JOURNAL OF FUNCTION SPACES AND APPLICATIONS Volume 9, Numbe 3 2), 245 282 c 2, Scienific Hoizon hp://www.jfsa.ne oundedness of Lusin-aea and gλ funcions on localized Moey-Campanao spaces ove doubling meic measue spaces Haibo Lin, Eiichi Nakai and Dachun Yang Communicaed by Fenando Cobos) 2 Mahemaics Subjec Classificaion. Pimay 4225; Seconday 4235, 463. Keywods and phases. Doubling meic measue space, Popey P ), admissible funcion, Schödinge opeao, localized Moey-Campanao space, Lusin-aea funcion, gλ funcion. Absac. Le X be a doubling meic measue space and ρ an admissible funcion on X. In his pape, he auhos esablish some equivalen chaaceizaions fo he localized Moey-Campanao spaces Eρ X ) and Moey-Campanao-LO spaces Ẽ ρ X ) when α, ) and p, ). If X has he volume egulaiy Popey P ), he auhos hen esablish he boundedness of he Lusin-aea funcion, which is defined via kenels modeled on he semigoup geneaed by he Schödinge opeao, fom Eρ X )oẽ ρ X ) wihou invoking any egulaiy of consideed kenels. The same is ue fo he gλ funcion and, unlike he Lusin-aea funcion, in his case, X is even no necessay o have Popey P ). These esuls ae also new even fo R d wih he d-dimensional Lebesgue measue and have a wide applicaions.. Inoducion The heoy of Moey-Campanao spaces plays an impoan ole in hamonic analysis and paial diffeenial equaions; see, fo example,, 5, 6, 22, 23, 25, 27, 29, 3] and hei efeences. I is well-known ha Coesponding auho
246 oundedness of Lusin-aea and g λ funcions he dual space of he Hady space H p R d ) wih p, ) is he Moey- Campanao space E /p, R d ). Noice ha he Moey-Campanao spaces on R d ae essenially elaed o he Laplacian Δ d j= 2. x 2 j On he ohe hand, hee exiss an inceasing inees on he sudy of Schödinge opeaos on R d and he sub-laplace Schödinge opeaos on conneced and simply conneced nilpoen Lie goups wih nonnegaive poenials saisfying he evese Hölde inequaliy; see, fo example, 6, 7, 8, 9, 7, 8, 26, 35, 37]. Le L Δ+V be he Schödinge opeao on R d, whee he poenial V is a nonnegaive locally inegable funcion. Denoe by q R d ) he class of nonnegaive funcions saisfying he evese Hölde inequaliy of ode q. Fo V d/2 R d ) wih d 3, Dziubański e al. 6, 7, 8] sudied he MO-ype space MO L R d ) and he Hady space H p L Rd ) wih p d/d +), ] and, especially, poved ha he dual space of HL Rd )ismo L R d ); moeove, hey obained he boundedness on hese spaces of he Lilewood-Paley g -funcion associaed o L. Le X be a doubling meic measue space, which means ha X is a space of homogeneous ype in he sense of Coifman and Weiss 2, 3], bu X is endowed wih a meic insead of a quasi-meic. Le ρ be a given admissible funcion modeled on he known auxiliay funcion deemined by V d/2 R d ) see 35] o 2.4) below). The localized aomic Hady space Hρ p, q X ) wih p, ] and q, ] p, ], he localized Moey- Campanao space Eρ X ) and localized Moey-Campanao-LO space X ) wih α R and p, ) wee inoduced in 34]. Moeove, he Ẽ ρ boundedness fom Eρ X )oẽα, ρ p X ) of seveal maximal opeaos and he Lilewood-Paley g -funcion, which ae defined via kenels modeled on he semigoup geneaed by he Schödinge opeao, was obained in 34]. Meanwhile, he boundedness fom localized MO-ype space MO ρ X ) o LO-ype space LO ρ X ) of he Lusin-aea and gλ funcions was esablished in 9]. The pupose of his pape is o invesigae behavios of he Lusinaea and gλ funcions on Moey-Campanao spaces ove doubling meic measue spaces. Pecisely, le X be a doubling meic measue space and ρ an admissible funcion on X. In his pape, we fis esablish some equivalen chaaceizaions fo Eρ X )andẽα, ρ p X )whenα, ) and p, ). To obain he boundedness of he Lusin-aea funcion on he Moey-Campanao spaces, we need o assume ha X has he volume egulaiy Popey P ), which was inoduced in 9], moivaed by Colding-Minicozzi II 4] and Tessea 3]. We emak ha he volume egulaiy popey is elaed o he Følne sequence of a compac geneaing se of a compacly geneaed locally compac goup wih polynomial gowh
H. Lin, E. Nakai, D. Yang 247 in 3] and used o esablish he genealized Liouville heoems fo hamonic secions of Hemiian veco bundles ove a complee meic space in 4]. In his pape, if X has Popey P ), we hen esablish he boundedness of he Lusin-aea funcion fom Eρ X )oẽρ X ) wihou invoking any egulaiy of consideed kenels. The coesponding boundedness of gλ funcion fom Eρ X )oẽα, ρ p X ) is also esablished in his pape. oh he Lusin-aea funcion and he gλ funcion ae defined via kenels modeled on he semigoup geneaed by he Schödinge opeao. Moeove, an ineesing phenomena is ha unlike he Lusin-aea funcion, he boundedness of he gλ funcion needs neihe he egulaiy of he kenels no Popey P ) of X, which eflecs he specialiy of he sucue of he gλ funcion. These esuls ae new even on Rd wih he d-dimensional Lebesgue measue and he Heisenbeg goup, and apply in a wide ange of seings, fo insance, o he Schödinge opeao o he degeneae Schödinge opeao on R d, o he sub-laplace Schödinge opeao on Heisenbeg goups o conneced and simply conneced nilpoen Lie goups. This pape is oganized as follows. Le X be a doubling meic measue space and ρ an admissible funcion on X. InSecion2,weesablishsome equivalen chaaceizaions fo Eρ X ) and Ẽα, ρ p X ) when p, ) and α, /p) oα /p, ); see Theoems 2. and 2.2 below. Moeove, unde he assumpion ha sup x X μx, ρx))) =, wepove ha he Moey-Campanao-LO space Ẽα, ρ p X ) is a pope subspace of he Moey-Campanao space Eρ X )whenp, ) andα /p, ); see Theoem 2.2iii) below. In Secion 3, assuming ha X has Popey P ) and he Lusin-aea funcion Sf) is bounded on L p X ) wih p, ), we pove ha if f Eρ X ), hen Sf)] 2 p/2 Ẽ2α, ρ X ) wih nom no moe han C f 2 Eρ X ),wheec is a posiive consan independen of f ; see Theoem 3. below. As a coollay, we obain he boundedness of he Lusin-aea funcion fom Eρ X ) o Ẽα, ρ p X ); see Coollay 3. below. If he gλ funcion gλ f) is bounded on Lp X ) wih p, ), he coesponding esuls fo gλ f) ae also esablished, and moeove, in his case, X is no necessay o have Popey P ); see Theoem 3.2 and Coollay 3.2 below. We poin ou ha Theoems 3. and 3.2 and Coollaies 3. and 3.2 ae ue o he Schödinge opeao o he degeneae Schödinge opeao on R d, o he sub-laplace Schödinge opeao on Heisenbeg goups o conneced and simply conneced nilpoen Lie goups; see 34] fo he deailed explanaions. Noice ha Eρ,p X ) = MO ρ X ) and Ẽρ,p X )=LO ρ X ) when p, ). Thus, he esuls in his secion when α = wee aleady obained in 9].
248 oundedness of Lusin-aea and g λ funcions We emak ha he esuls obained in Secion 3 ae also new even on R d wih he d-dimensional Lebesgue measue and he Heisenbeg goup, since we do no need any egulaiy of involved kenels. Howeve, o esablish he boundedness of Lusin-aea funcion on a doubling meic measue space X, we need ceain egulaiy of X, namely, he volume egulaiy Popey P ), which eflecs he specialiy of he Lusin-aea funcion, compaing wih he coesponding esuls of he gλ funcion. Moeove, Rd wih he Lebesgue measue and he Heisenbeg goup have he volume egulaiy Popey P ); see 9]. Finally, we make some convenions. Thoughou his pape, we always use C o denoe a posiive consan ha is independen of he main paamees involved bu whose value may diffe fom line o line. Consans wih subscips, such as C and K, do no change in diffeen occuences. If f Cg, wehenwief g o g f ;andiff g f,wehen wie f g. We also use o denoe a ball of X,andfoλ>, λ denoes he ball wih he same cene as, bu adius λ imes he adius of. Moeove, se X\. Also, fo any se E X, χ E denoes is chaaceisic funcion. Fo all f L loc X ) and balls,wealwaysse f μ) fy) dμy). 2. Some chaaceizaions of localized Moey-Campanao spaces Le X beadoublingmeicmeasuespaceandρ an admissible funcion on X. In his secion, we esablish some equivalen chaaceizaions fo Eρ X ) and Ẽα, ρ p X ) when α, ) and p, ). Moeove, unde he assumpion ha sup x X μx, ρx))) =, we pove ha he Moey-Campanao-LO space Ẽα, ρ p X ) is a pope subspace of he Moey-Campanao space Eρ X )whenp, ) andα /p, ). We begin wih ecalling he noion of doubling meic measue spaces 2, 3]. Definiion 2.. Le X,d) be a meic space endowed wih a oel egula measue μ such ha all balls defined by d have finie and posiive measues. Fo any x X and, ), se he ball x, ) {y X : dx, y) <}. The iple X,d,μ) is called a doubling meic measue space if hee exiss a consan C, ) such ha fo all x X and, ), 2.) μx, 2)) C μx, )) doubling popey).
H. Lin, E. Nakai, D. Yang 249 Fom Definiion 2., i is easy o see ha hee exis posiive consans C 2 and n such ha fo all x X,, ) andλ, ), 2.2) μx, λ)) C 2 λ n μx, )). Now we ecall he noion of admissible funcions inoduced in 35]. Definiion 2.2 35]). A posiive funcion ρ on X is called admissible if hee exis posiive consans C and k such ha fo all x, y X, 2.3) ρx) C ρy) + ) k dx, y). ρy) Obviously, if ρ is a consan funcion, hen ρ is admissible. Moeove, le x X be fixed. The funcion ρy) + dx,y)) s fo all y X wih s, ) also saisfies Definiion 2.2 wih k = s/ s) whens, ) and k = s when s, ). Anohe non-ivial class of admissible funcions is given by he well-known evese Hölde class q X,d,μ), which is wien as q X ) fo simpliciy. Recall ha a nonnegaive poenial V is said o be in q X ) wih q, ] if hee exiss a posiive consan C such ha fo all balls of X, ) /q V y)] q dy C V y) dy wih he usual modificaion made when q =. I is known ha if V q X ) fo ceain q, ], hen V is an A X ) weigh in he sense of Muckenhoup, and also V q+ɛ X ) fo ceain ɛ, ); see, fo example, 27] and 28]. Thus q X )= q q, ] q X ). Fo all V q X ) wih q, ] andallx X,se { 2.4) ρx) sup >: 2 } V y) dy ; μx, )) x, ) see, fo example, 26] and also 35]. I was also poved in 35] ha ρ in 2.4) is an admissible funcion if n, ), q max{, n/2}, ] and V q X ). The following localized Moey-Campanao space and localized Moey- Campanao-LO space associaed o he admissible funcion ρ wee fis inoduced in 34]. Definiion 2.3. Le ρ be an admissible funcion on X, D {x, ) : x X, ρx)}, p, ) andα R. Denoeby any ball of X.
25 oundedness of Lusin-aea and g λ funcions i) A funcion f L p loc X ) is said o be in he localized Moey- Campanao space Eρ X )if { f E ρ X ) sup / D μ)] +pα +sup D { μ)] +pα } /p fy) f p dμy) fy) p dμy)} /p <. ii) A funcion f L p loc X ) is said o be in he localized Moey- Campanao-LO space Ẽα, ρ p X )if { f Ẽ ρ X ) sup / D μ)] +pα +sup D { μ)] +pα fy) essinf ] } p /p f dμy) fy) p dμy)} /p <. iii) Le α, ). A funcion f on X is said o be in he localized Lipschiz space Lip ρ α; X ) if hee exiss a nonnegaive consan C such ha fo all x, y X and balls conaining x and y wih / D, fx) fy) Cμ)] α, and ha fo all balls D, f L ) Cμ)] α. The minimal nonnegaive consan C as above is called he nom of f in Lip ρ α; X ) and denoed by f Lipρ α; X ). iv) Le α, ). A funcion f L p loc X ) is said o be in he Moey space L X )if { f L X ) sup X μ)] +αp fx) p dμx)} /p <. Remak 2.. i) Fo all α R and p, ), Ẽρ X ) Eρ X ). ii) When α = and p, ), we denoe Eρ,p X )bymo p ρx )and MO ρx )bymo ρ X ). And we also denoe Ẽ,p ρ X )bylo p ρx )and LO ρx )bylo ρ X ). The localized LO space was fis inoduced in 3] in he seing of R d endowed wih a nondoubling measue. iii) If X is he Euclidean space R d and ρ, hen MO ρ X )isjus he localized MO space of Goldbeg ], and Lip ρ α; X ) wih α, ) is jus he inhomogeneous Lipschiz space see also ]). iv) When α, ), equivalen noms; see 34] fo deails. Ẽ ρ X ) = E ρ X ) = Lip ρ α; X ) wih Theoem 2.. Le X be a doubling meic measue space and ρ an admissible funcion on X.Lep, ) and α, /p).
H. Lin, E. Nakai, D. Yang 25 i) If μx )=, henẽα, ρ p X )=Eρ X )=L X )={}. ii) If μx ) <, henẽα, ρ p X )=Ẽ /p, ρ p X ), Eρ X )=Eρ /p, p X ) and L X )=L /p, p X )=L p X ) wih equivalen noms, especively. Poof. i) In his case, since μx ) =, hen, fo all x X, μx, )) as, which ogehe wih α, /p) implies ha if f L X ), hen fo all x, ) X, fy) p dμy) f p L X ) μx, ))]+αp, x, ) when,andiff Eρ X ), hen fo all x, ) X wih ρx), fy) p dμy) f p Eρ X ) μx, ))]+αp, x, ) as. Thus, in boh cases, we have ha X fy) p dμy), which implies ha fy) = fo almos all y X. Theefoe, L X ) = Eρ X ) = {}, which ogehe wih he fac ha Ẽα, ρ p X ) Eρ X ) yields i). ii) Since μx ) <, by 24, Lemmma 5.], hee exiss > such ha X x, ) fo all x X, which ogehe wih ha ρ is admissible and 2.2) implies ha < inf μx, ρx)/2)) = inf μx, )) x X x X,ρx)/2 <ρx) sup μx, )) μx ) <. x X,ρx)/2 <ρx) If <<ρx)/2, by +αp <, we hen have ] p μx, ))] +αp fy) essinf f dμy) x, ) x, ) μx, ρx)/2))] +αp fy) essinf and x, ρx)/2) fy) μx, ))] +αp f x, ) p dμy) x, ) μx, ρx)/2))] +αp 2 p μx, ρx)/2))] +αp x, ρx)/2) x, ρx)/2) x, ρx)/2) f fy) fx, ) p dμy) ] p dμy), fy) fx, ρx)/2) p dμy).
252 oundedness of Lusin-aea and g λ funcions Hence, sup x X, <<ρx) and = sup μx, ))] +αp fy) essinf x, ) x X,ρx)/2 <ρx) sup x X,ρx)/2 <ρx) sup x X, <<ρx) sup x X, <<ρx) μx, ))] +αp x, ) x, ) x, ) fy) essinf x, ) f fy) essinf x, ) f ] p f dμy) x, ) fy) essinf ] p dμy) ] p dμy), fy) fx, μx, ))] +αp ) p dμy) x, ) sup x X,ρx)/2 <ρx) sup x X,ρx)/2 <ρx) sup x X, <<ρx) which imply ha Ẽα, ρ p Similaly, μx, ))] +αp x, ) x, ) x, ) fy) fx, ) p dμy) fy) f x, ) p dμy), X )=Ẽ /p, ρ p ] p f dμy) x, ) fy) fx, ) p dμy) X ), and Eρ X )=Eρ /p, p X ). < inf μx, )) sup μx, )) μx ) <, x X,ρx) x X,ρx) and moeove, sup x X,> = sup μx, ))] +αp x X,ρx) sup x X,ρx) x, ) μx, ))] +αp fy) p dμy) x, ) fy) p dμy) x, ) fy) p dμy) sup x X,> x, ) fy) p dμy) f Lp X ), which leads o ha L X )=L /p, p X )=L p X ).
H. Lin, E. Nakai, D. Yang 253 Theoem 2.2. Le X be a doubling meic measue space and ρ an admissible funcion on X. Ifp, ) and α /p, ), hen he followings hold. i) Eρ X )=L X ) wih equivalen noms. ii) Fo all f, f Eρ X ) if and only if f Ẽα, ρ p X ) and moeove, f Ẽ ρ X ) f Eρ X ). iii) If M sup x X μx, ρx))) <, hen hee exiss a posiive consan C such ha fo all f saisfying < essinf X f<, ] f E ρ X ) f Ẽ ρ X ) C f E ρ X ) + M α essinf X f). iv) If sup x X μx, ρx))) =, hen hee exiss a funcion f Eρ X ) such ha < essinf X f< and f/ Ẽα, ρ p X ). Remak 2.2. i) I uns ou ha Theoem 2.2i), ii) and iii) hold fo α, ) and p, ) by Theoem 2.. ii) If X is an RD-space, Theoem 2.2i) & ii) ae aleady obained in 34], which ae used o pove Theoem 2.2i) & ii). Also we show Theoem 2.2iii) by fis assuming ha i is ue fo RD-space X,whichis poved in Poposiion 2. below. Recall ha he space X is said o have he evese doubling popey if hee exis consans κ, n]andk, ] such ha fo all x X,, 2diamX )) and λ, 2diamX )/), 2.5) K λ κ μx, )) μx, λ)). If X,d,μ) saisfies he condiions 2.2) and 2.5), hen X,d,μ) is called an RD-space, which was fis inoduced in 2] see also 2, 36] fo some equivalen chaaceizaions of RD-spaces). iii) y an agumen simila o ha used in he poof of Theoem 2.2i) & ii) when X,d,μ) is an RD-space wih d being a meic in 34], i is easy o see ha if X,d,μ) is an RD-space wih d being a quasi-meic, Theoem 2.2i) & ii) ae also ue. Moeove, a sligh modificaion of he poof below shows ha he whole Theoem 2.2 holds fo X wih d being a quasi-meic. iv) I was poved in 9] ha Theoem 2.2ii) is no ue when α =. To pove Theoem 2.2, we need some echnical lemmas. Following Macías and Segovia 2], we call a doubling meic measue space o be nomal if hee exis posiive consans K 2 and K 3 such ha fo all x X and μ{x}) <<μx ), 2.6) K 2 μx, )) K 3.
254 oundedness of Lusin-aea and g λ funcions Fo a doubling meic measue space X,d,μ), le { inf{μ) : is a ball conaining x and y} if x y, 2.7) δx, y) if x = y. Macías and Segovia 2] showed ha X,δ,μ) is a nomal space of homogeneous ype, namely, δ is a quasi-meic and μ saisfies 2.) and 2.6). Moeove, he opologies induced on X by d and δ coincide. In his secion, se d x, ) {y X : dx, y) <} and δ x, ) {y X : δx, y) <} fo all x X and >. Fo all x X,le 2.8) x) μ d x, ρx))). Lemma 2.. Le X be a doubling meic measue space and ρ an admissible funcion on X, and le be as in 2.8). If α, ) and p, ), henl X,d,μ)=L X,δ,μ) and Eρ X,d,μ)= X,δ,μ) wih equivalen noms, especively. E Lemma 2.2. Le X be a doubling meic measue space and ρ an admissible funcion on X,andle be as in 2.8). Le α, ) and p, ). If f, hen f Ẽ ρ X,d,μ) and f Ẽρ δ X,δ,μ) ae equivalen wih equivalen consans independen of f. Lemma 2.3. Le X be a doubling meic measue space and ρ an admissible funcion on X. Le be as in 2.8), α, ) and p, ). If M sup x X μ d x, ρx))) < and < essinf X f <, hen f Ẽ ρ X,d,μ) + M α essinf X f) and f Ẽ X,δ,μ) + M α essinf X f) ae equivalen wih equivalen consans independen of f. To pove Lemmas 2., 2.2 and 2.3, we fis sae some basic facs. Fo any d-ball d x, ), le μ d x, )). y he definiion of δ,wehave ha 2.9) d x, ) δ x, ) and μ δ x, )) K 3 μ d x, )). Moeove, 2.) <ρx) x), = ρx) = x), >ρx) x). Convesely, by 22, Lemma 3.9] o 4, Poposiion 2.], fo any δ -ball δ x, ), hee exiss a posiive consan, which may depend on x and
H. Lin, E. Nakai, D. Yang 255, such ha 2.) δ x, ) d x, ) and μ d x, )) C 3 μ δ x, )) fo some consan C 3, ), which is independen of x, and.inhis case, if < x)/c 3 K 3 ), hen μ d x, )) C 3 μ δ x, )) C 3 K 3 < x) =μ d x, ρx))). If > x)/k 2,hen μ d x, )) μ δ x, )) K 2 > x) =μ d x, ρx))). Tha is, 2.2) { < x)/c 3 K 3 ) <ρx), > x)/k 2 >ρx). Poof of Lemma 2.. y 2.9) and 2.), i is easy o see L X,d,μ)= L X,δ,μ) wih equivalen noms. Now, we pove ha Eρ X,d,μ) = Eρ δ X,δ,μ) wih equivalen noms. Pa ) Fo any d-ball = d x, ), le 2 = δ x, ), whee = μ d x, )). Fom 2.9), i follows ha μ ) μ 2 ). Case. <ρx). In his case, by 2.), < x) o = x), which ogehe wih 2.9) implies ha 2.3) { } /p μ )] α fy) f p dμy) μ ) { } /p 2 μ )] α fy) f 2 p dμy) μ ) { } /p μ 2 )] α fy) f 2 p dμy) f μ 2 ) E X,δ,μ). 2 Case 2. ρx). In his case, by 2.), x), which ogehe wih 2.9) leads o ha 2.4) { μ )] α μ 2 )] α μ ) { μ 2 ) } /p fy) p dμy) } /p fy) p dμy) f E 2 X,δ,μ).
256 oundedness of Lusin-aea and g λ funcions Theefoe, f E ρ X,d,μ) f E X,δ,μ). Pa 2) Fo any δ -ball δ x, ), le 2 d x, ), whee is as in 2.). Fom 2.), i follows ha μ ) μ 2 ). Case. < x)/c 3 K 3 ). In his case, by 2.2), <ρx). y an agumen simila o he esimae of 2.3), we have { } /p μ )] α fy) f p dμy) f μ ) E ρ X,d,μ). Case 2. > x)/k 2. In his case, by 2.2), > ρx). y an agumen simila o he esimae of 2.4), we have { } /p μ )] α fy) p dμy) f μ ) E ρ X,d,μ). Case 3. x)/c 3 K 3 ) x)/k 2. In his case, le δ x, ) and δ x, 2C 3 K 3 /K 2 )). Then μ ) μ ). Hence { } /p μ )] α fy) f p dμy) μ ) { } /p 2 μ )] α fy) f μ ) p dμy) { } /p μ )]α μ ) fy) f p dμy) { /p μ )]α μ ) fy) dμy)} p, and { } { /p μ )] α fy) p dμy) μ ) μ )]α μ ) Since 2C 3 K 3 /K 2 )> x)/k 2, using Case 2, we have { /p μ )]α μ ) fy) dμy)} p f E ρ X,d,μ). fy) p dμy)} /p. Theefoe, f E X,δ,μ) f E ρ X,d,μ) and we ae done.
H. Lin, E. Nakai, D. Yang 257 Poof of Lemma 2.2. Pa ) Fo any d-ball d x, ), le 2 δ x, ), whee μ d x, )). Fom 2.9), i follows ha μ ) μ 2 ). Case. <ρx). In his case, by 2.), < x) o = x), which implies ha 2.5) μ )] α { μ ) { μ ) { μ 2 ) μ )] α μ 2 )] α ] p /p fy) essinf f dμy)} 2 fy) essinf f 2 fy) essinf f 2 ] p dμy)} /p ] p } /p dμy) f E X,δ,μ). Case 2. ρx). In his case, by 2.), x). y an agumen simila o he esimae of 2.4), we have { } /p μ )] α fy) p dμy) f μ ) E X,δ,μ). Theefoe, f E ρ X,d,μ) f E X,δ,μ). Pa 2) Fo any δ -ball = δ x, ), le 2 = d x, ), whee is as in 2.). Fom 2.), i follows ha μ ) μ 2 ). Case. < x)/c 3 K 3 ). In his case, by 2.2), <ρx). y an agumen simila o he esimae of 2.5), we have { ] p } /p μ )] α fy) essinf f dμy) f μ ) E ρ X,d,μ). Case 2. > x)/k 2. In his case, by 2.2), > ρx). y an agumen simila o he esimae of 2.4), we have { } /p μ )] α fy) p dμy) f μ ) E ρ X,d,μ). Case 3. x)/c 3 K 3 ) x)/k 2. In his case, le δ x, ) and δ x, 2C 3 K 3 /K 2 )). Then μ ) μ ). Hence, if f,
258 oundedness of Lusin-aea and g λ funcions hen { μ )] α μ ) μ )] α μ )]α { μ ) { μ ) ] p /p fy) essinf f dμy)} } /p fy) p dμy) fy) p dμy)} /p, and { } { /p μ )] α fy) p dμy) μ ) μ )]α μ ) Since 2C 3 K 3 /K 2 )> x)/k 2, using Case 2, we have { /p μ )]α μ ) fy) dμy)} p f E ρ X,d,μ). fy) p dμy)} /p. Theefoe, if f, hen f E X,δ,μ) f E ρ X,d,μ). PoofofLemma2.3. Le M sup x X μ d x, ρx))) = sup x X x) <. y he same way as in he poof of Lemma 2.2, we divide he poof ino Pa ) and Pa 2). Then we have he same conclusions as in Case 2 of Pa ) and in Cases and 2 of Pa 2) of he poof of Lemma 2.2. So we only need o conside Case of Pa ) and Case 3 of Pa 2) heein. Pa ) Fo any d-ball d x, ), le 2 δ x, ), whee μ d x, )). Fom 2.9), i follows ha μ ) μ 2 ). Case. <ρx). In his case, by 2.), < x) o = x). If < x), hen we have he same inequaliy as 2.5). If = x), hen μ 2 ) x). Hence, if < essinf X f<, hen 2.6) { μ )] α μ ) μ 2 )] α μ 2 )] α { μ 2 ) { μ 2 ) ] p /p fy) essinf f dμy)} ] p /p fy) essinf f dμy)} 2 2 f Ẽ X,δ,μ) + M α essinf f X } /p fy) p dμy) + essinf X f 2 x)] α ).
Theefoe, if < essinf X f<, hen f Ẽ ρ H. Lin, E. Nakai, D. Yang 259 X,d,μ) f Ẽρ δ X,δ,μ) + M α essinf f X Pa 2) Fo any δ -ball δ x, ), le 2 d x, ), whee is as in 2.). Fom 2.), i follows ha μ ) μ 2 ). Case 3. x)/c 3 K 3 ) x)/k 2. In his case, le δ x, ) and δ x, 2C 3 K 3 /K 2 )). Then μ ) μ ) x). Hence, if < essinf X f<, hen μ )] α { μ ) { μ )]α { μ )]α μ )]α μ ) μ ) { μ ) ] p /p fy) essinf f dμy)} } p /p fy) essinf f] dμy) } /p fy) p dμy) + essinf X f x)] α ). fy) p dμy)} /p + M α essinf X and { } { /p μ )] α fy) p dμy) μ ) μ )]α μ ) ) f, fy) p dμy)} /p. Since 2C 3 K 3 /K 2 )> x)/k 2, using Case 2, we have { /p μ )]α μ ) fy) dμy)} ) p f Ẽ ρ X,d,μ) +M α essinf f. X Theefoe, if < essinf X f<, hen f Ẽ X,δ,μ) f Ẽρ X,d,μ) + M α essinf f X ). Lemma 2.4 c.f. 2, Lemma 3.3]). Le α /p, ). Then fo all X, χ L X ) =μ)] α.
26 oundedness of Lusin-aea and g λ funcions Poof. Fom he equaliy { /p μ)] α χ x) dμx)} p =μ)] α, μ) i follows ha χ L X ) μ)] α. Fo any balls z, ), if μz, )) <μ), hen { μz, ))] α μz, )) /p χ x) dμx)} p μz, )) α μ)] α. z, ) If μz, )) μ), hen { } /p μz, ))] α χ x) p dμx) μz, )) z, ) { } /p μ z, )) = μz, ))] α μz, )) ) α+/p μ z, )) = μ z, ))] α μ)] α. μz, )) Theefoe, χ L X ) μ)] α, which implies ha χ L X ) = μ)] α. Poof of Theoem 2.2. Since X,δ,μ) is nomal, X,δ,μ) is also an RD-space. i) y Lemma 2. and Remak 2.2iii), we have E ρ X,d,μ)=E X,δ,μ)=L X,δ,μ)=L X,d,μ). ii) If f, hen, by Remak 2.2iii), we obain ha f E, which ogehe wih Lemmas 2. and 2.2 yields ha f Ẽ X,δ,μ) f E ρ f E X,d,μ) f E X,δ,μ) f Ẽ X,δ,μ) X,δ,μ) f Ẽ X,δ,μ) f Ẽρ X,d,μ). iii) In he case M sup x X μx, ρx))) <, if < essinf X f<, hen, by Poposiion 2. below, we obain ha ), X,δ,μ) f Eρ δ X,δ,μ) + M α essinf f X
H. Lin, E. Nakai, D. Yang 26 which ogehe wih Lemmas 2. and 2.3 yields ha f E ρ X,d,μ) f Ẽ ρ X,d,μ) f Ẽρ δ X,δ,μ) + M α essinf X ) f ). f E X,δ,μ) + M α essinf X f E ρ X,d,μ) + M α essinf f X ) f iv) Since sup x X μx, ρx))) =, wechoose j z j,ρz j )/2), j N, sohaμ j ) as j. Then, we have wo siuaions ha I) fo all j, j i= i) j+ =, o II) hee exiss j N such ha j j2 = fo j <j 2 j, and ha j i= i) j fo all j>j. Le b>. Case I). Fo each j, choose j > so ha j < ρz j )/2 and μz j, j ))] α < /2 j. In his case, μz j, j )) < /2 j ) /α <. Le f b j f j and f j χ zj, j). Then essinf X f = b and by Lemma 2.4, we have f E ρ X ) 2 f L X ) 2b f j L X ) =2b μz j, j ))] α 2b. j= On he ohe hand, μ j )] α μ j ) j = μ j )] α μ j ) j b μ j )] α μ j ) j= ] p /p fx) essinf f dμx)) j bf j x) b)] p dμx) ) /p as j. ) /p Case II). Le f b j j= f j and f j x) χ j x). Then essinf X f = b and by Lemma 2.4, we have f E ρ X ) 2 f L X ) 2b j j= f j L X ) =2b j j= μ j )] α <.
262 oundedness of Lusin-aea and g λ funcions On he ohe hand, fo j>j, μ j )] α μ j ) = = μ j )] α b μ j )] α b μ j )] α j μ j ) μ j ) ] p /p fx) essinf f dμx)) j j j b j i= μ j ) μ i= μ i) μ j ) f i x) b)] p dμx) ) /p j ) ) ] ) /p i j i= ) /p as j. Combining he esimaes fo Cases I) and II) yields vi), which complees he poof of Theoem 2.2. In he poof of Theoem 2.2iii) above, we used he following poposiion. Poposiion 2.. Theoem 2.2 iii) holds if X is an RD-space. To pove Poposiion 2., we begin wih some echnical lemmas. A saighfowad compuaion via 2.5) leads o he following echnical lemma. Lemma 2.5. Le X be an RD-space and θ, ). Then,heeexiss a posiive consan C such ha fo all z X and <<s<, s d μz, ))] θ C μz, ))] θ. Le MOf,) fy) f dμy). μ) Then, by Lemma 2.4 in 24], hee exiss a posiive consan C such ha fo all z X and <<s<, 2.7) fz, ) f z, s) C 2s MOf,z, )) d. Lemma 2.6. Le X be an RD-space and α /p, ). Then hee exiss a posiive consan C such ha fo all f Eρ X ), z X and <<ρz), fz, ) f z, ρz)) C f E ρ X )μz, ))] α.
H. Lin, E. Nakai, D. Yang 263 Poof. Case. ρz)/2 <ρz). y 2.) and he Hölde inequaliy, we have fz, ) f z, ρz)) fx) dμx)+ fz, ρz)) μz, )) z, ) C +) fx) dμx) μz, ρz))) z, ρz)) ) /p C +) fx) p dμx) μz, ρz))) z, ρz)) C +) f E ρ X )μz, ))] α. Case 2. <<ρz)/2. Using 2.7) and Lemma 2.5, we have ρz) MOf,z, )) f z, ) f z,ρz)/2) d ρz) μz, ))] α f E ρ X ) d f E ρ X )μz, ))] α. Combining he esimaes fo Case and Case 2 complees he poof of Lemma 2.6. Poof of Poposiion 2.. Le M sup x X μx, ρx))) <. y Lemma 2.6, we have ha if <<ρz), hen fz, ) f z, ρz)) f E ρ X )μz, ))] α. Now, fo z, ), if <<ρz), hen μ)] α μ) μ)] α μ)] α μ) μ) p ) /p fx) essinf f] dμx) ] p /p fx) essinf dμx)) z, ρz)) f + μ)] α f z, ) f z, ρz)) + f E ρ X ) + μ)] α f z, ρz)) ) /p fx) f p dμx) μ)] α f z, ρz)) essinf f z, ρz)) essinf. z, ρz)) f
264 oundedness of Lusin-aea and g λ funcions If < essinf X f<, hen μ)] α f z, ρz)) essinf f z, ρz)) fz, μ)] α ρz)) + essinf μ)] α f X μz, ρz)))]α μz, ρz)))]α essinf X f μ)] α f E ρ X ) + μ)] α μz, ρz)))] α f E ρ X ) + M α essinf X f. Theefoe, if < essinf X f<, hen f E ρ X ) f Ẽ ρ X ) f Eρ X ) + M α essinf X f), which complees he poof of Poposiion 2.. 3. oundedness of Lusin-aea and g λ funcions Le X be a doubling meic measue space and ρ an admissible funcion. In his secion, we conside he boundedness of ceain vaians of Lusinaea and gλ p funcions fom Eα, ρ X )oẽα, ρ p X ). The boundedness fom MO ρ X )olo ρ X ) of hese opeaos wee obained in 9]. We emak ha unlike he boundedness of he gλ funcion, o obain he boundedness of he Lusin-aea funcion, we need o assume ha X has he following volume egulaiy Popey P ), which was inoduced in 9]; see also 4, 3]. Definiion 3. 9]). A doubling meic measue space X,d,μ)issaid o have Popey P ), if hee exis posiive consans δ and C such ha fo all x X, s, ) and s, ), s ) δ 3.) μx, + s)) μx, )) C μx, )). Remak 3.. Thee ae many examples of doubling meic measue spaces having Popey P ), such as R d,,dx), he d-dimensional Euclidean space endowed wih he Euclidean nom and he Lebesgue measue dx; R d,, wx)dx), he d-dimensional Euclidean space endowed wih he Euclidean nom and he measue wx)dx, wheew is an A 2 R d )weighanddx is he Lebesgue measue; H n,d,dx), he 2n +)- dimensional Heisenbeg goup H n wih a lef-invaian meic d and he
H. Lin, E. Nakai, D. Yang 265 Lebesgue measue dx; G, d,μ), he nilpoen Lie goup G wih a Cano- Caahéodoy conol) disance d and a lef invaian Haa measue μ and so on; see 9] fo moe deails. In wha follows, we always se V x) μx, )) and V x, y) μx, dx, y))) fo all x, y X and, ). Le ρ be an admissible funcion on X and {Q } > a family of opeaos bounded on L 2 X ) wih inegal kenels {Q x, y)} > saisfying ha hee exis consans C, δ, ), δ 2, ) and γ, ) such ha fo all, ) andx, y X, Q) i Q x, y) C V x)+v x, y) +dx, y) )γ ρx) +ρx) )δ ; Q) ii X Q x, z) dμz) C +ρx) )δ2. Fo all f L loc X )andx X, define he Lilewood-Paley g -funcion by seing { 3.2) gf)x) Q f)x) 2 d } /2, and Lusin-aea and g λ funcions, especively, by seing { 3.3) Sf)x) and 3.4) g λf)x) whee λ, ). { X, ) dx, y)< Q f)y) 2 dμy) V y) + dx, y) } /2 d, ) λ Q f)y) 2 dμy) V y) } /2 d, Theoem 3.. Le X be a doubling meic measue space having Popey P ). Le p, ), ρ be an admissible funcion on X, he Lusin-aea funcion Sf) as in 3.3) and α, min{γ/n, δ / + k )n], δ 2 /n, δ/2n)}). If Sf) is bounded on L p X ), hen hee exiss a posiive consan C such ha fo all f Eρ X ), Sf)] 2 p/2 Ẽ2α, ρ X ) and Sf)] 2 Ẽ 2/2 ρ X ) C f 2 Eρ X ). To pove Theoem 3., we begin wih he following wo echnical lemmas, which wee obained in 34]. Lemma 3. 34, Lemma 2.4]). Le α R, p, ), ρ be an admissible funcion on X and D as in Definiion 2.3. Then hee exiss a posiive consan C such ha fo all f E ρ X ),
266 oundedness of Lusin-aea and g λ funcions i) fo all balls x,) D, ) αn C ρx) μ)] α f E ρ X ), α >, fz) dμz) ) μ) C μ)] α f E ρ X ), α ; +log ρx) ii) fo all x X and < < 2, C f x, ) f x, 2) C 2 ) αn μx, )] α f E ρ X ), α >, ) +log 2 μx, )] α f E ρ X ), α. Lemma 3.2 34, Lemma 4.]). Le α, min{γ/n, δ 2 /n}), p, ) and ρ be an admissible funcion on X. Then hee exiss a posiive consan C such ha fo all f Eρ X ), x X and >, ) δ ρx) Q f)x) C μx, ))] α f + ρx) E ρ X ). PoofofTheoem3.. y similaiy, we only pove he case when α>. Le f Eρ X ). y he homogeneiy of E ρ X ) and Ẽ ρ X ),we may assume ha f E ρ X ) =. Le x,). We pove Theoem 3. by consideing he following wo cases. Case I. x,) D.Inhiscase, ρx ). We need o pove ha 3.5) Sf)x)] p dμx) μ)] +αp. Fo all x X,wie 8 Sf)x)] 2 = Q f)y) 2 dμy) d dx, y)< V y) + S f)x)] 2 +S 2 f)x)] 2. 8 dx, y)< y he L p X )-boundedness of Sf) and 2.), we have 3.6) S fχ 2 )x)] p dμx) fx) p dμx) μ)] +αp. Fix x. Noice ha if dx, y) <, hen fo all z X, 3.7) + dy, z) + dx, z) and V y)+v y, z) V x)+v x, z). 2
H. Lin, E. Nakai, D. Yang 267 Fom Q) i, 3.7), 2.2), he Hölde inequaliy and γ>αn, i follows ha fo all <8 and y X wih dx, y) <,wehave 3.8) ) γ fz) Q fχ 2) )y) dμz) 2) V y)+vy, z) + dy, z) ) γ fz) dμz) 2) V x)+vx, z) + dx, z) ) γ 2 jγ μ2 j+ fz) dμz) ) j= 2 j+ ) γμ)] α ) γμ)] 2 jγ αn) α. j= Noice ha fo all x, y X saisfying dx, y) <,wehave 3.9) V x) V y). I hen follows fom 3.8) and 3.9) ogehe wih γ, ) ha 3.) 8 ) ]p/2 2γ d S fχ 2) )x)] p dμx) μ)] +αp μ)] +αp, which ogehe wih 3.6) ells us ha 3.) S f)x)] p dμx) μ)] +αp. Obseve ha fo all y X wih dx, y) <, by 2.3), we have 3.2) ρy) + ρy) ρx) ) +k, and ha fo all x wih ρx ), by 2.3), we also have ha ρx). Combining hese wo obsevaions yields ha fo all x and y X wih dx, y) <, 3.3) ρy) ) + ρy) +k. I hen follows fom Lemma 3.2, 3.3) and 2.2) ha fo all x, 8 and y X wih dx, y) <,
268 oundedness of Lusin-aea and g λ funcions 3.4) ρy) ) δμy, Q f)y) ))] α + ρy) ) δ +k μx, ))] α ) δ +k αn μ)] α, which ogehe wih he assumpion ha δ > + k )αn implies ha ] ) p/2 2δ S 2 f)x)] p dμx) μ)] +αp +k 2αn d μ)] +αp. 8 y his and 3.), we obain 3.5). Moeove, i follows fom 3.5) ha Sf)x) < fo almos evey x X. Case II. x,) D.Inhiscase,<ρx ). We need o pove ha { 3.5) Sf)x)] 2 essinf Sf)]2} p/2 dμx) μ)] +αp. To his end, fo all x X,wie 8 Sf)x)] 2 = Q f)y) 2 dμy) d dx, y)< V y) + 8 S f)x)] 2 +S, x f)x)] 2 +S f)x)] 2. 8ρx) + 8ρx ) Then { Sf)x)] 2 essinf Sf)]2} p/2 dμx) { S f)x)] p dμx)+ S, x f)x)] 2 essinf S, x f)] 2} p/2 dμx) { + S f)x)] 2 essinf S f)] 2} p/2 dμx) S f)x)] p dμx)+μ) sup S, x f)x)] 2 S, x f)x )] 2 p/2 x, x +μ) sup S f)x)] 2 S f)x )] 2 p/2 I +I 2 +I 3. x, x Wie f f +f 2 +f, whee f f f )χ 2 and f 2 f f )χ 2). y he L p X )-boundedness of Sf) and 2.2), we have
H. Lin, E. Nakai, D. Yang 269 3.6) S f )x)] p dμx) f f p dμx) μ)] +αp. 2 I follows fom Q) i, 3.7), 2.2), he Hölde inequaliy, Lemma 3.ii) and γ>αnha fo all x and y X wih dx, y) < 8, ) γ Q f 2 )y) fz) f dμz) 2) V y)+vy, z) + dy, z) ) γ fz) f dμz) V x)+vx, z) + dx, z) 2) ) γ 2 jγ μ2 j+ ) j= ) γ μ)] α 2 jγ αn) j= which ogehe wih 3.9) leads o ha 3.7) S f 2 )x)] p dμx) μ)] +αp 8 2 j+ fz) f 2 j+ + f 2 j+ f ] dμz) ) γ μ)] α, ) ] 2γ p/2 d μ)] +αp. Obseve ha by 2.3), fo any a, ), hee exiss a consan C a, ) such ha fo all x, y X wih dx, y) aρx), 3.8) ρy)/ C a ρx) C a ρy). y his, we obain ha fo all x and y X saisfying dx, y) < wih, 8) and<ρx ), ρy) ρx ). Hence, by Q) ii and Lemma 3.i), we have ) δ2 ) δ2 ) αn ρx ) Q f )y) f μ)] α, ρy) ρx ) which ogehe wih δ 2 >αn, <ρx ) and 3.9) implies ha S f )x)] p dμx) μ)] +αp ρx ) μ)] +αp. ) αpn 8 Combining his, 3.6) and 3.7) yields I μ)] +αp. ) 2δ2 ] p/2 d ρx )
27 oundedness of Lusin-aea and g λ funcions Now we un ou aenion o pove ha fo all x, x, S, x f)x)] 2 S, x f)x )] 2 μ)] 2α. Wie S, x f)x)] 2 S, x f)x )] 2 = 8ρx) 8 8ρx) + dx, y)< 8 x, ) x,) 8ρx) 8 x, ) x,) Q f)y) 2 dμy) d V y) 8ρx) 8 Q f f )y) 2 dμy) d V y) dx,y)< Q f )y) 2 dμy) d V y) J +J 2, whee x, ) x,) x, ) \ x,)] x,)\ x, )]. y he facs ha x, x and 8, we have x, 2) x, ) x,)]. Since X has he volume egulaiy Popey P ), we obain ) δμx, μx, ) \ x,)) μx, )) μx, 2)) )). y symmey, we also have μx,)\ x, )) ) δμx,)), which ogehe wih 2.) implies ha ) δμx, 3.9) μx, ) x,)) )). y Q) i, 3.7), 3.9), 3.9), 2.2), he Hölde inequaliy and Lemma 3.ii), we obain J 8ρx) 8 8ρx) 8 + j= 8ρx) 8 ) δ ) δ X μ2) V x)+vx, z) fz) f dμz) 2 γ +2 j ) γ μ2 j+ ) ) δ + j= 2 j+ γ 2 jαn +2 j ) γ + dx, z) ) γ ] 2 d fz) f dμz) ] 2 d fz) f dμz) ] 2 d μ)]2α.
H. Lin, E. Nakai, D. Yang 27 Noice ha γ > αn and δ > 2αn. Choosing ɛ αn, min{γ,δ/2}), we have J + { + 8ρx) 8 8 ) δ ) δ 2ɛ d j= γ 2 jαn ] 2 d γ ɛ 2 j ) ɛ μ)]2α } μ)] 2α μ)] 2α. Thus, J μ)] 2α. Noice ha <ρx )and 8, 8ρx )). y 3.8), we have ha fo any x and y X wih dx, y) <, ρx ) ρx) ρy). Choosing η, ) such ha ηδ 2 = αn, hen by Lemma 3.i), Q) ii and 3.9), we have J 2 8ρx) 8 8 ) δ + ρx ) ) δ2 ρx ) ) αn μ)] α ] 2 d 8ρx) ) ) δ ηδ2 ) ] αn 2 ρx ) d μ)] α 8 ρx ) ) δ 2αn d μ)] 2α μ)]2α. Combining he esimaes fo J and J 2 yields I 2 μ)] +αp. To pove Theoem 3., i emains o esimae he em I 3. Fo all x, x,wie S f)x)] 2 S f)x )] 2 = Q f)y) 2 dμy) d 8ρx ) dx, y)< V y) 8ρx ) dx,y)< Q f)y) 2 dμy) d V y). 8ρx ) x, ) x,) Noice ha <ρx ). Hence, fo 8ρx ), ), 3.9) sill holds. On he ohe hand, fo all x and y X wih dx, y) <, by 3.2), Lemma 3.2, 2.2) and he fac ha ρx ) ρx), we obain ha Q f)y) ) δ ρy) μy, ))] α + ρy) ρx ) ) δ ) +k αn μ)] α.
272 oundedness of Lusin-aea and g λ funcions I follows fom his, 3.9), 3.9) and δ>2αn ha fo all x, x, S f)x)] 2 S f)x )] 2 8ρx ) μ)] 2α, ρx ) ) 2δ +k ) δ 2αn μ)] 2α d so ha I 3 μ)] +αp. This finishes he poof of Theoem 3.. As a consequence of Theoem 3., we have he following conclusion, which can be poved by an agumen simila o ha used in he poof of 34, Coollay 4.]. We omi he deails. Coollay 3.. Wih he assumpions same as in Theoem 3., hen hee exiss a posiive consan C such ha fo all f Eρ X ), Sf) Ẽα, ρ p X ) and Sf) Ẽ ρ X ) C f Eρ X ). Remak 3.2. i) If α =, Theoem 3. and Coollay 3. wee aleady obained in 9]. p/2 ii) If α >, hen by Remak 2.iv), he space Ẽ2α, ρ X ) in Theoem 3. and he space Ẽα, ρ p X ) in Coollay 3. ae exacly he spaces 2/2 Eρ X )andeρ X ), especively. iii) If α<, hen by Theoem 2.2 and he fac ha he Lusin-aea p/2 funcion is nonnegaive, we know ha if he space Ẽ2α, ρ X )intheoem 3. and he space Ẽα, ρ p X ) in Coollay 3. ae eplaced, especively, by 2/2 he spaces Eρ X )andeρ X ), we obain he same esuls. Now we sudy he boundedness of gλ funcion in localized Moey- Campanao spaces. In his case, X is no necessay o have Popey P ). Theoem 3.2. Le X be a doubling meic measue space. Le p, ), ρ be an admissible funcion on X,hegλ funcion g λ f) as in 3.4) wih λ 3n, ) and α, min{γ/n, δ / + k )n], δ 2 /n, λ 3n)/2 + k )n], /2n)}). If gλ f) is bounded on Lp X ), hen hee exiss a posiive consan C such ha fo all f Eρ X ), gλ f)]2 p/2 Ẽ2α, ρ X ) and gλ f)]2 Ẽ 2/2 ρ X ) C f 2 Eρ X ). Poof. y similaiy, we only pove he case when α >. Le f Eρ X ), by he homogeneiy of E ρ X ) and Ẽ 2/2 ρ X ),we may assume ha f E ρ X ) =. Le x,). Fo any nonnegaive inege k,le Jk) {y, ) X, ) : dy, x ) < 2 k+ and <<2 k+ }.
Fo any f Eρ X )andx X,wie gλ f)x)]2 = J) H. Lin, E. Nakai, D. Yang 273 ) λ Q f)y) 2 dμy) d + dx, y) V y) + X, )]\J) g λ, f)x)] 2 +g λ, f)x)] 2. We now conside he following wo cases. Case I. x,) D. Hee, ρx ). We fis pove ha 3.2) gλ, f)x)]p dμx) μ)] +αp. Fo any x,wie gλ, f)x)]2 +2 +2 J) dx, y)< J) dx, y) J) dx, y) I x)+i 2 x)+i 3 x). ) λ Q f)y) 2 dμy) d + dx, y) V y) ) λ Q fχ 8)y) 2 dμy) + dx, y) V y) ) λ Q fχ + dx, y) d 8) )y) 2 dμy) V y) Noe ha fo all x,i x) Sf)x)] 2 and hen 3.5) gives 3.2) I x)] p/2 dμx) μ)] +αp. We emak ha in he poof of 3.5), we did no use Popey P )ofx. y he L p X )-boundedness of gλ f) and 2.), we have 3.22) I 2 x)] p/2 dμx) fx) p dμx) μ)] +αp. dy, x )<2 dx, y) 8 To deal wih I 3 x), noicing ha fo all z 8) and y X wih dy, x ) < 2,wehavehady, z) dx,z) and V y, z) V x,z). Hence, fom he assumpions ha λ>n and γ>αn, i follows 2 ) λ I 3 x) + dx, y) 8) V y)+v y, z) d ) ] γ fz) 2 dμy) d dμz) + dy, z) V y)
274 oundedness of Lusin-aea and g λ funcions 2 dy, x )<2 dx, y) dμy) V y) 2 d dy, x )<2 dx, y) j=3 ) λ ) ] γ fz) 2 dμz) + dx, y) 8) V x,z) dx,z) + dx, y) ) λ ) γ 2 j μ2 j+ ) 2 j+ fz) dμz) 2 dμy) d V y) 2 λ ) γ 2 dy, x )<2 + dx, y)) jγ αn) μ)] α dx, y) j=3 2 ) 2γ μ)] 2α k= 2 k dx, y)<2 k+ 2 dμy) d V y) 2 kλ kn dμy) d 2 V 2 k+ x) μ)]2α, which implies ha I 3x)] p/2 dμx) μ)] +αp. Combining his, 3.2) and 3.22) poves 3.2). Now we pove ha 3.23) gλ, f)x)]p dμx) μ)] +αp. Noice ha fo y, ) Jk) \ Jk ) wih k N and x, + dx, y) 2 k.thus, gλ, f)x)]2 k= 2 k+4 + k= Jk)\Jk ) ) λ 2 k V y)+v y, z) Jk)\Jk ) ) γ ρy) + dy, z) + ρy) ) λ 2 k dμz) 2 k+4 ) ) ] δ fz) 2 dμy) d dμz) V y) ] 2 dμy) d V y) E x)+e 2 x).
H. Lin, E. Nakai, D. Yang 275 The fac ha ρx ) and 2.3) imply ha fo all y X dy, x ) < 2 k+, wih +k 3.24) ρy) ρx )] 2 k ) k +k. y he assumpions ha λ 3n, ), δ > + k )αn and λ 3n) > 2 + k )αn, wechooseη, δ ) + k )αn, λ 3n)/2)]. Theefoe, λ 2η 3n >. I hen follows fom 3.24) ha E x) k= 2 k+ dy, x )<2 k+ 2 2kαn 2α dμy) μ)] V y) μ)] 2α μ)] 2α k= k= 2 2kαn 2 d ) λ 2 k ) 2n ρx +k )] 2 k ) k 2 k k+ +k ) 2η ) λ 2 k ) 3n ρx +k )] 2 k ) k 2 k 2 2kαn ρx ) ] 2η +k μ)] 2α 2 k. Choosing η 2, δ ) +k )αn, λ 3n)/2)], hen λ+2γ 2η 2 n>. Noice ha fo z 2 k+4 ) and y X wih dy, x ) < 2 k+, dy, z) dx,z) and V y, z) V x,z). This ogehe wih 3.24) and γ>αnimplies ha E 2 x) k= 2 k+ j= μ)] 2α μ)] 2α 2 j+k+ k= k= dy, x )<2 k+ ) γ μ2 j+k+4 ) 2 2kαn 2 ) λ ρx +k )] 2 k ) k +k ) η2 2 k ] 2 dμy) fz) dμz) V y) k+ 2 j+k+4 ) λ+2γ n ρx +k )] 2 k ) k 2 k 2 2kαn ρx ) ] 2η 2 +k μ)] 2α 2 k, which ogehe wih he esimae of E x) yields 3.23). Combining 3.2) and 3.23) yields ha 3.25) gλf)x)] p dμx) μ)] +αp. d +k +k ) 2η d ) 2η2 d
276 oundedness of Lusin-aea and g λ funcions Moeove, fom 3.25), i follows ha gλ f)x) < fo almos evey x X. Case II. x,) D.Inhiscase,<ρx ). We need o pove ha { gλ f)x)]2 essinf g λ f)]2} p/2 dμx) μ)] +αp. To his end, fo all x X,wie { gλ f)x)]2 essinf g λ f)]2} p/2 dμx) { gλ, f)x)]p dμx)+ gλ, f)x)]p dμx)+μ) sup x, x g λ, f)x)]2 essinf Now we pove ha 3.26) gλ, f)x)] p dμx) μ)] +αp. g λ, f)]2} p/2 dμx) g λ, f)x)] 2 g λ, f)x )] 2 p/2. To his end, wie f f + f 2 + f, whee f f f )χ 8 and f 2 f f )χ 8). y he L p X )-boundedness of gλ f), 2.2) and Lemma 3.ii), we have 3.27) gλ, f )x)] p dμx) f f p dμx) μ)] +αp. 8 Noice ha fo z 8) and y X wih dy, x ) < 2, dy, z) dx,z) and V y, z) V x,z). This ogehe wih Q) i, 2.2), he Hölde inequaliy, Lemma 3.ii) and γ>αnyields ha ) γ Q f 2 )y) fz) f dμz) 8) V y)+vy, z) + dy, z) ) γ fz) f dμz) 8) V x,z) dx,z) ) γ ] 2 k μ2 k+3 fz) f ) 2 dμz)+ f k+3 2 k+3 f k= 2 k+3 ) γ ) γ μ)] α 2 kγ αn) μ)] α. k= y an agumen simila o he esimaes of 3.) and I 3 x), we obain
H. Lin, E. Nakai, D. Yang 277 3.28) gλ, f 2)x)] p dμx) 2 2 dx, y)< 2 + μ)] +αp. dx,y)<2 dx,y)<2 dx, y) + dx, y) ) 2γ μ)] + dx, y) ) λ ) 2γ μ)] 2α dμy) V y) 2α dμy) V y) ] p/2 d dμx) ) λ ) 2γ μ)] 2α dμy) V y) ] p/2 d dμx) ] p/2 d dμx) Fo y X wih dx,y) < 2 < 2ρx ), by 3.8), we have ha ρx ) ρy), which ogehe wih Q) ii, Lemma 3.i) and δ 2 >αn leads o ha ) δ2 ) δ2 ) αn ρx ) ) δ2μ)] Q f )y) f μ)] α α. ρy) ρx ) Then, similaly o he esimae of 3.28), we obain gλ, f )x)] p dμx) μ)] +αp, which ogehe wih 3.27) and 3.28) yields 3.26). The poof of Theoem 3.2 is educed o show ha fo all x, x, 3.29) gλ, f)x)]2 gλ, f)x )] 2 μ)] 2α. Wie g λ, f)x)] 2 gλ, f)x )] 2 k= + X, )\J) k= Jk)\Jk ) Jk)\Jk ) + dx, y) ) λ + dx,y) λ 2 k ) λ+ Q f f )y) 2 dμy) V y) λ 2 k ) λ+ Q f )y) 2 dμy) V y) ) λ d Q f)y) 2 dμy) V y) d G +G 2. d
278 oundedness of Lusin-aea and g λ funcions The assumpions ha λ 3n, ) andγ>αn, ogehe wih Q) i and Lemma 3.ii), imply ha G k= + Jk)\Jk ) 2 k+4 k= Jk)\Jk ) 2 k+ k= + k= λ 2 k ) λ+ ) γ fz) f dμz) V y)+vy, z) + dy, z) ] 2 λ dμy) 2 k ) λ+ dμz) 2 k+4 ) V y) λ 2 k ) 2n2 2kαn 2 k ) λ+ μ)] dy, x )<2 k+ 2 k+ μ)] 2α +μ)] 2α μ)] 2α k= k= k= dy, x )<2 k+ 2 k+ 2 2kαn 2 k+ 2 2kαn 2 2kαn 2 k μ)] 2α, λ ) 2γ2 2kαn 2 k ) λ+ 2 k μ)] λ 2 k ) 3n d 2 k ) λ+ λ 2 k ) n 2γ d 2 k ) λ+ ] 2 dμy) V y) d 2α dμy) V y) d 2α dμy) V y) whee in he las inequaliy, we used he assumpion ha 2αn <. Noice ha < ρx ), hee exiss a posiive inege k such ha 2 k <ρx ) 2 k.ifk {,,k },henfoy X wih dy, x ) < 2 k+, by 3.8), we have ha ρx ) ρy); if k {k +,k +2, }, hen by 2.3), fo, 2 k+ )andy X wih dy, x ) < 2 k+,wehave + ρy) ρy) + dy, x ) k ) k ) 2k + 2k 2 k ) +k. ρx ) ρx ) ρx ) ρx ) ρx ) Fom Q) ii, Lemma 3.i), δ 2 >αn, λ n, ) and 2αn <, i hen follows ha G 2 k= 2 k+ + ρy) dy, x )<2 k+ ) 2δ2 dμy) d V y) λ 2 k ) λ+ ) 2αn ρx ) μ)] 2α d d
H. Lin, E. Nakai, D. Yang 279 { k μ)] 2α + k=k + μ)] 2α k= 2 k+ k= 2 k+ λ 2 k ) λ+ ) 2αn ρx ) ) 2αn 2 k ) n d ρx ) ) 2αn ρx ) 2 k ) 2αn 2 k ) } n d ρx ) λ 2 k ) λ+ 2 k 2αn) μ)] 2α. Combining he esimaes fo G and G 2 yields 3.29), which complees he poof of Theoem 3.2. As a consequence of Theoem 3.2, we have he following conclusion. Coollay 3.2. Wih he assumpions same as in Theoem 3.2, hen hee exiss a posiive consan C such ha fo all f Eρ X ), gλ p f) Ẽα, ρ X ) and gλ f) Ẽρ X ) C f Eρ X ). We poin ou ha Remak 3.2 is also suiable o Theoem 3.2 and Coollay 3.2. The following is a simple coollay of Theoems 3. and 3.2, and Coollaies 3. and 3.2. We omi he deails hee; see 34, Secion 5]. Poposiion 3.. Theoems 3. and 3.2, and Coollaies 3. and 3.2 ae ue if Q 2 de sl ds s= 2, whee L = Δ +V is he Schödinge opeao o he degeneae Schödinge opeao on R d, o he sub-laplace Schödinge opeao on Heisenbeg goups o conneced and simply conneced nilpoen Lie goups, and V is a nonnegaive funcion saisfying ceain evese Hölde inequaliy; see he deails in 34, Secion 5]. Acknowledgemen. The fis auho was suppoed by Chinese Univesiies Scienific Funf Gan No. 2JS43) and he Mahemaical Tianyuan Youh Fund Gan No. 262) of Naional Naual Science Foundaion of China, he second auho was suppoed by Gan-in-Aid fo Scienific Reseach C), No. 25467, Japan Sociey fo he Pomoion of Science, and he hid auho was suppoed by he Naional Naual Science Foundaion Gan No. 8725) of China.
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282 oundedness of Lusin-aea and g λ funcions 3] H. Tiebel, Theoy of Funcion Spaces. II, ikhäuse Velag, asel, 992. 32] N. Th. Vaopoulos, L. Saloff-Cose and T. Coulhon, Analysis and Geomey on Goups, Cambidge Univesiy Pess, Cambidge, 992. 33] Da. Yang, Do. Yang and Y. Zhou, Localized MO and LO spaces on RD-spaces and applicaions o Schödinge opeaos, Commun. Pue Appl. Anal., 9 2), 779 82. 34] Da. Yang, Do. Yang and Y. Zhou, Localized Moey-Campanao spaces on meic measue spaces and applicaions o Schödinge opeaos, Nagoya Mah. J., 98 2), 77 9. 35] D. Yang and Y. Zhou, Localized Hady spaces H elaedoadmissible funcions on RD-spaces and applicaions o Schödinge opeaos, Tans. Ame. Mah. Soc., 363 2), 97 239. 36] D. Yang and Y. Zhou, New popeies of esov and Tiebel-Lizokin spaces on RD-spaces, Manuscipa Mah., 34 2), 59 9. 37] J. Zhong, The Sobolev esimaes fo some Schödinge ype opeaos, Mah. Sci. Res. Ho-Line 3:8 999), 48. College of Science, China Agiculual Univesiy eijing 83 People s Republic of China and School of Mahemaical Sciences, eijing Nomal Univesiy Laboaoy of Mahemaics and Complex Sysems Minisy of Educaion, eijing 875 People s Republic of China E-mail : linhaibo@mail.bnu.edu.cn) Depamen of Mahemaics, Osaka Kyoiku Univesiy Kashiwaa, Osaka 582-8582 Japan E-mail : enakai@cc.osaka-kyoiku.ac.jp) Cuen Addess : Depamen of Mahemaics, Ibaaki Univesiy Mio Ibaaki 3 582, Japan, E-mail : enakai@mx.ibaaki.ac.jp) School of Mahemaical Sciences, eijing Nomal Univesiy Laboaoy of Mahemaics and Complex Sysems Minisy of Educaion, eijing 875 People s Republic of China E-mail : dcyang@bnu.edu.cn) Received : June 29 )
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