =, 2. i In his cse we hve A i (, c i ) = A (c i i, b) = T ci (, b) <, i =, 2. Since f () = f (b) = for f AC ( ) ( ) ρ, I, he resricion of AC ρ, I o (, ci ) ( nd (c i, b) belongs o AC,l ρ, (, ci ) ) ( nd AC,r ρ, (ci, b) ), resecively. Theorem 8 imlies h ( γ i A i (, c i ) ) υ() f () d = c i ( γ i T ci (, b) ) ci ρ(s) f (s) ds c i υ() f () d + = ( γ i T ci (, c i ) ) b c i + ( γ i A i (c i, b) ) ρ(s) f (s) ds + b c i υ() f () d ρ(s) f (s) ds c i ρ(s) f (s) ds, ρ(s) f (s) ds i.e., (2) holds nd he bes consn C = J (, b) in (2) mees he esime J (, b) min { γ T c (, b), γ 2 T c2 (, b) }, which defines he righ-hnd side of (25). Theorem is roven. Remr. Theorem imroves he esime for J (, b) in []. For emle, in Theorem 8.8 of [], under he ssumion A (, ) = A (b, b) = (he ler is euivlen o he condiions lim A (, c, ) = nd lim b A (c, b, ) = ), i is esblished h 2 A J (, b) ( + ) ( ) + A, where A = inf <c<b m { A (, c), A (c, b)}. Under our condiions, i is esily seen h A = T (, b). Le c ρ (s)ds <, ρ (s)ds =, c I, (45) c

47 7 or c ρ (s)ds =, ρ (s)ds <, c I. (46) c Theorem 2. Le < <. If (45) or (46) holds hen he bes consn J (, b) in (2) mees he esime m {A (, b), A 2 (, b)} J (, b) min { γ A (, b),γ 2 A 2 (, b) } (47) m { A (, b), A 2 (, b)} J (, b) min { γ A (, b),γ 2A (, b)} (48) PROOF OF THEOREM 2. ( ) W ρ, I, Since AC ( ρ, I ) is dense everywhere in J (, b) = su f AC (ρ,i) υ() f () d ρ() f () d = su f W (ρ,i) υ() f () d. (49) ρ() f () d ( ) { Le (45) hold. In view of iem (ii) of Theorem A, W ρ, I = ( ) } ( ) f W ρ, I : f () = = AC,l ρ, I. Hence, J (I) = J l (I) nd (47) follows from Theorem B. By nlogy we jusify (48) in he cse (46). Theorem 2 is roven. Finlly, le c ρ (s)ds =, c ρ (s)ds =, c I. (5) Theorem 3. Assume h < < nd (5) is vlid. Then (2) fils on he se W ( ) ρ, I, i.e., J (, b) =. PROOF OF THEOREM 3. By condiion, (5) holds. By Theorem A (iem (iv)) W ( ) ( ) ( ) ρ, I = W ρ, I. Since f () W ρ, I, (49) yields J (, b) =. Theorem 3 is roven The cse of < < <.

48 8 Definiion 2. A oin c I is clled midoin for (B, B ) if B (, c) = B (c, b) T (, b) <. Obviously, for midoin for (B, B ) o eis, i is necessry nd sufficien h B (,β ) <, β I, nd B (α, b) <, α I. Theorem 4. Assume h < < <, >, nd (24) holds. Then (2) is fulfilled on AC ( ) ρ, I if nd only if here eiss midoin c I for (B, B ) ; in his cse he bes consn J (, b) in (2) sisfies he esime ( ) T (, b) J (, b) 2 μ + T (, b). (5) PROOF. Necessiy: Assume h < < <, >, nd (2) holds on ( ) AC ρ, I wih C = J (, b). Le <α<β<b. In view of he condiions on he weighed funcions υ nd ρ, he uniies B (α, c), c (α, b), B ( c,β ), nd c (,β ) re finie. Hence, here eiss midoin c = c ( α, β ) ( α, β ) for B ( α, β ) nd B ( α, β ), i.e., c α c υ α ρ Inroduce he funcion where f () = b α ( ) ρ ()d = α β c c υ β ρ ( ), < α, ( c ) ( ) ( ) υ ρ ρ ()d, α c, ( ) β ( ) β b 2 υ ρ ρ ()d, c β, c, β b, b = b 2 = c α β c c c υ υ ρ α β ρ ( ) ( ) ρ ()d, ρ ()d. ρ ()d. (52)

49 9 Obviously, f AC ( ) ρ, I. For funcion f we hve ρ() f () d = b (B(α, c)) + ( B b (c,β) ) 2 Since c = α f () b f c υ() f () d = ( f () ) c υ α for α c nd similrly for c β, we infer β c Hence, (54) yields c α α = ( T(α, β) ) b c α β υ(s)dsd + ρ + b 2 υ() ( β f () ) d + f () b f () ( f () ) c ( f ()) ( f () ) υ() f () d c c c, (53) υ() ( f () ) d ( f ()) ( f () ) ρ ()d = b c υ υ(s)dsd υ(s)dsd β ρ c c υ(s)dsd. (54) υ ( ) (B(α, c)), b ρ ( ) ( B (c,β) ). b 2 ( ) b (B(α, c)) + b (B (c,β)) 2 α

50 2 = [ ( ) ] Since >, we hve + b b 2 υ() f () d ( ) ( ) T(α, β) b ( b Relions (2), (53) nd (55) imly h + b. 2 + b 2 ). Hence, ( ) ( ) T(α, β) b + b 2. (55) ( ) T(α, β) J (, b). (56) The bsolue coninuiy of he inegrl ensures he coninuiy of T ( α, β ) in α nd β for α<β b. In view of he indeendence of he righ-hnd side (56) of α nd β, <α<β<b, we hve ( ) T(, b) J (, b), (57) i.e., here eiss midoin c I for (B, B ) nd (57) is rue. Sufficiency: Le midoin c I for (B, B ) eis, i.e., B (, c) = B (c, b) = T(, b) <. Arguing s in he sufficiency r of Theorem nd involing Theorem C, we derive h ( μ + B(, c) ) υ() f () d = c ( μ + T(, b) ) 2 c α ρ(s) f (s) ds c α υ() f () d + β + ( μ + B (c, b) ) ρ(s) f (s) ds + c β c υ() f () d c ρ(s) f (s) ds ρ(s) f (s) ds i.e., (2) is fulfilled nd J (, b) μ + 2 T(, b); he ls esime nd (57) ensure (5). Theorem 4 is roven.,

51 2 Remr 2. The comrision of (5) nd he esime ( ) 2 B J (, b) 2 ( ) B, where B = min <c<b m {B(, c), B(b, c)}, obined in he cse of < < in Theorem 8.7 of [], shows h he esime in (5) is beer hn h of []. Theorem 5. Le < < <, >. If (26) or (27) holds hen he bes consn J (, b) in (2) sisfies he esime μ B(, b) J (, b) μ + B(, b) or μ B (, b) J (, b) μ + B (, b), resecively. Theorem 6. Assume h < < <, >, nd (5) holds. Then (2) fils on AC ( ) ρ, I ; i.e., J (, b) =. Theorems 5 nd 6 re roven by nlogy wih Theorem 2 nd 3. References [] Oic B. nd Kufner A. Hrdy-Tye Ineuliies, Longmn, Hrlow (99). [2] Kufner A. nd Persson L.-E. Weighed Ineuliies of Hrdy Tye, Word Sci., New Jersey, London, Singore nd Hong Kong (23). [3] Kufner A., Mligrnd L., nd Persson L-E. The Hrdy Ineuliy: Abou Is Hisory nd Some Reled Resuls, Pilseń (27). [4] Gogishvili A., Kufner A., Persson L.-E., nd Wedesig A. An euivlence heorem for inegrl condiions reled o Hrdy s ineuliy, Rel Anl. Echnge, 29, (23/24), No.2, [5] Kufner A., Persson L.-E. nd Wedesig A. A sudy of some consns chrcerizing he weighed Hrdy ineuliy, in: Bnch Cener Publ. Vol.64. Orlicz Cenenry Volume, Polish Acd. Sci., Wrsw, 24, [6] Drbe P. nd Kufner A. Hrdy ineuliy nd roeries of he usiliner Surm-Liouville roblem, Rend. Lincei M. Al. 8 (27), [7] Oinrov R. nd Rhimov S.Y. Weighed Hrdy ineuliies nd heir licion o oscillion heory of hlf-liner differenil euions, Eursin Mh. J. (2), No.2, -24. [8] Oinrov R. nd Rhimov S.Y. Oscillion nd nonoscillion of wo erms liner nd hlf-liner euions of higher order, Elecron. J. Qul. Theory Differ. Eu. 49 (2), -5.

52 22 [9] Kudbyev S.Y. nd Oinrov R. The crieri of disconjuge of hlfliner second order differenil euions, Mh. J. Kzhsn, (2), No.2, [] Lizorin P.I. On he closure of he se of comcly suored funcions in he weighed sce W, l, Dol. Ad. Nu SSSR, 239 (978), No.4, [] Kudryvsev L.D. On he densiy of comcly suored funcions in weighed sces, Sovie Mh. Dol. 9 (978), No., [2] Oinrov R. On weighed norm ineuliies wih hree weighs, J. London Mh. Soc. 48 (993) No.2, 3-6. [3] Persson L.-E. nd Senov V.D. Weighed inegrl ineuliies wih he geomeric men oeror, J. Ineul. Al. 7 (22), No.5, [4] Sinnmon G. nd Senov V.D. The weighed Hrdy ineuliy: new roofs nd he cse =, J.Mh. Soc. 54 (996), No.2, 89-. A.M. Abylyev; A.O. Birysnov; R.Oinrov L.N. Gumilyov Eursin Nionl Universiy, Asn, Kzhsn E-mil ddress: bylyev b@mil.ru; osr 62@mil.ru; o rysul@mil.ru

53 Per 2 Boundedness, comcness for clss of frcionl inegrion oerors of Weyl ye. Eursin Mhemicl Journl 7 (26), No., Remr: The e is he sme bu he form hs been modified o fi he syle in his PhD hesis.

54

55 EURASIAN MATHEMATICAL JOURNAL ISSN Volume 7, Number (26), 9 27 BOUNDEDNESS, COMPACTNESS FOR A CLASS OF FRACTIONAL INTEGRATION OPERATORS OF WEYL TYPE A.M. Abylyev Communiced by E.D. Nursulnov Key words: frcionl inegrion oeror, Weyl oeror, Riemnn- Liouville oeror, Hdmrd oeror, Erdelyi-Kober oeror, boundedness, comcness. AMS Mhemics Subjec Clssificion: 26A33, 26D, 47G. Absrc. We esblish crieri for he boundedness nd comcness for clss of oerors of frcionl inegrion involving he Weyl oeror. Inroducion Le I = (, b), < b,<, <, + =. Le u, v be lmos everywhere osiive nd loclly inegrble funcions on I. ByL,u L (u, I) we denoe he se of ll mesurble funcions f on I such h f,u = f () u()d <. In he cse u we wrie L L (I). Le W be osiive sricly incresing nd loclly bsoluely coninuous funcion on I. Suose dw() w() for d lmos everywhere I. Le >α>. We consider he oeror u(s)w β (s) f (s)w(s)ds K α,β f () =, I. () α (W(s) W()) In he cse β =, u he dul oeror o oeror () hs he form K α,β f () = f (s)w(s)ds, I. (2) α (W() W(s))

56 2 A.M. Abylyev Oeror (2) is clled [5] he oeror of frcionl inegrion of he funcion f of he funcion W. Weighed esimes for oeror (2) were reviously considered in [2], []. When W() =, u, β = oeror () is he Weyl oeror I α f () = which is dul o he Riemnn-Liouville oeror I α g(s) = s f (s)ds, (s ) α I, (3) g()d, (s ) α s I. (4) Oerors (3) nd (4) cing from he weighed sce L,u o he weighed sce L,v re invesiged in ers [3], [3], [7], [5], [2], [2] nd ohers, where necessry nd sufficien condiions for heir boundedness, comcness re obined for vrious relions beween he rmeers α,, nd under vrious ssumions regrding he weigh funcions u nd v. Two-sided esimes of heir norms re lso obined. We invesige oeror () cing from he sce L,w o L,v. From he obined resuls new sserions follow, in simle erms, for oerors (3) nd (4), generlizing he resuls of [7], [5], [2]. The osiiviy nd monooniciy of W imlies he eisence of he nonnegive limi lim + W() W(). Fuher, we ssume W() = nd oherwise, we consider he oeror K α,β in he form, where funcion W() is relced by he funcion W () = W() W(), I. Furher, he norm of he liner oeror T from normed sce o noher one is denoed briefly by T. Which sces re men will be cler from he cone. Throughou he er he roducs of he form re suosed be eul o zero. Relions A B, A B men A cb wih consn c deending only on,, α which cn be differen in differen lces. If A B nd A B hen we wrie A B. By Z we denoe he se of ll ineger numbers, χ E denoes he chrcerisic funcion of he se E. 2 Auiliry sserions To rove he min resuls we need some well-nown sserions. Along wih oeror () we consider he Hrdy oeror

57 Boundedness, comcness of clss frcionl inegrion oeror of Weyl ye 3 H α,β f () = u(s)w β+α (s) f (s)w(s)ds. () I is esy o see h for f K α,β f () H α,β f (), I. (2) Issues of boundedness nd comcness of oeror () in weighed Lebesgue sces were sudied uie comleely. A summry of he resuls cn be found in [5]. The following Theorem A nd Theorem B re corollries of Theorem 5 nd Theorem 6 in [5]. Theorem A. Le < <. The oeror H α,β is bounded from L,w o L,v if nd only if z A α,β = su v()d z I b u (s)w (α+β ) (s)w(s)ds <. Moreover, H α,β A α,β. z Theorem B. Le < < <, >. The oeror H α,β is bounded from L,w o L,v if nd only if B α,β = z u (s)w (α+β ) w(s)ds ( ) ( z v()d ) v(z)dz <. Moreover, H α,β B α,β. Remr 2.. In he cse < < <,> he vlue B α,β is euivlen o he vlue B α,β (, b) = z u (s)w (α+β ) (s)w(s)ds ( z v()d ) ( ) u (z)w (α+β ) (z)w(z)dz.