Research Article Weak Type Inequalities for Some Integral Operators on Generalized Nonhomogeneous Morrey Spaces

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1 Hindawi Publishing Copoaion Jounal of Funcion Spaces and Applicaions Volume 23, Aicle ID 8974, 2 pages hp://dx.doi.og/.55/23/8974 Reseach Aicle Weak Type Inequaliies fo Some Inegal Opeaos on Genealized Nonhomogeneous Moey Spaces Henda Gunawan, Denny Ivanal Hakim, Yoshihio Sawano, 2 and Idha Sihwaningum 3 Depamen of Mahemaics, Bandung Insiue of Technology, Bandung 432, Indonesia 2 Depamen of Mahemaics and Infomaion Sciences, Tokyo Meopolian Univesiy, Minami-Ohsawa -, Hachioji,Tokyo92-397,Japan 3 Depamen of Mahemaics, Jendeal Soediman Univesiy, Puwokeo 5322, Indonesia Coespondence should be addessed o Denny Ivanal Hakim; dnny hkm@yahoo.com Received 9 Sepembe 23; Acceped 2 Novembe 23 Academic Edio: Vagif Guliyev Copyigh 23 Henda Gunawan e al. This is an open access aicle disibued unde he Ceaive Commons Aibuion License, which pemis unesiced use, disibuion, and epoducion in any medium, povided he oiginal wok is popely cied. We pove weak ype inequaliies fo some inegal opeaos, especially genealized facional inegal opeaos, on genealized Moey spaces of nonhomogeneous ype. The inequaliy fo genealized facional inegal opeaos is poved by using wo diffeen echniques: one uses he Chebyshev inequaliy and some inequaliies involving he modified Hady-Lilewood maximal opeao and he ohe uses a Hedbeg ype inequaliy and weak ype inequaliies fo he modified Hady-Lilewood maximal opeao. Ou esuls genealize he weak ype inequaliies fo facional inegal opeaos on genealized non-homogeneous Moey spaces and exend o some singula inegal opeaos. In addiion, we also pove he boundedness of genealized facional inegal opeaos on genealized non-homogeneous Olicz-Moey spaces.. Inoducion In his pape, we pove ha he heoy of genealized Moey spaces can be saged on he nondoubling seing on R d,so ha we assume ha μ is a posiive Boel measue on R d saisfying he gowh condiion; ha is, hee exis n [,d] and C μ >such ha μ (B (a, )) μ n () fo any ball B(a, ) ceneed a a R d wih adius > (see [ 4]). Fo < n d and a measuable funcion ρ : (, ) (, ), we define he genealized facional inegal opeao I ρ by I ρ f (x) = R d x y n f(y)dμ(y) (2) fo any suiable funcion f on R d. This opeao daes back o he book when ρ() = α fo <α<n[5, Secion6.]. Noe ha if μ is he Lebesgue measue, hen I ρ =I α is he facional inegal opeao inoduced in [6, 7]. See also [8, 9] fo exhausive and compehensive explanaion abou he opeao. Below, we will always assume he Dini condiion, ha is, (ρ()/)d < and we also assume ha ρ saisfies he so-called gowh condiion; namely, hee exis consans C>and <2k <k 2 <such ha sup (/2)<s k 2 ρ (s) k ρ () d (3) fo evey >.Foconvenience,wieρ () = k 2 k (ρ()/)d. Noe ha if ρ saisfies he doubling condiion, ha is, hee exiss a consan C > such ha /C ρ()/ρ(s) C wheneve /2 /s 2, hen ρ saisfies he gowh

2 2 Jounal of Funcion Spaces and Applicaions condiion. See [ 3] fo discussion abou I ρ, whee ρ saisfies he doubling condiion. Now, we say ha a funcion f belongs o he genealized nonhomogeneous Moey space L p,φ (μ) = L p,φ (R d,μ)fo a funcion φ : (, ) (, ) and p<if f := sup B(a,) R φ () ( n f(y) p /p dμ (y)) <. d B(a,) (4) Noe ha his definiion is a special case of [4, Definiion.], whee diffeen ypes of opeaos ae consideed. In his pape, we will assume he following wo condiions. (.a) The funcion φ is almos deceasing; ha is, hee exiss a consan C >such ha φ() C φ(s) fo evey s. (.b) The funcion n φ() p is almos inceasing; ha is, hee exiss a consan C 2 >such ha n φ() p C 2 s n φ(s) p fo evey s. These wo condiions imply ha φ saisfies he doubling condiion. Noe ha if φ() = n/p,henl p,φ (μ) = L p (μ) is he nonhomogeneous Lebesgue space. The sudy of he boundedness of he facional inegal opeao I α on genealized Moey spaces was iniiaed in [5, Theoem 3]. The following heoem pesens he weak ype inequaliies fo I α on genealized nonhomogeneous Moey spaces. Theoem (see [6, Theoem2.4]). Le p < q <. Suppose ha inf > φ() =, sup > φ() =,andheeexis posiive consans C and C such ha (5) φ() p d C φ() p, α φ () + α φ () d C φ() p/q fo evey >. Then, hee exiss a consan C>such ha, fo any funcion f L p,φ (μ) and any ball B(a, ) R d,one has μ(x B(a, ) : I αf (x) >})n φ() p ( fo evey >. f q ), (6) Remak 2. Noe ha we can obain he weak ype inequaliies fo I α on nonhomogeneous Lebesgue spaces which ae poved in [7, 8] byakingφ() = n/p and /q = (/p) (α/n) in Theoem. Bysubsiuing/q = (α/(n λ)) fo some λ [,n α) o α φ() Cφ() /q,wehave α φ()d C λ+α n fo evey >,whichisoneof he hypoheses in he weak ype inequaliies fo I α in [9]. The poof of Theoem employssomeinequaliiesinvolving he modified Hady-Lilewood maximal opeao M n (see [8]),whichisdefinedfoanylocallyinegablefuncion f by M n f (x) = sup > n f(y) dμ (y) (7) B(x,) andhechebyshevinequaliywhichispesenedinhe following heoem. Theoem 3 (see [2]). Le E be a measuable subse of R d.if f is an inegable funcion on E,hen,foevey>,onehas μ (x E : f (x) > }) f (x) dμ (x). (8) E One of he easons why we ae fascinaed wih he genealized facional inegal opeaos is ha hese opeaos appea naually in he conex of diffeenial equaions; see [2, Secion 6.4] fo a nice explanaion in connecion wih he holomophic calculus of opeaos and see [22, (4.3)] and [23, Lemma 2.5] fo a deailed accoun ha ( Δ) α/2 wih α > saisfies he equiemen of ρ in he pesen pape. In addiion, invesigaing genealized Moey spaces is no a mee ques o he absac heoy; i aises naually in he conex of Sobolev embedding. In [24], he following poposiion is poved. Poposiion 4 (see [24, Theoem5.]). Le <p<and <λ<n. Then, hee exiss a posiive consan C p,λ such ha f (x) dx B p,λ B (+ B ) /p log (e + B ) ( Δ)λ/2p f L p,λ holds fo all f L p,λ (R N ) wih ( Δ) λ/2p f L p,λ (R N ) and fo all balls B,wheeL p,λ is he abbeviaion of L p,φ wih φ() = λ. Lae Poposiion 4 is senghened by [25, Example 5]. An example in [24] aswellashenecessayandsufficien condiion obained in [25, Theoem.3] implicily shows ha helogfacoaboveisabsoluelynecessay. In his pape, we will pove he weak ype inequaliies fo I ρ which is a genealizaion of Theoem.InSecion 2,wewill pove he weak ype inequaliies fo I ρ by using he Chebyshev inequaliy and some inequaliies involving opeao M n. In Secion 3, we will pove a Hedbeg ype inequaliy on genealized nonhomogeneous Moey space by adaping he poof of a Hedbeg ype inequaliy on homogeneous seing in [25]. Though he weak ype inequaliies fo M n,we hen pove he weak ype inequaliies fo I ρ on genealized nonhomogeneous Moey spaces. In Secion 4, we exend ou esuls o he singula inegal opeaos defined in []. Finally, in Secion 5, wepoveheboundednessofi ρ (9)

3 Jounal of Funcion Spaces and Applicaions 3 on genealized nonhomogeneous Olicz-Moey spaces. See [26 28]foelaedesuls. Thoughou he pape, C denoes a posiive consan which is independen of he funcion f and he vaiable x and may have diffeen values fom line o line. We also denoe by C k (k N) he fixed consans ha saisfy ceain condiions. 2. Weak Type Inequaliies fo I ρ via he Chebyshev Inequaliy Now, we give an inequaliy which is used in he poof of he weak ype inequaliies fo I ρ in he following lemma. If we use () and he doubling condiion of φ,henwehave j= k 2 2 j+ R k 2 j+ R ρ () d Cφ(k 2 R) (p/q) φ(r) (p/q). Hence, I(x) CM n f(x)φ(r) (p/q). By leing f o f χ B(a,),wehavehefollowing. (4) Coollay 6. Le p<q<and B(x, R) be any ball in R d.ifhefuncionsρ and φ saisfy inequaliy (),hen Lemma 5. Le p<q<.ifρ and φ saisfy ρ () d Cφ() (p/q) () x y n dμ (y) Cφ(R) (p/q) and, fo any ball B(a, ) R d,onehas (5) fo evey >,hen,foanyballb(x, R) R d and evey locally inegable funcion f,one has ρ( x y ) x y n f(y) dμ (y) CMn f (x) φ(r) (p/q). () Poof. Le I(X) = (B,) (ρ( x y )/ x y n ) f(y) dμ(y), fo any ball B(x, R) R d. By he dyadic decomposiion of he ball B(x, R) and he gowh condiion of ρ,wehave I (x) = ρ( x y ) j= B(x,2 j+ R)\B(x,2 j R) x y n f(y) dμ (y) j= M n f (x) ρ (2 j+ R) (2 j R) n B(x,2 j+ R) j= k 2 2 j+ R k 2 j+ R f(y) dμ (y) ρ () d. (2) Then,weuseheovelappingpopey(see[29, 3]) o obain j= k 2 2 j+ R k 2 j+ R = j= k 2 R = ρ () d k 2 R χ [k 2 j+ R, k 2 2 j+ R] () ( j= (+log 2 ( k 2 χ [k 2 j+ R, k 2 2 j+ R] ()) k 2 R )) k ρ () d. ρ () d ρ () d (3) x y n χ B(a,) (y) dμ (y) M n χ B(a,) (x) φ(r) (p/q). (6) Remak 7. These wo inequaliies will be used lae o pove one of ou main heoems. The nex lemma pesens an inequaliy involving he modified Hady-Lilewood maximal opeao M n. This inequaliy is an impoan pa of he poof of he weak ype inequaliies fo I α in [6, 9]. See [8] fo simila esuls. Lemma 8 (see [6]). Le p <. If φ saisfies (φ() p /)d Cφ() p fo evey >,hen,foanyfuncion f L p,φ (μ) and any ball B(a, ) R d,onehas R d f(y) p M n χ B(a,) (y) dμ (y) C n φ() p f p L p,φ (μ). (7) Wih Theoem 3, Coollay 6, andlemma 8, weaenow eady o pove he weak ype inequaliies fo I ρ on genealized nonhomogeneous Moey spaces. Theoem 9 (see [3]). Le p<q<and assume ha sup > φ() =.Ifρ and φ saisfy φ () ρ () φ() p d Cφ() p, d + (8) ρ () φ () d Cφ() p/q fo evey >,hen,foanyfuncionf L p,φ (μ) and any ball B(a, ) R d,onehas μ (x B(a, ) : I ρf (x) >}) n φ() p ( fo evey >. f q ), (9)

4 4 Jounal of Funcion Spaces and Applicaions Poof. Le B(a, ) be any ball in R d.foeveyx B(a, ) and R>,le By using he doubling condiion of φ and he ovelapping popey, we have I (x) = I 2 (x) = R d \ x y n f(y)dμ(y), x y n f(y)dμ(y). (2) Le E = x B(a, ) : I ρ f(x) > }, foany>.since I ρ f(x) I (x) + I 2 (x),wehave μ(e ) μ(x B(a, ) : I (x) > 2 }) +μ(x B(a, ) : I 2 (x) > (2) 2 }). ρ (2 j+ R) φ (2 j+ R) = φ(2 j+ R) =C 2k R 2k R χ [k 2 j+ R, k 2 2 j+ R] () χ [k 2 j+ R, k 2 2 j+ R] () 2k R ( χ [k 2 j+ R, k 2 2 j+ R] ()) (+log 2 ( k 2 )) k 2k R ρ () d ρ () φ () d ρ () φ () d. ρ () φ () d Now, we invoke he inegal assumpion on ρ φ: (24) By he dyadic decomposiion of R d \ B(x, R) and he gowh condiion of ρ,we have I 2 (x) ρ( x y ) B(x,2 j+ R)\B(x,2 j R) x y n f(y) dμ (y) ρ (2 j+ R) (2 j R) n B(x,2 f(y) dμ (y). j+ R) (22) We use Hölde s inequaliy, he gowh condiion of μ,andhe definiion of f L p,φ (μ) o obain I 2 (x) ρ (2 j+ R) (2 j R) n (B(x,2 j+ R) (μ(b(x,2 j+ R))) (/p) ρ (2 j+ R) f ( (2 j+ R) n B(x,2 j+ R) ρ (2 j+ R) φ (2 j+ R). f(y) p /p dμ(y)) /p f(y) p dμ (y)) (23) Hence, ρ (2 j+ R) φ (2 j+ R) Cφ(2k R) p/q φ(r) p/q. (25) I 2 (x) 3 f φ(r)p/q. (26) Le = (/2C 3 f L p,φ (μ) )q/p.remakha (φ()p /) d C φ() p implies ha inf > φ() =.Ohewise, =inf > φ( ) p d φ() p d Cφ() p, (27) which is impossible. Now, fom inf > φ() = < <= sup > φ(), wecanfindk Z such ha, fo R =2 k and R 2 =2 k,wehave φ(r ) φ(r 2 ). (28) Since R /R 2 =2, hee exiss C>such ha φ(r 2 ) Cφ(R ). Hence, Taking R=R,weobain φ(r ) φ(r ). (29) I 2 (x) 3 f φ(r ) p/q 3 f p/q 2. (3) Consequenly, μ(e ) μ(x B(a, ) : I (x) > }). (3) 2

5 Jounal of Funcion Spaces and Applicaions 5 We combine Hölde s inequaliy and he inequaliy (5) o obain I (x) ( B(x,R ) ( B(x,R ) /p x y n f(y) p dμ(y)) 4 φ(r ) ((p/q) )( (/p)) (/p) x y n dμ (y)) (32) ρ( ( x y ) /p B(x,R ) x y n f(y) p dμ(y)). Finally, by using he las inequaliy and he Chebyshev inequaliy, we ge μ(e ) μ x B(a, ) : B(x,R ) (/2) p } > C p 4 φ(r ) ((p/q) )(p ) } } 2p C p 4 φ(r ) ((p/q) )(p ) p B(a,) B(x,R ) = C p φ(r ) ((p/q) )(p ) R d B(x,R ) x y n f(y) p dμ (y) x y n f(y) p dμ (y) dμ (x) f(y) p x y n χ B(a,) (χ) (x) dμ (y) dμ (x). (33) By viue of he inequaliies (6), (7), and (29)aswellashe definiion of,we ge μ(e ) C p φ(r ) ((p/q) )(p ) R d f(y) p B(y,R ) x y n χ B(a,) (x) dμ (x) dμ (y) C p φ(r ) ((p/q) )p R d f(y) p M n χ B(a,) (y) dμ (y) Remak. Noe ha ρ() = α,whee<α<nsaisfies he condiion of Theoem 9 and, fo his ρ, weobainheweak ype inequaliies fo I α in Theoem. 3. Weak Type Inequaliies fo I ρ via a Hedbeg Type Inequaliy and Weak Type Inequaliies fo M n In his secion, we will pove weak ype inequaliies fo I ρ usingadiffeenechnique,namely,viaahedbegype inequaliy and weak ype inequaliies fo M n.iunsou ha some hypoheses can be emoved. The Hedbeg ype inequaliy is pesened in he following poposiion. Poposiion (see [25, 3]). Le p<q<.ifρ and φ saisfy φ () ρ () d + ρ () φ () d Cφ() p/q (35) fo evey >,hen,foeveyf L p,φ (μ) and x R d,one has I ρf (x) (Mn f(x) p/q f (p/q) L p,φ (μ) + f inf > φ()p/q ). (36) Poof. WeadaphepoofofaHedbegypeinequaliyon genealized Moey space in [25]. Fo evey x R d and R>, wiei ρ f(x) = I (x) + I 2 (x), wheei (x) and I 2 (x) ae defined in he poof of Theoem 9. Byusingheinequaliies ()and(26), we ge I ρf (x) (Mn f (x) φ(r) (p/q) + f φ(r)p/q ). (37) Nex, we sepaae he poof ino he following wo cases. Fis Case (M n f(x) 2C (/p) μ f L p,φ (μ) inf >φ()). Inhis case, we have I ρf (x) C ((2C (/p) μ f φ (R))φ(R)(p/q) + f φ(r)p/q ) f φ(r)p/q fo evey R>.Hence, I ρ f(x) C f L p,φ (μ) inf >φ() p/q. (38) Second Case. M n f(x) > 2C (/p) μ f L p,φ (μ) inf >φ(). We use Hölde s inequaliy, he gowh condiion of μ, andhe definiion of f L p,φ (μ) o obain as desied. C p (p2 /q) p n φ() p f p L p,φ (μ) n φ() p ( f q ), (34) R n f(y) dμ (y) R n ( f(y) p /p dμ(y)) μ(b (x, R)) (/p) (/p) μ f φ (R) (39)

6 6 Jounal of Funcion Spaces and Applicaions fo evey R>.Hence, M n f (x) (/p) μ f sup >φ (). (4) Since sup > φ() >,wehave M inf φ () < > n f (x) 2C (/p) μ f 2 sup > φ () < sup φ (). > (4) Thus, hee exiss j Z such ha M n f (x) φ(r ) 2Cμ (/p) f φ(r 2 ) (42) fo R =2 j and R 2 =2 j.sincer /R 2 =2, hee exiss C>such ha M n f (x) φ(r ) 2Cμ (/p) f φ(r ). (43) By choosing R = R in he inequaliy (37) andusinghe inequaliy (43), we have I ρf (x) C ((2C (/p) μ f Cφ (R )) φ(r ) (p/q) + f φ(r ) p/q ) Cφ(R ) p/q f ( M n f (x) 2Cμ (/p) f ) M n f(x) p/q f (p/q) L p,φ (μ). p/q f Fom hese wo cases, we obain he inequaliy (36). (44) Sihwaningum e al. [9] povedheweakypeinequal- iies fo M n on genealized nonhomogeneous Moey space by assuming ha φ p saisfies he inegal condiion; ha is, (φ() p /)d Cφ() p fo evey >.In[9], he weak ype inequaliies fo I α ae also poved by using he weak ype inequaliies fo M n.inhispape,weemovehe inegal condiion of φ p in he hypohesis of ou poposiion below. See [32, Theoem 2.3] and [33, Theoem 2.3] fo such aemps. Poposiion 2 (see [3]). Le p<;hen,heeexiss aconsanc>such ha, fo any funcion f L p,φ (μ) and any ball B(a, ) R d,onehas μx B(a, ) :M n f (x) >} n φ() p ( fo evey >. f p ), (45) When p>, we have he song boundedness; see [5, Theoem ] and [34, Lemma 2.4] fo he Lebesgue case and see [35, Theoem 4.3] fo he song L p,φ (μ) o L p,φ 2 (μ) esul and he weak L p,φ (μ) o L p,φ 2 (μ) esul wih μ equal o he Lebesgue measue. Poof. The poof is simila o ha of song boundedness of maximal opeao on genealized nonhomogeneous Moey spaces which is discussed in [34]. The diffeence is ha in he final sep we use he Chebyshev inequaliy, as we will see below. Conside he ball B(a, ) R d.lex B(a, ) and le beanyposiiveealnumbe.foy R d, define f (y) = χ B(a,2) (y)f(y) and f 2 (y) = f(y) f (y).noeha M n f 2 (x) = sup R n R> ( χ B(a,2) (y)) f (y) dμ (y). Since B(x, R) B(a, 2),foeveyR<,wehave (46) R n ( χ B(a,2) (y)) f(y) dμ (y) = (47) fo evey R<.Hence, M n f 2 (x) sup R n f(y) dμ (y). (48) R> Obseve ha, fo evey R>,wehave φ () R n C φ (R) R n ( Thus, f(y) dμ (y) f(y) p /p dμ (y)) (μ (B (x, R))) (/p) C φ (R) ( R n f(y) p /p dμ (y)) f. (49) M n f 2 (x) φ() f. (5) Since M n f(x) M n f (x) + M n f 2 (x),wehave μx B(a, ) :M n f (x) >} μx B(a, ) :M n f (x) > 2 } +μx B(a, ) :M n f 2 (x) > 2 }. (5)

7 Jounal of Funcion Spaces and Applicaions 7 Fohefisem,weuseheweakypeinequaliiesfoM n on he nonhomogeneous Lebesgue space L p (μ) (see [8]) o obain μx B(a, ) :M n f (x) > 2 } Poof. This poof is adaped fom [25]. We eplace by 2. Conside he ball B(a, ) R d. By applying Poposiion, we have μx B(a, ) : I ρf (x) >2} ( f p L p (μ) ) C p f(y) p dμ (y) B(a,2) (52) μx B(a, ) :C(M n f(x) p/q f (p/q) L p,φ (μ) + f inf > φ()p/q )>2} μx B(a, ) :CM n f(x) p/q f (p/q) L p,φ (μ) >} (55) C p φ(2)p (2) n f p L p,φ (μ) n φ() p ( f p ). Meanwhile, fo he second em, by using he Chebyshev inequaliy and he inequaliy (5), we have μx B(a, ) :M n f 2 (x) > 2 } =μx B(a, ) :M n f 2 (x) p >( 2 )p } 2p p M n f 2 (x) p dμ (x) B(a,) 2p p C p φ() p f p dμ (x) L B(a,) p,φ (μ) 2p p Cp φ() p f p μ (B (a, )) L p,φ (μ) n φ() p ( f p ). (53) Finally, by combining hese wo esimaes, we obain inequaliy (45). Wih Poposiions and 2, we ae now eady o pove he weak ype inequaliies fo I ρ on genealized nonhomogeneous Moey spaces. Theoem 3 (see [3]). Le p<q<.ifρ and φ saisfy inequaliy (35),hen,foanyfuncionf L p,φ (μ) and any ball B(a, ) R d,onehas μ (x B(a, ) : I ρf (x) >}) n φ() p ( f ) (54) fo evey >. q +μx B(a, ) :C f inf > φ()p/q >}. Obseve ha he second em in he mos igh-hand side of he above inequaliy vanishes, when C f inf > φ()p/q. (56) So,oesimaeheem,wecansupposeha Wih his in mind, we calculae C f inf > φ()p/q >. (57) μx B(a, ) :C f inf > φ()p/q >} =μ(b (a, )) n φ() p ( Meanwhile, by using Poposiion 2, we have f q ). μx B(a, ) :CM n f(x) p/q f (p/q) L p,φ (μ) >} μ x B(a, ) :M n f (x) >( C f ) (p/q) L p,φ (μ) n φ() p f p L p,φ (μ) ( n φ() p ( f q ). C f (p/q) L p,φ (μ) ) q q/p} (58) } } (59) By summing he wo pevious esimaes, we ge he desied inequaliy. Remaks. (i) Noe ha he hypoheses (φ() p /)d Cφ() p in Theoem 9 ae no included in Theoem 3, since we can pove he weak ype inequaliies fo M n wihou his condiion.

8 8 Jounal of Funcion Spaces and Applicaions (ii) The condiions on φ, namely,inf > φ() = and sup > φ() =, ae no included in he hypoheses in Theoem 3. Howeve, we have o use he weak ype inequaliies fo M n on genealized nonhomogeneous Moey spaces and a Hedbeg ype inequaliy fo I ρ in he poof of Theoem Boundedness of Singula Inegal Opeaos Poposiion 2 caies ove o he singula inegal opeao whose definiion is given in []. Recall ha he singula inegal opeao T is a bounded linea opeao on L 2 (μ) fo which hee exiss a funcion K ha saisfies hee popeies lised below. (4.a) Thee exiss C>such ha K(x, y) C/ x y n fo all x =y. (4.b) Thee exis ε>and C>such ha K (x, y) K(z, y) + K (y, x) K(y, z) x z ε x y n+ε (6) if x y 2 x z wih x =y. (4.c) If f is a bounded μ-measuable funcion wih a compac suppo, hen we have Tf (x) = R d K (x, y) f (y) dμ (y) x supp (f). (6) As fo his singula inegal opeao T, hefollowing esulisdueonazaov,teil,andvolbeg. Poposiion 4 (see [, 2]). The singula opeao T is bounded on L p (μ) fo <p<.moeove,heeexissaconsan C>such ha μx R d : Tf (x) >}f L (μ) fo evey f L (μ) and evey >. (62) Theoem 5. Le T be a singula inegal opeao. Le p<.inaddiionohedoublingcondiion,assumeha φ () d Cφ () (63) fo evey >. Then, hee exiss a consan C>such ha, fo any funcion f L p,φ (μ) and any ball B(a, ) R d,one has μ x B(a, ) : Tf (x) >} n φ() p ( fo evey >. f p ) (64) Poof. The poof is a modificaion of ha of Poposiion 2. We decompose f=f +f 2 as befoe. The eamen of f is he same as ha in Poposiion 2 bu by using he weak ype inequaliy fo T in Poposiion 4. We need o ake cae of f 2.Byhecondiion(4.a),Hölde s inequaliy, and he gowh condiion of μ,we have Tf 2 (x) R d \B(x,) f(y) x y n dμ (y) (2 j ) n B(x,2 j+ ) (2 j ) n (B(x,2 j+ ) f(y) dμ (y) (μ(b(x,2 j+ ))) (/p) ( (2 j+ ) n B(x,2 j+ ) φ(2 j+ ) f f φ () d. f(y) p /p dμ(y)) /p f(y) p dμ(y)) (65) If we use ou inegabiliy assumpion, hen we have a poinwise esimae: So, we ae done. Tf 2 (x) φ() f. (66) Remak 6. If we define he genealized weak Moey space of nonhomogeneous ype wl p,φ (μ) o be he se of all μmeasuable funcions f such ha f wl p,φ (μ) (μ x B (a, ) : := sup f (x) >})/p <, B(a,) R d,> n/p φ () (67) hen he inequaliy (64) amouns o he boundedness of T fom L p,φ (μ) o wl p,φ (μ). Similaly, ou pevious esuls can be anslaed ino his language. In he following secion, we will use hese noaions fo convenience. 5. Genealized Nonhomogeneous Olicz-Moey Spaces Ou esuls above can be caied ove o genealized nonhomogeneous Olicz-Moey spaces. We fis fomulae ou main esuls and hen pove hem lae in Secions

9 Jounal of Funcion Spaces and Applicaions 9 Recall ha Φ : [,) [,) is a Young funcion, if Φ is bijecive and convex. We define he Φ-aveage of f ove a ball B(a, ) as follows: f Φ,B(a,) = inf λ > : n Φ( f(y) )dμ(y) }. B(a,) λ (68) The genealized nonhomogeneous Olicz-Moey space L Φ,φ (μ) = L Φ,φ (R d,μ)is he se of all f L loc (μ) fo which he nom f L Φ,φ (μ) = sup B(a,) R φ () f Φ,B(a,) (69) d is finie. Noe ha if Φ() = p,henl Φ,φ (μ) = L p,φ (μ).abou he sucue of his funcion space, we have he following. Theoem 7. Le Φ : [, ) [, ) be a Young funcion and le φ saisfy he wo condiions (.a) and (.b) as usual. Then, L Φ,φ (μ) is a Banach space. We define he genealized weak Olicz-Moey spaces of nonhomogeneous ype as follows. Fo a Young funcion Φ, he genealized weak Olicz-Moey space of nonhomogeneous ype wl Φ,φ (μ) = wl Φ,φ (R d,μ) is he se of all μmeasuable funcions f fo which he nom f wl Φ,φ (μ) = sup B(a,) R d,> φ () χ f >}Φ,B(a,) (7) is finie. Wie wl p,φ (μ) = wl Φ,φ (μ),whenφ() = p.iisno so had o pove f wl Φ,φ (μ) f L Φ,φ (μ) (7) fo all μ-measuable funcions f fom he inequaliy χ f >} f. (72) By aking f(x) = (/ x )χ [,] (x) and φ() = (/) log(3 + ) and μ is he Lebesgue measue on R, weseehaf wl,φ (R,μ)\L,φ (R,μ),showinghawL,φ (R,μ)is a pope supese of L,φ (R,μ). Theoem 8. Le Φ : [,) [,) be a Young funcion and le φ saisfy he wo condiions (.a) and (.b) as usual. Then, wl Φ,φ (μ) is a quasi-banach space. Moe pecisely, () f wl Φ,φ (μ) =if and only if f=; (2) cf wl Φ,φ (μ) = c f wl Φ,φ (μ) fo all c C and f wl Φ,φ (μ); (3) if f k } k= is a sequence in wlφ,φ (μ) such ha lim k,k 2 f k f k2wl Φ,φ (μ) =. (73) We pove he following boundedness esul on genealized nonhomogeneous Olicz-Moey spaces. Theoem 9. Le Φ : [,) [,) be a Young funcion and le φ saisfy he wo condiions (.a) and (.b) as usual. Then, he maximal opeao M is bounded fom L Φ,φ (μ) o wl Φ,φ (μ). If we assume ha R φ () d Cφ (R) (R>) (75) and ha Φ saisfies he doubling condiion, hen he singula inegal opeao T is bounded fom L Φ,φ (μ) o wl Φ,φ (μ). Theoem 2. Le ρ : (, ) (, ) and, fo some b (, ], φ : (, ) (, ),saisfy φ () ρ () d + φ () ρ () d Cφ() b (76) fo evey >.SupposehaΦ : [, ) [, ) is a Young funcion wih he doubling condiion. Se Then, ψ () =φ() b, Ψ() =Φ( /b ), [, ). (77) I ρf wl Ψ,ψ (μ) f L Φ,φ (μ). (78) 5.. Poof of Theoems 7 and 8. We sa wih a lemma. Lemma 2. Le Φ : [, ) [, ) be a Young funcion wih he doubling popey: Φ (2) Φ(), [, ). (79) Fo μ-measuable funcions f and g and a ball B(a, ),onehas Then, hee exiss g wl Φ,φ (μ) such ha lim k g f kwl Φ,φ (μ) =. (74) f+g Φ,B(a,) f Φ,B(a,) + g Φ,B(a,). (8)

10 Jounal of Funcion Spaces and Applicaions Poof. If f=μ-a.e. o g=μ-a.e., hen we have he equaliy ivially; so le us assume ha f =μ-a.e. and g =μ-a.e. Then, by viue of he convexiy, we have n B(a,) Φ( f (x) +g(x) f Φ,B(a,) + g )dμ(x) Φ,B(a,) f Φ,B(a,) f Φ,B(a,) + g Φ,B(a,) n Φ( f (x) B(a,) f )dμ(x) Φ,B(a,) + g Φ,B(a,) f Φ,B(a,) + g Φ,B(a,) n Φ( g (x) B(a,) g )dμ(x) Φ,B(a,) f Φ,B(a,) f Φ,B(a,) + g Φ,B(a,) + g Φ,B(a,) f Φ,B(a,) + g =. Φ,B(a,) (8) Fom he definiion of he quaniy f + g Φ,B(a,),weobain he inequaliy. Lemma 22. If Φ : [, ) [, ) is a Young funcion, hen n f(y) dμ (y) C f Φ,B(a,) (82) B(a,) fo any ball B(a, ) and μ-measuable funcion f. Poof. A nomalizaion allows us o assume ha f Φ,B(a,) = ;ouagewillbeopove n f(y) dμ (y) C. (83) B(a,) In view of he gowh condiion, we may suppose ha f assumes is value in } [, ).SinceΦis a Young funcion, we have Theefoe, Φ () Φ(), } [, ). (84) Φ () n f(y) dμ (y) B(a,) n B(a,) Φ( f(y) )dμ(y) = lim ε n B(a,) So, we ae done. Φ( f(y) f Φ,B(a,) +ε)dμ(y). Now, we ae eady fo he poof of Theoem 7. (85) Poof of Theoem 7. In view of Lemma 2, L Φ,φ (μ) is a nomed space. So, we need o pove he compleeness. To his end, we choose a sequence f k } k= of μ-measuable funcion such ha f kl Φ,φ (μ) <. (86) k= Denoe by O he oigin. Then, we have φ (L) L n f kl (B(O,L),μ) f kl Φ,φ (B(O,L),μ) < (87) k= fom Lemma 22. This implies ha k= f k(x) is finie μ-a.e. on B(O, L) fo all L N.Hence, k= f k(x) is finie μ-a.e. on R d.wihhisinmind,leusseg(x) = k= f k(x) wheneve he seies is absoluely convegen; ohewise se g(x) =. We fix a ball B(a, ).Then,wehave g f f 2 f kφ,b(a,) = inf λ > inf λ> inf λ> φ() k= : Φ( g (x) f (x) f 2 (x) f k (x) ) B(a,) λ dμ(x) n } lim inf Φ( B(a,) K λ : lim inf Φ( K B(a,) λ j=k+ f jl Φ,φ (μ). As a esul, g L Φ,φ (μ) and K j=k+ K j=k+ f j (x) )dμ(x) n} } } f j (x) )dμ(x) n} } } (88) g f f 2 f kl Φ,φ (μ) f jl. Φ,φ (μ) (89) j=k+ So j= f j conveges o g in L Φ,φ (μ). The poof of Theoem 8 is simila; we use he embedding f wl (B(a,),μ) f wl Φ,φ (μ) (9) which follows fom Lemma 22.

11 Jounal of Funcion Spaces and Applicaions 5.2. Poof of Theoem 9. We fis concenae on he maximal opeao; we modify he agumen o pove he boundedness of singula inegal opeaos lae. The poof hinges upon he decomposiion in Poposiion 2, keeping he same noaion as befoe. As fo f 2, we have a poinwise esimae, so ha a small modificaion woks. Also, we nomalize f L Φ,φ (μ) =. Le us concenae on f. Le us esablish χ Mf (9) >2}Φ,B(a,) fo any >,wheeheconsanc>is independen of and f.wieλ= χ Mf >2} Φ,B(a,).Then,wehave μmf >2} n Φ( Λ ) = n B(a,) Φ( χ Mf >2} (y) )dμ(y)= Λ (92) by viue of he dominaed convegence heoem. So, we have n Φ( Λ ) fχ f >}L. (B(a,2),μ) (93) So Since n B(a,2) f (x) χ f >} (x) we have (by he convexiy of Φ) Φ( Λ ) f (x) χ f >} (x) dμ (x). (94) } [, ), (95) n Φ( f (x) χ f >} (x) )dμ(x). (96) B(a,2) Λ In view of he doubling popey, we ae done wih he maximal opeao. As fo he singula inegal opeao, we combine he above poof and ha of Theoem 5.Wemimicheagumen above fo f, while we use esimae (66)obainedinhepoof of Theoem 5. We omi he fuhe deails Poof of Theoem 2. WesawihhepoofofaHedbeg ype inequaliy. Le R>.Then,asin(37), we have I ρf (x) R d x y n f(y) dμ (y) ρ( x y ) x y n f(y) dμ (y) ρ( + x y ) R d \ x y n f(y) dμ (y) (M n f (x) φ(r) b + f L Φ,φ (μ) φ(r)b ). (97) So we ae led o I ρf (x) (Mn f(x) b f b L Φ,φ (μ) + f L Φ,φ (μ) inf > φ()b ), (98) as we did in Poposiion. So we have o pove ψ () χ x B(a,) : f L Φ,φ (μ) inf > φ()b >} f L Φ,φ (μ), Ψ,B(a,) ψ () χ x B(a,) : M n f(x) b f b L Φ,φ >} (μ) f L Φ,φ (μ). Ψ,B(a,) (99) As fo he fis inequaliy, we use he following obsevaion: χ ψ () x B(a,) : f L Φ,φ (μ) inf φ( ) b >} > Ψ,B(a,) ψ () f L Φ,φ (μ) inf φ( ) b. > () In view of he definiion of ψ, we ae done wih he esimae. As fo he second inequaliy, we poceed as follows: ψ () χ x B(a,) : M n f(x) b f b L Φ,φ (μ) >} Ψ,B(a,) = φ() b ( χ x B(a,): M n f(x) f /b L Φ,φ (μ) >/b } Φ,B(a,) b ) =( /b b φ () χ x B(a,): M n f(x) f /b L Φ,φ (μ) >/b } ) Φ,B(a,) f L Φ,φ (μ). () Hee, fo he las inequaliy, we used Theoem 9. Conflic of Ineess The auhos declae ha hee is no conflic of ineess egading he publicaion of his pape. Acknowledgmens The auhos hank Pofesso E. Nakai of Ibaaki Univesiy fo his useful commens on he oiginal pape. The fis and second auhos ae suppoed by ITB Reseach and Innovaion Pogam 23. The hid auho is paially suppoed by Gan-in-Aid fo Scienific Reseach (C), no , Japan Sociey fo he Pomoion of Science. The fouh auho is suppoed by Fundamenal Reseach Pogam 23, Diecoae Geneal of Highe Educaion, Minisy of Educaion and Culue, Indonesia.

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