336 ERIDANI kfk Lp = sup jf(y) ; f () jj j p p whee he supemum is aken ove all open balls = (a ) inr n, jj is he Lebesgue measue of in R n, () =(), f

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1 TAMKANG JOURNAL OF MATHEMATIS Volume 33, Numbe 4, Wine 2002 ON THE OUNDEDNESS OF A GENERALIED FRATIONAL INTEGRAL ON GENERALIED MORREY SPAES ERIDANI Absac. In his pape we exend Nakai's esul on he boundedness of a genealized facional inegal opeao fom a genealized Moey space o anohe genealized Moey o ampanao space.. Inoducion Main Resuls Fo a given funcion :(0 )! (0 ), le T be he genealized facional inegal opeao, given by f(y) T f(x) = pu ~T f(x) = R n R f(y) n ; (jyj)( ; 0 (y)) jyj n he modied vesion of T, whee 0 is he uni ball abou he oigin, 0 is he chaaceisic funcion of 0. In [4], Nakai poved he boundedness of he opeaos ~T T fom a genealized Moey space M o anohe genealized Moey space M o genealized ampanao space L. Moe pecisely, hepoved ha kt fk M kfk M k ~T fk L kfk M whee > 0, wih some appopiae condiions on. Using he echniques developed by Kuaa e.al.[], we invesigae he boundedness of hese opeaos fom genealized Moey spaces M p o genealized Moey spaces M p o genealized ampanao spaces L p fo <p<. The genealized Moey ampanao spaces ae dened as follows. Fo a given funcion :(0 )! (0 ), <p<, le kfk Mp =sup () jj jf(y)j p p Received Sepembe 9, 200 evised Januay 29, Mahemaics Subjec lassicaion. Pimay 4220, 46E30, 43A5, 26A33. Key wods phases. Genealized facional inegals, genealized Moey spaces, genealized ampanao spaces. 335

2 336 ERIDANI kfk Lp = sup jf(y) ; f () jj j p p whee he supemum is aken ove all open balls = (a ) inr n, jj is he Lebesgue measue of in R n, () =(), f = jj f(y). We dene he genealized Moey space M p by M p = ff 2 L p loc (Rn ):kfk Mp < g R he genealized ampanao space L p by Ou esuls ae he following: L p = ff 2 L p loc (Rn ):kfk Lp < g: Theoem. If :(0 ) ;! (0 ) saisfying he condiions below: 2 2 ) A () A A 2 () A 2 (.) p d < fo all >0 we have d A 3 () p (.2) 0 () d + d A 4 () fo all >0 (.3) 0 whee A i > 0 ae independen of > 0, hen fo each <p< hee exiss p > 0 such ha kt fk Mp p kfk Mp : Theoem 2. If :(0 )! (0 ) saisfying he condiions below: 2 2 ) A () A A 2 () A 2 (2.) p d < fo all >0 we have d A 3 () p (2.2) 0 () ; A 4 j ; j () n n n+ fo 2 2 (2.3) () d d A 5 () fo all >0 (2.4) whee A i > 0 ae independen of > 0, hen fo each <p< hee exiss p > 0 such ha k ~T fk Lp p kfk Mp :

3 THE OUNDEDNESS OF A GENERALIED FRATIONAL INTEGRAL Poof of he Theoems To pove he heoems, we shall use he following esul of Nakai [2] (in a slighly modied vesion) abou he boundedness of he sad maximal funcion Mf on a genealized Moey space M p. The sad maximal funcion Mf is dened by Mf(x) =sup 3x jj jf(y)j x 2 R n whee he supemum is aken ove all open balls conaining x. Theoem(Nakai). If :(0 ) ;! (0 ) saisfying he condiions below: (a) 2 2 ) A () A d A 2 () p fo all >0, (b) R p whee A i > 0 ae independen of > 0, hen fo each <p< hee exiss p > 0 such ha kmfk Mp p kfk Mp : Fom now on, p will denoe posiive consans, which mayvay fom line o line. In geneal, hese consans depend on n. Poof of Theoem. Fo x 2 R n,>0, wie T f(x) = jx;yj< f(y) + jx;yj f(y) = I jx ; yj (x)+i 2 (x): n Noe ha, fo 2 [2 k 2 k+ ], hee exis consans i > 0such ha So, we have ji (x)j (2 k ) 2 k+ 2 k (2 k )(2 k ) 2 2 k+ jx;yj< ; X 2 k d k=; 2 k jx;yj<2 k+ X; k=; d: (2 k ) jf(y)j (2 k ) jx;yj<2 n k+

4 338 ERIDANI Meanwhile, ji 2 (x)j X; (2 k )Mf(x) k=; X; 2 k+ Mf(x) k=; 2 k Mf(x) 0 () () Mf(x): X jx;yj k=0 X k=0 X 2 k jx;yj<2 k+ (2 k+ ) jf(y)j (2 k ) jx;yj<2 n k+ (2 k+ )(2 k+ )kfk Mp k=0 X 2 k+2 kfk Mp d k=0 2 k+ kfk Mp d ()kfk Mp : Now, fo p<, wehave jt f(x)j p 2 p; (ji (x)j p + ji 2 (x)j p ) by Nakai's Theoem, we have fo all balls = (a ) () p jj ji (x)j p dx () p jj Mf(x) p dx kmfk p M p ji () p jj 2 (x)j p dx kfk p M p : ombining he wo esimaes, we obain () p jj jt f(x)j p dx p kfk p M p p kfk p M p

5 THE OUNDEDNESS OF A GENERALIED FRATIONAL INTEGRAL 339 he esul follows. whee Poof of Theoem 2. Le ~ = (a 2). Fo x 2 = (a ), we have ~T f(x) ; = E (x)+e 2 (x) = E (x) = E 2 (x) = ( ; R f(y) ~ (y)) n ja ; yj n Fom (2.3), we have ~ j j f(y) ~ c f(y) jx ; yj n ja;yj<k ; ja ; yj n jf(y)j + jaj ja;yjk ; (jyj)( ; 0 (y)) jyj n :! jf(y)j ja ; yj n+ whee k =max(2jaj 2), so we knowha is nie fo evey ball = (a ). Wih he same echnique as in he poof of he pevious heoem, we have by (2.3) je 2 (x)j je (x)j ja;yj<2 jx;yj<3 3 Mf(x) 0 Mf(x) 0 ja;yj2 jx ; aj he esul follows as befoe. kfk Mp jf(y)j d d ja;yj2 2 d ; ja ; yj n jf(y)j ja ; yj n+ Remak. We also suspec ha ~T, he modied vesion of T,isboundedfomL p o L p unde he same hypohesis on, as in Theoem 2. Howeve, we have no obained he poof he eseach in his diecion is sill ongoing.

6 340 ERIDANI Acknowledgemens The auho would like o hank D. Henda Gunawan D. Eiichi Nakai fo hei useful commens suggesions duing he pepaaion of his noe. He also hanks he efeee(s) fo hei emaks on he ealie vesion of his noe. Refeences [] K. Kuaa, S. Nishigaki S. Sugano, oundedness of inegal opeaos on genealized Moey spaces is applicaion o Schodinge opeaos, Poc. Ame. Mah. Soc. 28(999), [2] E. Nakai, Ha-Lilewood maximal opeao, singula inegal opeaos, he Riesz poenials on genealized Moey spaces, Mah. Nach. 66(994), [3] E. Nakai, On genealized facional inegals, Poceedings of he Inenaional onfeence on Mahemaical Analysis is Applicaions 2000 (Kaohsiung, Taiwan), Taiwanese J. Mah. 5(200), [4] E. Nakai, On genealized facional inegals on he weak Olicz spaces, MO, he Moey sapces he ampanao spaces, Poceedings of he onfeence on Funcion Spaces, Inepolaion Theoy elaed opics in honou of Jaak Peee on his 65h bihday, Lund Univesiy Sweden, o appea. Depamen of Mahemaics, ung Insiue of Technology, ung 4032, Indonesia. Pemanen addess: Depamen of Mahemaics, Univesiy of Ailangga, Suabaya 605, Indonesia.