ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS. ( ρ( x y ) T ρ f(x) := f(y) R x y n dy, R x y n ρ( y )(1 χ )

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1 Scieiae Mahemaicae Japoicae Olie, Vol., 24), ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS ERIDANI, HENDRA GUNAWAN 2 AND EIICHI NAKAI 3 Received Augus 29, 23; evised Apil 7, 24 Absac. We pove he boudedess of he geealized facioal iegal opeaos ad hei modified vesios o Moey spaces ad o Campaao spaces especively. Ou appoach ivolves he Hady-Lilewood maximal fucio ad Youg fucios.. Ioducio Le R + :=, ). Associaed o a fucio ρ : R + R +, we defie he mappig f T ρ f by ρ x y ) T ρ fx) := fy) R x y dy, fo ay suiable fucio f o R. We also defie is modified vesio T ρ by ρ x y ) T ρ fx) := fy) R x y ρ y ) χ ) y)) y dy, whee is he ui ball aoud he oigi ad χ is he chaaceisic fucio of. Fo example, if ρ) = α, <α<, he T ρ = I α he facioal iegal opeao o he Riesz poeial. Hece T ρ may be viewed as a geealizaio of he facioal iegal opeao. Nex, fo p< ad a suiable fucio φ : R + R +, we defie he geealized Moey space M p,φ = M p,φ R ) o be he se of all fucios f L p loc R ) fo which /p f Mp,φ := sup fy) dy) p <, φ) ad he geealized Campaao space L p,φ = L p,φ R ) o be he se of all fucios f L p loc R ) fo which f Lp,φ := sup /p fy) f dy) p <. φ) Hee he supemums ae ake ove all ope balls = a, ) ir, deoes he Lebesgue measue of i R, φ) = φ), ad f := fy)dy. Fo M p,φ, he fucio φ) is usually equied o be oiceasig ad φ p ) o be odeceasig. Fo L p,φ,iis φ) ha is equied o be oiceasig. Oe may obseve ha f belogs o L p,φ if hee exis a cosa C< ad, fo evey ball, a cosa c < such ha /p fy) c dy) p <C, φ) 2 Mahemaics Subjec Classificaio. 4235, 26A33, 46E3, 422, 43A5. Key wods ad phases. Facioal iegals, Campaao spaces, Moey spaces.

2 38 ERIDANI, HENDRA GUNAWAN AND EIICHI NAKAI fo we he have f Lp,φ < 2C. Accodigly, M p,φ L p,φ. Fuhe, if p q<, he M p,φ M q,φ ad L p,φ L q,φ. Ulike MO he space of ouded Mea Oscillaio fucios), he Campaao space L p,φ is geeally depede of he expoe p see [] o [4]). Fo ceai fucios φ, M p,φ ad L p,φ educe o some classical spaces. Fo a bief hisoy of hese spaces, see [], whee fuhe efeeces ae lised. Fo ece applicaios, see e.g. [6]. I [8, 9], Nakai showed ha T ρ is bouded fom M,φ o M,ψ, while T ρ is bouded fom L,φ o L,ψ, ude some appopiae codiios o ρ, φ ad ψ. I [3], Eidai showed ha, fo <p<, T ρ is bouded fom M p,φ o M p,ψ, while T ρ is bouded fom M p,φ o L p,ψ, ude simila codiios o ρ, φ ad ψ. I his pape, we pove ha, ude some ohe codiios o ρ, φ ad ψ, he opeao T ρ is bouded fom M p,φ o M q,ψ, while T ρ is bouded fom L p,φ o L q,ψ, fo <p q<. Relaed esuls may be foud i a ece wok of Sugao ad Taaka [2]. 2. asic assumpios ad facs Le us begi wih a few assumpios, paiculaly o he associaed fucio ρ, ad some eleva facs ha follow. Heeafe, C, C i, C p ad C p,q deoe posiive cosas, which ae o ecessaily he same fom lie o lie. I he defiiio of T ρ, we always assume ha ρ saisfies he followig codiios: ρ) 2.) d < ; 2.2) 2 s 2 C ρ) ρs) C. Fo T ρ, we assume ha ρ also saisfies wo addiioal codiios, amely: 2.3) ρ) 2 d C 2 ρ) 2.4) 2 s 2 ρ) fo all >; ρs) s C 3 s ρs) 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 fo evey iege k ad >we have 2 k+ ρ) d ρ2 k ). 2 k Fuhe, i follows fom he doublig codiio ha ρ) ρ) C d, fo evey >. Nex, if ρ saisfies 2.) 2.4), he we have Nakai s lemma which saes ha R ρ x y ) x y ρ x ) 2 y ) x 2 y dy = fo evey choice of x ad x 2 see [8]). Fo such a fucio ρ, he opeao T ρ maps a cosa o a cosa, ad hece i is well-defied fom oe geealized Campaao space o aohe.

3 ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS 39 I he ex secio, we shall ivolve he so-called Hady-Lilewood maximal opeao M, which is defied by Mfx) := sup fy) dy. x A classical esul fo M is ha i is bouded o L p fo <p see e.g. [3]). Now, if φ saisfies he doublig codiio ad φ) 2.5) 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 see [7]). We shall also ivolve Youg fucios ad Olicz spaces i ou discussio. A fucio Φ:[, ] [, ] is called a Youg fucio if Φ is covex, lim Φ) = Φ) = ad + Φ) =Φ ) =. A Youg fucio is always odeceasig. Fo a Youg fucio lim Φ, we defie Φ ) = if{s :Φs) >} wih if = ). If Φ is coiuous ad bijecive, he Φ is 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 R Φ fx) ) ɛ dx < fo some ɛ>. We equip L Φ wih he om { ) } fx) f L Φ := if ɛ>: Φ dx. R ɛ 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]. Oe moe emiology. A fucio θ : R + R + is said o be almos deceasig if hee exiss a cosa C>such ha θ) Cθs) fo s. Almos iceasig fucios ca be defied aalogously. 3. The boudedess of T ρ o Moey spaces We shall hee coside he geealized facioal iegal opeao T ρ. Fo <p<q<, i is well-kow ha he facioal iegal opeao I α is bouded fom L p o L q povided ha α/ =/p /q see e.g. [3], p. 354). Moe geeally, I α is bouded fom he Moey space L p,λ o L q,µ whee α/ =/p /q, λ< αp ad pµ = qλ. I ou oaio, L p,λ = M p,φ wih φ) = λ )/p.) This esul is due o Spae see [], Theoem 5.4) ad is epoved by Chiaeza ad Fasca [2]. Acually, Chiaeza ad Fasca obaied a soge esul saig ha I α is bouded fom L p,λ o L q,λ whee α/ λ) =/p /q ad <λ< αp, fom which Spae s esul follows as a coollay. Thei poofs ae valid fo he case λ =.) The classical esul ca be ecoveed fom Spae s by akig λ = because L p, = L p ). A fuhe geealizaio of he above esul is obaied by Nakai [7], who showed ha I α is bouded fom M p,φ o M q,ψ fo appopiae fucios φ ad ψ) = α φ). Hee Spae s esul ca be ecoveed fom Nakai s by akig φ) = λ )/p wih λ< αp ad α/ =/p /q. Fo T ρ, we have he esuls of Nakai [9] ad Eidai [3] meioed ealie. While T ρ is a geealizaio of I α, hese esuls fo T ρ cao, ufouaely, be viewed as a aual

4 3 ERIDANI, HENDRA GUNAWAN AND EIICHI NAKAI geealizaio of hose fo I α i he sese ha we cao ecove he L p L q beoudedess of I α fom hem). Recely, Eidai ad Guawa [4] obais a geealizaio of Chiaeza-Fasca s esul, which has bee efomulaed by Guawa [5] as follows. Noice ha Chiaeza-Fasca s esul ca be ecoveed by akig ρ) = α ad φ) = λ )/p wih λ<ad α/ λ) =/p /q. Theoem 3.. [4, 5] Suppose ha ρ ad φ saisfies he doublig codiio. Suppose also ha φ is sujecive ad saisfies he iequaliy 2.5) ad φ) ρ) d + ρ)φ) d Cφ) p/q, fo <p<q<. The hee exiss C p,q > such ha T ρ f C Mq,φ p/q p,q f Mp,φ ha is, T ρ is bouded fom M p,φ o M q,φ p/q. fo all >, Skech of Poof. The idea is o spli he iegal io wo pas, amely ρ x y ) T ρ fx) = fy) x y <R x y dy + ρ x y ) fy) x y R x y dy = I x)+i 2 x). The we esimae each pa, by decomposig he iegal fuhe, diadically. Fo I x), we use he hypoheses o ρ ad φ ad he popey of he Hady-Lilewood maximal opeao M o ge I x) CMfx) φr) p q)/q. Fo I 2 x), we use he hypoheses o ρ ad φ ad he fac ha f M p,φ o obai I 2 x) C f Mp,φ φr) p/q. y he sujeciviy of φ, we ca choose R> such ha φr) =Mfx). f M p,φ, assumig ha f is o ideically ad ha Mfx) < fo evey x R. Wih his value of φr), he wo esimaes equal ad hece, fo evey x R, we have T ρ fx) q CMfx) p f q p M p,φ. The desied iequaliy he follows fom his ad he fac ha he maximal opeao M is bouded o M p,φ. QED) Ou ew esul fo T ρ is he followig heoem, which may be cosideed as a geealizaio of Spae s esul. Theoem 3.2. Suppose ha ρ saisfies 2.) ad 2.2). Suppose fuhe ha ρ) ad /p ρ) d ae almos deceasig, ρ) /p d C /p ρ) d, ad hee 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 M p,φ o M q,ψ. ρ)φ) d Cψ), fo all >,

5 ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS 3 Poof. Le = a, ) be ay ball i R ad = a, 2). Fo evey x R, wie ρ x y ) T ρ fx) = fy) x y dy + ρ x y ) fy) x y dy = I x)+ix). 2 C To esimae I, we se f = fχ. The we have /q ) /q I dx) x) q = T ρ fx)χ x) q dx C T ρ f LΦ χ LΦ2 R see []). u T ρ is bouded fom L p o L Φ see Coollay 3.2 of [8]) ad χ L Φ 2 Φ 2 ) ). Hece we obai /q Ix) dx) q C f L p Φ 2 ) ) C /p φ) f Mp,φ Φ ) /q) C /q ρ) φ) f Mp,φ d C /q ψ) f Mp,φ. Now we esimae I 2. Obseve ha fo evey x we have I 2 x) ρ x y ) fy) x y x y dy. Hece, as i [3], we obai Ix) 2 ρ)φ) C f Mp,φ d Cψ) f Mp,φ, whece /q I dx) 2 x) q C /q ψ) f. Mp,φ Combiig he wo esimaes, we ge he desied iequaliy fo T ρ. QED) 4. The boudedess of T ρ o Campaao spaces We ow u o he modified facioal iegal opeao T ρ.foρ) = α, he opeao T ρ = Ĩα is well-defied fo <α<+ ad is kow o be bouded fom L p o MO whe p> ad α = /p, fom L p o Lip β whe p> ad <α /p = β<, fom MO o Lip α whe <α<, ad fom Lip β o Lip γ whe <α+ β = γ<. Fo a geeal fucio ρ, Nakai [8, 9] poved ha T ρ is bouded fom L,φ o L,ψ fo appopiae fucios φ ad ψ. Fo φ) = β wih β, he space L,φ educes o MO whe β = ) o Lip β whe <β ). I his case, Nakai s esul coves he MO Lip α ad Lip β Lip γ esuls fo Ĩα. Fo φ) = β wih /p β<, <p<, we have Eidai s esul [3] which coves he ohe esuls fo Ĩα. The followig heoem is a exesio of Eidai s. Theoem 4.. Suppose ha ρ saisfies 2.) 2.4), ad ha φ saisfies he doublig codiio ad φ) d <. If φ) ρ) ρ)φ) d d + 2 d Cψ) fo all >,

6 32 ERIDANI, HENDRA GUNAWAN AND EIICHI NAKAI he T ρ is bouded fom L p,φ o L p,ψ fo <p<. Poof. Le f L p,φ. Fo ay ball = a, ) ir, le = a, 2) ad, fo evey x, wie ) ρ x y ) ρ a y ) χ I x) := fy) f ) y)) R x y a y dy I x) := ρ x y ) fy) f ) x y dy ) ρ x y ) Ix) 2 ρ a y ) := fy) f ) x y a y dy C ρ a y ) χ C R y)) := fy) f ) a y ρ y ) χ ) y)) y dy ρ x y ) C 2 := f R x y ρ y ) χ ) y)) y dy. The clealy T ρ fx) C + C2 )=I x) =I x)+i2 x), ad oe may obseve ha C ad C2 ae well-defied cosas see [8]). To esimae I, wie f := f f )χ ad φ) := φ) d. The, as i [3], we have I x) ρ x y ) ρ) fy) x y dy CM fx) d C ψ) φ) M fx). y L p boudedess of M ad Fac 6.2 see Appedices), we obai /p I C p dx) ψ) x) p φ) f L /p p /p C [M dx) φ) fx)] p /p C p f σf))χ L φ) /p p + /p f σf) C p f σf) Mp, φ + f Lp,φ ) C p f Lp,φ, whee σf) = lim f,). I he emais o esimae I 2. y 2.2) ad 2.4), we have I 2 x) fy) f C C x a = C x a C x a C x a y a 2 k=2 k=2 k=2 ρ x y ) ρ y a ) x y y a dy ρ y a ) fy) f + dy y a 2 k y a <2 k ρ2 k ) 2 k ) + ρ2 k ) 2 k y a <2 k 2 k ) fy) f ρ y a ) y a + dy fy) f dy y a <2 k fy) f p dy) /p. )

7 ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS 33 u, fo each k 2, we have a, 2 k ) a,2 k ) fy) f p dy) /p C f Lp,φ k=2 2 k+ 2 k see Fac 6. i Appedices). Hece, by 2.2), 2.3), ad ou assumpio o φ ad ψ, we obai Ix) 2 ρ2 k ) 2 k φs) C x a f Lp,φ 2 k ds s φs) s ) ds d ρ) φs) C x a f Lp,φ k=2 2 k 2 2 s ρ) ) φs) C x a f Lp,φ 2 2 ds d 2 s ) ρ) φs) = C x a f Lp,φ 2 s 2 d ds s ρs)φs) C f Lp,φ s 2 ds Cψ) f Lp,φ, whece /p I dx) 2 ψ) x) p C f. Lp,φ This complees he poof. QED) 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 appopiae fucios φ ad ψ. Ideed, we have he followig aalog of Theoem 3.2 fo T ρ. Theoem 4.2. Suppose ha ρ saisfies 2.) 2.4). Suppose fuhe ha ρ) ad /p ρ) d ae almos deceasig, ρ) /p d C /p ρ) d, ad hee 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 ρ) φ) d + he T ρ is bouded fom L p,φ o L q,ψ. ρ)φ) 2 d Cψ), fo all >, Poof. Le f L p,φ. Fo ay ball = a, ) i R, le = a, 2) ad defie I, I, I2,C ad C2 as i he poof of Theoem 4.. To esimae I, wie f := f f )χ as befoe. The I = T f ρ i is T ρ, o T ρ ), ad hece as i he poof of Theoem 3.2) we have /q Ix) dx) q C /q ψ) f. Lp,φ ds

8 34 ERIDANI, HENDRA GUNAWAN AND EIICHI NAKAI Meawhile, we have he same esimae fo I 2 as i he poof of Theoem 4., whece /q Ix) dx) 2 q C /q ψ) f. Lp,φ The desied iequaliy fo T ρ follows immediaely fom hese wo esimaes. QED) 5. Examples Le l be a coiuous fucio o, ) such ha { /log /) fo small >, l) = log fo lage >. We assume ha l is Lipschiz coiuous o evey closed ad bouded ieval coaied i, ). The l) l ) /l/ ). Le 5.) <p<, < α < /p, β ad ρ) = α l β ). The ρ saisfies he assumpio i Theoem 3.2. Moeove, if < α <, he ρ saisfies he assumpio i Theoem 4.2. I paicula, oe may obseve ha ρ) d α l β ). Example 5.. Take φ) = /p l) βq/p q) whee /q =/p α/. The φ) p q)/q = ρ) ad ρ)φ) d φ) p/q. Fom Theoem 3. i follows ha T ρ is bouded fom M p,φ o M q,φ p/q. The ad Now, fo β>, le Φ i i =, 2) be Youg fucios ad Fo β =, le Φ s) s q l β s) fo s>, /q =/p α/, { / exp/s /β ) fo small s>, Φ 2 s) = exps /β ) fo lage s>. Φ ) /q /l β ), /p ρ) Φ 2 ) l β ), ) ) /q d Φ l β ), Φ )Φ 2 ) /q. Φ s) =s q, /q =/p α/, Φ 2 s) = { fo s<, + fo s. The Φ ) /q, Φ 2 ).

9 ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS 35 Example 5.2. Ude he codiio 5.), le φ) /p be almos iceasig ad φ) α+ɛ be almos deceasig fo some ɛ>. The ρ)φ) d ρ)φ). Fom Theoem 3.2 i follows ha T ρ is bouded fom M p,φ o M q,ψ fo ψ) =ρ)φ). I he case β =, his boudedess also follows fom Theoem 3 i [7].) Example 5.3. Ude he codiio 5.) ad <α<, le φ) /p be almos iceasig ad φ) α +ɛ be almos deceasig fo some ɛ>. The ρ)φ) 2 d ρ)φ). Fom Theoem 4.2 i follows ha T ρ is bouded fom L p,φ o L q,ψ fo ψ) =ρ)φ). If ψ) is almos iceasig, he his boudedess also follows fom Theoem 3.6 i [9] sice L p,φ L,φ ad L q,ψ = L,ψ.) Le us ow coside he case whee { log /) β fo small >, 5.2) <p= q<, β> ad ρ) = log ) β fo lage >. We assume ha ρ is Lipschiz coiuous o evey closed ad bouded ieval coaied i, ). The ρ) d l β ) ad ρ saisfies he assumpios i Theoem 3.2 ad i Theoem 4.2. Now le Φ i i =, 2) be Youg fucios ad The ad Φ s) s p l β s), Φ 2 s) = Φ ) /p l β ), /p ρ) { / exp/s /β ) fo small s>, exps /β ) fo lage s>. Φ 2 ) l β ), ) ) /p d Φ l β ), Φ )Φ 2 ) /p. Example 5.4. Ude he codiio 5.2), le φ) = δ l γ ), fo /p < δ < ad <γ<+, o fo δ = /p ad γ<+. The ρ)φ) d ρ)φ) Cφ) ρ) d δ l β+γ ). Fom Theoem 3.2 i follows ha T ρ is bouded fom M p,φ o M p,ψ fo ψ) = δ l β+γ ). This boudedess also follows fom Theoem i [3].) Example 5.5. Ude he codiio 5.2), le φ) = δ l γ ), fo /p < δ < ad <γ<+, o fo δ = /p ad γ<+. The ρ)φ) 2 d ρ)φ) Cφ) ρ) d δ l β+γ ). Fom Theoem 4.2 i follows ha T ρ is bouded fom L p,φ o L p,ψ fo ψ) = δ l β+γ ). If δ<, he φ) d φ) ad so his boudedess also follows fom Theoem 4.. If δ>, o if δ = ad β + γ, he his boudedess also follows fom Theoem 3.6 i [9] sice L p,φ L,φ ad L p,ψ = L,ψ.)

10 36 ERIDANI, HENDRA GUNAWAN AND EIICHI NAKAI 6. Appedices Hee we pese facs ha we used ealie i he poof of Theoem 4. ad implicily i he poof of Theoem 4.2). Fac 6.. If f L p,φ fo some p< ad φ saisfies he doublig codiio, he fo ay ball = a, ) i R ad k =, 2, 3,..., we have a, 2 k ) a,2 k ) fy) f p dy) /p C f Lp,φ whee C> is depede oly o ad he doublig cosa of φ. 2 k φ) d, Poof. y Mikowski s iequaliy, we have ) /p a, 2 k fy) f p dy ) a,2 k ) /p k a, 2 k fy) f ) a,2 ) dy) p + f k a,2j ) f a,2 j+ ). a,2 k ) u, fo each j =,...,k, oe may obseve ha f a,2 j ) f a,2 j+ ) a, 2 j fy) f ) a,2 j+ ) dy a,2 j ) /p 2 a, 2 j+ fy) f ) a,2 j+ ) dy) p Cφ2 j+ ) f. Lp,φ a,2 j+ ) Summig up, we ge k a, 2 k fy) f dy C f Lp,φ φ2 j+ ) ) a,2 k ) k C f Lp,φ j= 2 j+ 2 j φ) sice φ saisfies he doublig codiio. QED) j= j= 2 k d = C f Lp,φ φ) d, Fac 6. ca acually be geealized as follows. Fac 6.. Le f L p,φ fo some p< ad φ saisfy he doublig codiio. If a, ) b, s) i R, he fa,) f b,s) C f Lp,φ 2s φ) whee C> is depede oly o ad he doublig cosa of φ. d,

11 ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS 37 Poof. Ideed, if 2 k s <2 k s, he, choosig balls j j =,, k) so ha he adius of j is 2 j s ad a, ) k k = b, s), we have k fa,) f b,s) fa,) f k + f j+ f j j= C f Lp,φ k 2s φ2 j s) C f Lp,φ Fac 6.2. Le p<, φ saisfy he doublig codiio ad he f,) coveges as eds o ifiiy ad f lim f,) Mp, C f φ Lp,φ, whee φ) = j= 2 k s φ) d. QED) φ) d <. Iff L p,φ, φ) d ad C> is depede oly o ad he doublig cosa of φ. Poof. Fom Fac 6. i follows ha hee exiss a cosa σf), idepede of a R, such ha lim f a,) = σf), ad fa,) σf) C f Lp,φ Hece we have, fo all = a, ), ) /p fx) σf) p dx /p fx) f dx) p + f σf) f Lp,φ φ)+c f Lp,φ φ) C f Lp,φ φ). QED) φ) d. Refeeces [] J.A. Aloso, The disibuio fucio i he Moey space, Poc. Ame. Mah. Soc ), [2] F. Chiaeza ad M. Fasca, Moey spaces ad Hady-Lilewood maximal fucio, Red. Ma ), [3] Eidai, O he boudedess of a geealized facioal iegal o geealized Moey spaces, Tamkag J. Mah ), [4] Eidai ad H. Guawa, O geealized facioal iegals, J. Idoesia Mah. Soc. MIHMI) 83) 22), [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, oudedess of iegal opeaos o geealized Moey spaces ad is applicaio o Schödige opeaos, Poc. Ame. Mah. Soc ), [7] E. Nakai, Hady-Lilewood maximal opeao, sigula iegal opeaos ad he Riesz poeials o geealized Moey spaces, Mah. Nach ), [8] E. Nakai, O geealized facioal iegals, Taiwaese J. Mah. 5 2), [9] E. Nakai, O geealized facioal iegals o he weak Olicz spaces, MO φ, he Moey spaces ad he Campaao spaces, Fucio spaces, iepolaio heoy ad elaed opics Lud, 2), 389-4, de Guye, eli, 22. [] J. Peee, O he heoy of L p,λ spaces, J. Fuc. Aal ), [] M. M. Rao ad Z. D. Re, Theoy of Olicz spaces, Macel Dekke, Ic., New Yok, 99.

12 38 ERIDANI, HENDRA GUNAWAN AND EIICHI NAKAI [2] S. Sugao ad H. Taaka, oudedess of facioal iegal opeaos o geealized Moey spaces, Sci. Mah. Jp. Olie 8 23), [3] E. M. Sei, Hamoic Aalysis: eal vaiable mehods, ohogoaliy, ad oscillaoy iegals, Piceo Uivesiy Pess, Piceo, New Jesey, 993. [4] C. Zoko, Moey space, Poc. Ame. Mah. Soc ), ,2 Depame of Mahemaics, adug Isiue of Techology, adug 432, Idoesia. addess: ei sai@ds.mah.ib.ac.id, 2 hguawa@ds.mah.ib.ac.id Pemae addess: Depame of Mahemaics, Campus C, Ailagga Uivesiy, Suabaya 65, Idoesia. 3 Depame of Mahemaics, Osaka Kyoiku Uivesiy, Kashiwaa, Osaka , Japa. addess: 3 eakai@cc.osaka-kyoiku.ac.jp