GENERALIZED FRACTIONAL INTEGRAL OPERATORS AND THEIR MODIFIED VERSIONS

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1 GENERALIZED FRACTIONAL INTEGRAL OPERATORS AND THEIR MODIFIED VERSIONS HENDRA GUNAWAN Absac. Associaed o a fucio ρ :(, ) (, ), le T ρ be he opeao defied o a suiable fucio space by T ρ f(x) := f(y) dy, R ad T ρ be he modified vesio of T ρ give by ( T ρ f(x) := R ρ( y )( χ ) B (y)) y f(y) dy. Fo ρ() = α, <α<, he opeao T ρ is ohig bu he facioal iegal opeao o he Riesz poeial, which is kow o be bouded fom L p (R ) o L q (R ) povided ha /p /q = α/. Nex, fo p< ad a fucio φ :(, ) (, ), we defie he geealized Moey space M p φ = Mp φ (R ) by ( /p M p φ {f := L p loc : sup } ) p < B φ(b) B B ad he geealized Campaao space L p φ = Lp φ (R ) by L p φ := {f L p loc : sup B φ(b) ( B B ) /p } f(y) f B p dy <, whee he supemum is ake ove all ope balls B = B(a, ) i R, B deoes he Lebesgue measue of B, φ(b) =φ(), ad f B is he aveage of f ove B. I his alk, we discuss he boudedess of T ρ ad T ρ o geealized Moey spaces ad o geealized Campaao spaces, especively. Ude some codiios o ρ, φ, ad ψ, we pove ha T ρ is bouded fom M p φ o Mq ψ, while T ρ is bouded fom L p φ o L q ψ fo <p<q<. Relaed esuls wee poved ealie by E. Nakai [8]. Some of he esuls peseed hee is joi wih Eidai ad E. Nakai, ad has bee published ecely i [4].. Ioducio Fo <α<, he (classical) facioal iegal opeao o he Riesz poeial I α, defied by I α f(x) = R f(y) dy, α 2 Mahemaics Subjec Classificaio. 42B35, 26A33, 46E3, 42B2, 43A5. Key wods ad phases. Facioal iegals, Moey spaces, Campaao spaces.

2 is kow o be bouded fom L p (R )ol q (R ) povided ha /p /q = α/, < p<q<. The associaed iequaliy I α f L q C p,q f L p is kow as he Hady-Lilewood-Sobolev iequaliy (see [2], p. 354). Lae, i [, ], i is show ha I α exeds o a bouded opeao fom he Moey space E p,β (R )oe q,γ (R ) whee /p /q = α/, α + β = γ, /p β<α. The Moey space E p,β (R ) is defied o be he se of all locally iegable fucios f o R fo which sup B ( /p ) p <, β B B whee he supemum is ake ove all balls B = B(a, ) ir ad B deoes he Lebesgue measue of B. The ealie esul ca be ecoveed fom he lae by akig β = /p, because E p,β (R )=L p (R ). A fuhe exesio of he above esul is obaied by E. Nakai [7], who showed ha I α is bouded fom he geealized Moey space M p,φ (R )om q,ψ (R ) fo /p /q = α/, <p<q< ad appopiae fucios φ ad ψ wih ψ() = α φ(). Hee he geealized Moey space M p,φ = M p,φ (R ) is defied by whee M p,φ := {f L p loc (R ): f Mp,φ < } f Mp,φ := sup B ( /p ) p. φ() B B Now he classical esul ca be ecoveed fom Nakai s by akig φ() = /p. I his oe, we shall discuss he geealized facioal iegal opeao T ρ, defied fo a suiable fucio ρ :(, ) (, ) by T ρ f(x) := R f(y) dy, wheeve his iegal makes sese. I paicula, we ae ieesed i he boudedess of T ρ fom he geealized Moey space M p,φ o M q,ψ. Noe ha fo ρ() = α, < α<, we have T ρ = I α he facioal iegal opeao meioed ealie. 2

3 Nex, we shall also pese some esuls fo a modified vesio T ρ, deoed by T ρ, which is defied by he fomula ( T ρ f(x) := f(y) R ρ( y )( χ ) B (y)) dy, y whee B is he ui ball aoud he oigi ad χ B is he chaaceisic fucio of B.Foρ() = α, he opeao T ρ = Ĩα is well-defied fo <α<+, ad is kow o be bouded fom L p o BMO whe p> ad α = /p, fom L p o Lip β whe p> ad <α /p = β<, fom BMO o Lip α whe <α<, ad fom Lip β o Lip γ whe <α+β = γ< (see [8] ad fuhe efeeces heei). Ou iees hee will be he boudedess of T ρ fom he geealized Campaao space L p,φ o L q,ψ. The space L p,φ = L p,φ (R ) is defied o be he se of all fucios f L p loc (R ) fo which ( /p f Lp,φ := sup f(y) f B dy) p <, B φ() B B whee f B deoes he aveage of f ove B, ha is, f B := f(y)dy. B B Noe ha fo he space M p,φ, he fucio φ() is usually equied o be oiceasig ad φ p () o be odeceasig, while fo he space L p,φ, i is φ() ha is equied o be oiceasig. The geealized facioal iegal opeao T ρ ad is modified vesio T ρ wee fis sudied by Nakai [8]. Some exesios of Nakai s esuls wee obaied by Eidai [2], Eidai ad Guawa [3], Guawa [5], ad Eidai, Guawa, ad Nakai [4]. Relaed esuls may also be foud i Kuaa e.al. [6]. Resuls peseed hee ae summaized fom [4, 5]. Thoughou his oe, C, C i,c p ad C p,q will deoe posiive cosas, which may vay fom lie o lie. 2. Pelimiaies I he defiiio of T ρ, we assume ha he fucio ρ saisfies he followig codiios: (2.) ρ() d < ; (2.2) 2 s 2 C ρ() ρ(s) C. 3

4 Meawhile, i he defiiio of T ρ, we assume ha ρ saisfies (2.) ad (2.2) ad he followig wo addiioal codiios: (2.3) ρ() 2 d C 2 ρ() (2.4) 2 2 s ρ() fo all >; ρ(s) C s 3 s ρ(s). s + Fo example, he fucio ρ() = α, <α<, saisfies (2.), (2.2) ad (2.4). If <α<, he ρ() = α also saisfies (2.3). A fucio ρ saisfyig (2.2) is said o saisfy he doublig codiio (wih a doublig cosa C ). If ρ saisfies he doublig codiio, he oe may obseve ha 2 k+ ρ() d ρ(2 k ) 2 k fo evey iege k ad >. Fuhe, i follows fom he doublig codiio ha ρ() ρ() C d, fo evey >. Nex, if ρ saisfies (2.) (2.4), he we have ( ρ() R x y ρ( x ) 2 y ) dy = x 2 y fo evey choice of x ad x 2 (see [8]). Fo such a fucio ρ, we see ha he opeao T ρ maps a cosa o a cosa, ad so T ρ is well-defied fom oe geealized Campaao space o aohe. I he ex secio, we shall ivolve he so-called Hady-Lilewood maximal opeao M, which is defied by Mf(x) := sup. B x B B A classical esul fo M is ha i is bouded o L p fo <p (see e.g. [2]). I [7], Nakai showed ha if φ saisfies he doublig codiio ad (2.5) φ() p d Cφ() p fo all >, fo some <p<, he hee exiss C p > such ha Mf Mp,φ C p f Mp,φ, ha is, M is bouded o M p,φ. 4

5 I ou appoach, we shall also ivolve Youg fucios ad Olicz spaces. A fucio Φ:[, ] [, ] is called a Youg fucio if Φ is covex, lim Φ() = Φ() = + ad lim Φ() =Φ( ) =. Noe ha a Youg fucio is always odeceasig. Give a Youg fucio Φ, we defie Φ () = if{s : Φ(s) >} (wih if = ). If Φ is coiuous ad bijecive, he Φ is ohig bu he usual ivese fucio. If a Youg fucio Φ saisfies (2.6) < Φ() < fo <<, he Φ is coiuous ad bijecive fom [, ) o iself. I his case, he ivese fucio Φ is iceasig, coiuous ad cocave, ad hece saisfies he doublig codiio. Fo a Youg fucio Φ, we defie he Olicz space L Φ = L Φ (R ) o be he se of all locally iegable fucio f o R fo which Φ ( f(x) ) dx < R ɛ fo some ɛ>. Hee L Φ is equipped wih he om { ( ) } f(x) f L Φ := if ɛ>: dx. ɛ R Φ Noe ha fo Φ() = p, p<, we have L Φ = L p. Fo fuhe popeies of Youg fucios ad Olicz spaces, see e.g. []. Fo hei elevace wih ou subjec, see [8, 9]. 3. The boudedess of T ρ I [9], Nakai poved ha T ρ is bouded fom M,φ o M,ψ, ude appopiae codiios o φ ad ψ, paiculaly he assumpio ha φ() ρ() d + ρ()φ() d Cψ(), fo all >. Lae, Eidai [2] showed ha T ρ is bouded fom M p,φ o M p,ψ fo <p<, ude simila assumpios o ρ, φ ad ψ. Noe, howeve, ha we cao ecove he kow esuls fo I α fom hese esuls. 5

6 Recely, Eidai ad Guawa [3] poved ha I ρ is bouded fom M p,φ o M p,φ p/q fo <p<q<, ude some assumpios o ρ ad φ. Pecisely, hey poved he followig heoem. Theoem 3. [3]. Suppose ha ρ is sujecive ad saisfies he doublig codiio. Suppose also ha φ saisfies he doublig codiio, (2.5), ad ρ() d + ρ() q/(q p) ρ()φ() d Cρ(), fo all >. The hee exiss C p,q > such ha T ρ f Mq,φ p/q C p,q f Mp,φ ha is, T ρ is bouded fom M p,φ o M q,φ p/q, fo <p<q<. Alhough Theoem 3. geealizes he esul fo I α, he assumpios o ρ ad φ seem o be diffee fom hose made by Nakai [9]. The followig heoem seves as a lik bewee Eidai ad Guawa s ad Nakai s esuls. Theoem 3.2 [5]. Suppose ha ρ ad φ saisfies he doublig codiio. Suppose also ha φ is sujecive, saisfies (2.5) ad φ() ρ() d + ρ()φ() d Cφ() p/q, fo all >. The hee exiss C p,q > such ha T ρ f Mq,φ p/q C p,q f Mp,φ ha is, T ρ is bouded fom M p,φ o M q,φ p/q, fo <p<q<. Poof [5]. Fo evey x R ad R>, we wie T ρ f(x) = f(y) dy + f(y) dy = I (x)+i 2 (x). x y <R x y R 6

7 Fo I (x), we have I (x) C x y <R k= k= CMf(x) CMf(x) = CMf(x) 2 k R x y <2 k+ R ρ(2 k R) (2 k R) x y <2 k+ R ρ(2 k R) k= 2 k+ R k= R ρ() 2 k R d CMf(x) φ(r) (p q)/q. ρ() d Meawhile, fo I 2 (x), we have I 2 (x) C C x y R k= k= k= 2 k R x y <2 k+ R ρ(2 k R) (2 k R) C f Mp,φ C f Mp,φ ρ(2 k R) (2 k ) /p x y <2 k+ R ( k= 2 k+ R k= x y <2 k+ R ρ(2 k+ R)φ(2 k+ R) 2 k R ρ()φ() = C f Mp,φ R C f Mp,φ φ(r) p/q. 7 ρ()φ() d ) /p f(y) p dy d

8 Summig he wo esimaes fo I ad I 2, we ge T ρ f(x) C[Mf(x) φ(r) (p q)/q + f Mp,φ φ(r) p/q ]. Sice φ is sujecive, we ca choose R>such ha φ(r) =Mf(x). f M p,φ, assumig ha f is o ideically ad ha Mf is fiie eveywhee. Hece, fo evey x R, we have T ρ f(x) q CMf(x) p f q p M p,φ. Fom his ad he boudedess of he maximal opeao M o M p,φ, we obai he desied iequaliy. (QED) The ex heoem is aohe geealizaio of he kow esuls fo I α (see [4] fo is poof). Theoem 3.3 [4] Suppose ha ρ saisfies (2.) ad (2.2). Suppose fuhe ha ρ() ad /p ρ() d ae almos deceasig, ρ() /p d C /p exis Youg fucios Φ saisfyig (2.6) ad Φ 2 such ha /p ρ() d Φ ( ) ad Φ ( )Φ 2 ( ) /q fo <p q<. Ifφ saisfies he doublig codiio ad φ() ρ() d + ρ()φ() he T ρ is bouded fom M p,φ o M q,ψ. ρ() d Cψ(), fo all >, d, ad hee Noe. A fucio θ : R + R + is said o be almos deceasig if hee exiss a cosa C> such ha θ() Cθ(s) fo s. 4. The boudedess of T ρ o Campaao spaces We ow u o he modified facioal iegal opeao T ρ. I [8, 9], Nakai poved ha T ρ is bouded fom L,φ o L,ψ fo appopiae fucios φ ad ψ. Foφ() = β wih β, he space L,φ educes o BMO (whe β = ) o Lip β (whe <β ). I his case, Nakai s esul coves he BMO Lip α ad Lip β Lip γ esuls fo Ĩα. Fo φ() = β wih /p β<, <p<, we have Eidai s esul 8

9 [2] which coves he ohe esuls fo Ĩα. The followig heoem is a exesio of Eidai s (see [4] fo is poof). Theoem 4. [4] Suppose ha ρ saisfies (2.) (2.4), ad ha φ saisfies he doublig codiio ad φ() d <. If φ() d ρ() d + he T ρ is bouded fom L p,φ o L p,ψ fo <p<. ρ()φ() 2 d Cψ() fo all >, The esuls fo T ρ idicae ha he modified facioal iegal opeao T ρ mus also be bouded fom L p,φ o L q,ψ fo <p q< ad appopiae fucios φ ad ψ. Ideed, we have he followig aalog of Theoem 3.3 fo T ρ. Theoem 4.2 [4] Suppose ha ρ saisfies (2.) (2.4). ad /p ρ() d ae almos deceasig, ρ() /p Suppose fuhe ha ρ() d C /p exis Youg fucios Φ saisfyig (2.6) ad Φ 2 such ha /p ρ() d Φ ( ) ad Φ ( )Φ 2 ( ) /q fo <p q<. Ifφ saisfies he doublig codiio ad φ() ρ() d + he T ρ is bouded fom L p,φ o L q,ψ. ρ() ρ()φ() 2 d Cψ(), fo all >, d, ad hee 5. Cocludig emaks Though ou wok we have bee able o exed he kow esuls fo he classical facioal iegal opeao I α ad is modified vesio Ĩα o he boudedess of T ρ o Moey spaces ad ha of T ρ o Campaao spaces. Ou esuls o oly cove he kow esuls fo I α, bu also eich he class of fuios of ρ, φ ad ψ fo which he opeao T ρ is bouded fom he Moey space M p,φ o M q,ψ, ad he epeao T ρ is bouded fom he Campaao space L p,φ o L q,ψ. To give a example, le <p<q<. Takeρ() = α l() β, whee α = /p /q, β>, ad l() = / log fo small ad l() = log fo lage, so ha ρ saisfies he 9

10 doublig codiio. The ρ() d ρ() (see [8]). Now ake φ() = /p l() βq/(p q). The φ() (p q)/q = ρ(), ad oe may check ha ρ ad φ saisfy he assumpios i Theoem 3.2. Hece he associaed opeao T ρ is bouded fom M p,φ o M q,φ p/q. Fuhe examples ha suppo ou esuls ca be foud i [4]. Refeeces [] F. Chiaeza ad M. Fasca, Moey spaces ad Hady-Lilewood maximal fucio, Red. Ma. 7 (987), [2] Eidai, O he boudedess of a geealized facioal iegal o geealized Moey spaces, Tamkag J. Mah. 33 (22), [3] Eidai ad H. Guawa, O geealized facioal iegals, J. Idoesia Mah. Soc. (MIHMI) 8(3) (22), [4] Eidai, H. Guawa ad E. Nakai, O geealized facioal iegal opeaos, Sci. Mah. Jp. 6 (24), [5] H. Guawa, A oe o he geealized facioal iegal opeaos, J. Idoesia Mah. Soc. (MIHMI) 9() (23), [6] K. Kuaa, S. Nishigaki ad S. Sugao, Boudedess of iegal opeaos o geealized Moey spaces ad is applicaio o Schödige opeaos, Poc. Ame. Mah. Soc. 28 (999), [7] E. Nakai, Hady-Lilewood maximal opeao, sigula iegal opeaos ad he Riesz poeials o geealized Moey spaces, Mah. Nach. 66 (994), [8] E. Nakai, O geealized facioal iegals, Taiwaese J. Mah. 5 (2), [9] E. Nakai, O geealized facioal iegals o he weak Olicz spaces, BMO φ, he Moey spaces ad he Campaao spaces, Fucio spaces, iepolaio heoy ad elaed opics (Lud, 2), 389-4, de Guye, Beli, 22. [] J. Peee, O he heoy of L p,λ spaces, J. Fuc. Aal. 4 (969), [] M. M. Rao ad Z. D. Re, Theoy of Olicz spaces, Macel Dekke, Ic., New Yok, 99. [2] E. M. Sei, Hamoic Aalysis: eal vaiable mehods, ohogoaliy, ad oscillaoy iegals, Piceo Uivesiy Pess, Piceo, New Jesey, 993. Depame of Mahemaics, Badug Isiue of Techology, Badug 432, Idoesia. addess: hguawa@ds.mah.ib.ac.id