BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS

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1 BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS XUAN THINH DUONG, JI LI, AND ADAM SIKORA Absrac Le M be a manifold wih ends consruced in [2] and be he Laplace-Belrami operaor on M In his noe, we show he weak ype, and L p boundedness of he Hardy-Lilewood maximal funcion and of he maximal funcion associaed wih he hea semigroup M fx = sup exp fx on L p M for < p The significance of hese resuls comes from he fac ha M does no saisfies he doubling condiion Inroducion The heory of Calderón-Zygmund operaors has played a crucial role in harmonic analysis and is wide applicaions in he las half a cenury or so We refer readers o he excellen book [7] and he references herein In he sandard Calderón-Zygmund heory, an essenial feaure is he so-called doubling condiion Le us recall ha a meric space X, d, µ equipped wih a meric d and a measure µ saisfies he doubling condiion if here exiss a consan C such ha for all x X and r > 0 µbx, 2r CµBx, r Many meric spaces in classical analysis saisfy he doubling condiion such as he Euclidean spaces and heir smooh domains wih Lebesgue measure, Lie groups and manifolds of polynomial growh However, here are significan applicaions for which underlying ambien spaces do no saisfy he doubling condiion, for example domains of Euclidean spaces wih rough boundaries, Lie groups and manifolds wih exponenial growh To hese non-doubling spaces, he sandard Calderón-Zygmund heory esablished in he 70 s and 80 s is no applicable Recen works of Nazarov, Treil, Volberg, Tolsa and ohers, see for example [3, 4, 5, 6, 8, 9] show ha a large par of he sandard Calderön-Zygmund heory can be adaped o he case of non-doubling spaces which saisfy a mild growh condiion In [], Duong and A McInosh also obain esimaes for cerain singular inegrals acing on some domains which do no necessarily saisfy he doubling condiion However, he heory of singular inegrals This work was sared during he second named auhor s say a Macquarie Universiy J Li was suppored by a scholarship from Macquarie Universiy during , and is suppored by China Posdocoral Science Foundaion funded projec Gran No and he Fundamenal Research Funds for he Cenral Universiies No lgpy Mahemaics Subjec Classificaion: Primary 42B5; Secondary 35P99 37

2 38 XUAN THINH DUONG, JI LI, AND ADAM SIKORA on non-doubling spaces is far from being complee and here are sill many significan open problems in his opic In his noe, we sudy he boundedness of cerain maximal funcions on non-doubling manifolds wih ends More specifically, we will show he weak ype, of he Hardy-Lilewood maximal funcion and he maximal funcion associaed wih he hea semigroup of he Laplace-Belrami operaor as well as L p boundedness for hese maximal operaors for < p Le us recall ha he maximal funcion associaed wih he hea semigroup is defined by he following formula M fx = sup exp fx for f L p M, p The behaviour of he kernels of he semigroup exp on manifolds wih ends was sudied in [2] For he convenience of reader, we recall he main resul of [2] in he nex secion as i plays a key role in our esimaes of he operaor M 2 Manifolds wih ends Le M be a complee non-compac Riemannian manifold Le K M be a compac se wih non-empy inerior and smooh boundary such ha M\K has k conneced componens E,, E k and each E i is non-compac We say in such a case ha M has k ends wih respec o K and refer o K as he cenral par of M In many cases, each E i is isomeric o he exerior of a compac se in anoher manifold M i In such case, we wrie M = M M 2 M k and refer o M as a conneced sum of he manifolds M i, i =, 2,, k Following [2] we consider he following model case Fix a large ineger N which will be he opological dimension of M and, for any ineger m [2, N], define he manifold R m by R m = R m S N m The manifold R m has opological dimension N bu is dimension a infiniy is m in he sense ha V x, r r m for r, see [2, 3] Thus, for differen values of m, he manifold R m have differen dimension a infiniy bu he same opological dimension N, This enables us o consider finie conneced sums of he R m s Fix N and k inegers N, N 2,, N k [2, N] such ha Nex consider he manifold N = max{n, N 2,, N k } M = R N R N 2 R N k In [2] Grigoryan and Saloff-Cose esablish boh he global upper bound and lower bound for he hea kernel acing on his model class Now we recall he firs par of heir resuls wih he hypohesis ha n := min i k N i > 2 Le K be he cenral par of M and E,, E k be he ends of M so ha E i is isomeric o he complemen of a compac se in R N i Wrie E i = R N i \K

3 BOUNDEDNESS OF MAXIMAL FUNCTIONS 39 Thus, x R N i \K means ha he poin x M belongs o he end associaed wih R N i For any x M, define x := sup dx, z, z K where d = dx, y is he geodesic disance in M One can see ha x is separaed from zero on M and x + dx, K For x M, le Bx, r := {y M : dx, y < r} be he geodesic ball wih cener x M and radius r > 0 and le V x, r = µbx, r where µ is a Riemannian measure on M Throughou he paper, we ake he simple case k = 2 for he model of meric spaces wih non-doubling measure, ie, we se M = R n R m wih 2 < n < m Then, from he consrucion of he manifold M, we can see ha a V x, r r m for all x M, when r ; b V x, r r n for Bx, r R n, when r > ; and b V x, r r m for x R n \K, r > 2 x, or x R m, r > I is no difficul o check ha M does no saisfy he doubling condiion Indeed, consider a sequence of balls Bx k, r k R n such ha r k = x k > and r k as k Then V x k, r k r k n However, V x k, 2r k r k m and he doubling condiion fails Le be he Laplace-Belrami operaor on M and e he hea semigroup generaed by We denoe by p x, y he hea kernel associaed o e We recall here he following heorem which is he main resuls obain in [2] Theorem A [2] Le M = R m R n wih 2 < n < m Then he hea kernel p x, y saisfies he following esimaes For and all x, y M, p x, y C V x, exp c 2 For x, y K and all >, p x, y C n/2 exp c 3 For x R m \K, y K and all >, p x, y C n/2 x m 2 + m/2 4 For x R n \K, y K and all >, p x, y C n/2 x n 2 + n/2 5 For x R m \K, y R n \K and all >, p x, y C n/2 x m 2 + m/2 y n 2 dx, y2 dx, y2 exp c exp c exp c dx, y2 dx, y2 dx, y2

4 40 XUAN THINH DUONG, JI LI, AND ADAM SIKORA 6 For x, y R m \K and all >, C n/2 p x, y x m 2 y m 2 exp 7 For x, y R n \K and all >, p x, y C n/2 x n 2 y n 2 exp c x 2 + y 2 c x 2 + y 2 + C m/2 exp c + C n/2 exp c dx, y2 dx, y2 3 The boundedness of Hardy-Lilewood maximal funcion In his secion we consider M = R m R n for m > n > 2 A main difficuly which we encouner in our sudy is ha he doubling condiion fails in his seing However, local doubling sill holds, ie he doubling condiion holds for a ball Bx, r under he addiional assumpion r Le us recall nex he sandard definiion of uncenered Hardy Lilewood Maximal funcion For any p [, ] and any funcion f L p le { } Mfx = fz dz : x By, r V y, r sup y M, r>0 By,r Also we have he cenered Hardy Lilewood Maximal funcion For any p [, ] and any funcion f L p we se M c fx = sup fz dz r>0 V x, r Bx,r I is sraighforward o see ha M c fx Mfx for all x Moreover in he doubling seing 2 Mf CM c f, where C is he same consan as in he doubling condiion However, we poin ou ha esimae 2 does no hold in he seing M = R m R n wih m > n > 2 More specifically, one has he following proposiion Proposiion In he seing M = R m R n wih m > n > 2, he esimae Mf CM c f fails for any consan C Proof Denoe he characerisic funcions of he ses R m \K, R n \K and K by χ, χ 2 and χ 3, respecively Le f = χ 2 Then for any fixed x R m, we firs noe ha χ 2 ydy V B B for any B x Furhermore, we can consruc balls B x such ha he ball B wih cenre z, radius r, lying mosly in R n by choosing z R n, r large enough and dz, x = r ɛ for ɛ sufficienly small This implies ha Mfx = sup χ 2 ydy = B x V B Now consider he cenered Hardy Lilewood Maximal funcion M c f By he definiion for any r > 0, fzdz = C V x, r r m dz Bx,r B Bx,r R n \K

5 BOUNDEDNESS OF MAXIMAL FUNCTIONS 4 This implies ha r > x and he erm C r m Bx,r R n \K dz is comparable r x n o r m I is easy o check ha he maximal value of he above erm is comparable o n x n / m x m, m n m n which shows ha Mf is no poinwise bounded by any muliple of M c f since he maximal value depends on x and ends o zero when x goes o This proves Proposiion Theorem 2 The maximal funcion Mf is of weak ype, and bounded on all L p spaces for < p Proof Here and hroughou he paper, for he sake of simpliciy we use o denoe he measure of he ses in M I is sraighforward ha he maximal funcion Mf is bounded on L We will show ha he weak ype, esimae {x: Mfx > α} C f α holds, hen he L p boundedness of Mf follows from he Marcinkiewicz inerpolaion heorem We consider wo cases: Case : f α < Following he sandard proof of weak ype for Maximal operaor we noe ha for any x {x: Mfx > α} here exis a ball such ha x By, r and 3 fz dz > α V y, r By,r This implies f = M fz dz By,r fz dz > αv y, r Therefore > f α > V y, r, hence r and he ball By, r saisfies doubling condiion so one can use sandard Viali covering argumen o prove weak ype, esimae in his case Case 2 : f α Firs we spli M ino hree componens R m \K, R n \K and K, and denoe heir characerisic funcions by χ, χ 2 and χ 3, respecively Since he maximal funcion Mf is sublinear, i is enough o show ha each of he hree erms Mχ f, Mχ 2 f and Mχ 3 f is of weak ype, We firs consider Mχ f Then {x : Mχ fx > α} {x R m \K : Mχ fx > α} + {x R n \K : Mχ fx > α} + {x K : Mχ fx > α} =: I + I 2 + I 3

6 42 XUAN THINH DUONG, JI LI, AND ADAM SIKORA The esimae for I follows from he classical weak ype, esimae since χ f is a funcion on R m \K and he measure on R m \K saisfies he doubling condiion To esimae I 2, we noe ha for all x R n \K, { sup By, r : r > dx, y and By, r Rm \K The above inequaliy implies ha 4 Mχ fx C χ f x n Hence x R n \K I 2 {x R n \K : C χ f x n > α} C χ f α } C f α C x n To esimae I 3, we noe ha he measure of K is finie Therefore I 3 K C f α To prove he weak, esimae of Mχ 2 f we noe ha {x : Mχ 2 fx > α} {x R m \K : Mχ 2 fx > α} + {x R n \K : Mχ 2 fx > α} + {x K : Mχ 2 fx > α} =: II + II 2 + II 3 II 2 and II 3 can be verified following he same seps as for I and I 3, respecively To esimaes II we observe ha 5 Mχ 2 fx C χ 2f x m Hence II 2 C f α Similarly, o deal wih Mχ 3 f we noe ha x R m \K {x : Mχ 3 fx > α} {x R m \K : Mχ 3 fx > α} + {x R n \K : Mχ 3 fx > α} + {x K : Mχ 3 fx > α} =: III + III 2 + III 3 The esimae of III follows immediaely since he measure on R m \K K saisfies he doubling condiion The esimae of III 3 is he same as ha of I 3 or II 3 Nex o esimaes III 2 we furher decompose {x R n \K} ino wo pars {x R n \K : x 2} and {x R n \K : x > 2} For he firs par we direcly have {x R n \K : x 2, Mχ 3 fx > α} C C f α For he second par, similar o he esimae of I 2, we noe ha for all x R n \K and x > 2, Hence, { sup : r > dx, y and By, r K By, r Mχ 3 fx C χ 3f x n x R n \K and x > 2, } C x n

7 which implies ha BOUNDEDNESS OF MAXIMAL FUNCTIONS 43 {x R n \K : x > 2, Mχ 3 fx > α} C f α Combining he esimaes of Mχ f, Mχ 2 f and Mχ 3 f we verify 3 The proof of Theorem 2 is now complee 4 The boundedness of he maximal funcion M In his secion we prove ha he hea maximal operaor saisfies weak ype, and is bounded on L p for < p We noe ha when he hea semigroup has a Gaussian upper bound, hen he maximal funcion corresponding o hea semigroup is poinwise dominaed by he Hardy-Lilewood maximal operaor In his case, he weak ype, esimae of M follows from he weak ype, esimae of he Hardy-Lilewood maximal funcion However, in considered seing his is no longer he case and he operaor M can no be conrolled by he Hardy-Lilewood maximal funcion We can see his via he esimaes of he hea semigroup in he proof of Theorem 3 below where we give a direc proof of he weak ype esimaes of he hea maximal operaor The following heorem is he main resul of his secion Theorem 3 Le M be he operaor defined by Then M is weak ype, and for any funcion f L p, < p, he following esimaes hold M f L p M C f L p M Proof We firs show ha M is weak ype,, ie, we need o prove ha here exiss a posiive consan C such ha for any f L M and for any λ > 0, 6 { x M : sup exp fx > λ } C λ f L M Fix f L M Similarly as in Secion 3 we se f x = fxχ R m \Kx, f 2 x = fxχ R n \Kx and f 3 x = fxχ K x, where K is he cener of M To prove 6, i suffices o verify ha he following hree esimaes hold: 7 { x R m \K : sup exp fx > λ } C λ f L M; 8 9 { x R n \K : sup exp fx > λ } C λ f L M; { x K : sup exp fx > λ } C λ f L M

8 44 XUAN THINH DUONG, JI LI, AND ADAM SIKORA We firs consider 7 Since M is a sublinear operaor, we have { x R m \K : sup exp fx > λ } { x R m \K : sup exp f x > λ } + { x R m \K : sup exp f 2 x > λ } + { x R m \K : sup exp f 3 x > λ } =: I + I 2 + I 3 To esimae I we consider wo cases Case : > By Theorem A Poin 6 + y 2 exp f x C R m \K n 2 x m 2 y m 2 exp c x 2 + cdx, y2 exp fy dy m 2 =: I + I 2 To esimae I we noe ha n/2 n/2 + y 2 x m 2 y m 2 exp c x 2 C x m 2 y m 2 + x 2 + y 2 n 2 x m 2+n x m since y and n > 2 Hence, I C R m \K x m 2+n fydy C f L M x m To esimae I 2 we noe ha if x R m \K hen cdx, y2 exp fy dy CM R m \Kfx R m \K m 2 where M R m \Kfx is he Hardy-Lilewood maximal funcion acing on R m \K Case 2: By Theorem A Poin exp f x exp R m \K m 2 n 2 cdx, y2 fy dy Again he righ-hand side of above esimae is bounded by M R m \Kfx These esimaes prove weak ype, for I since R m \K saisfies doubling condiion Nex we show weak ype esimaes for I 2 We also consider wo cases

9 BOUNDEDNESS OF MAXIMAL FUNCTIONS 45 Case : > By Theorem A Poin 5 exp f 2 x C n 2 x + m 2 m 2 y n 2 R n \K =: I 2 + I 22 exp cdx, y2 fy dy Similarly as in he esimae for I we ge n 2 I 2 C fy dy R n \K n 2 x m 2 + dx, y 2 n 2 C x m 2+n fy dy C f x m, R n \K since n > 2, x and in his case, dx, y x To esimae I 22 we noe ha m I 22 C R n \K m 2 y n 2 + dx, y 2 m fy dy m 2 C R n \K + dx, y 2 m fy dy m C R n \K fy dy + dx, y 2m m since y By decomposing he Poisson kernel ino + dx, y 2m annuli, i is easy o see ha he las erm of he above inequaliy is bounded by CM R n \Kfx Case 2: Again by Theorem A Poin cdx, y2 exp f 2 x C exp fy dy R n \K m 2 Hence i is bounded by CMfx Similar o I, we have I 2 C f λ Now we consider I 3 Case : > By Theorem A Poin 3 exp f 3 x C K n 2 x + cdx, y2 exp fy dy m 2 m 2 =: I 3 + I 32 To esimae I 3 we noe ha n 2 I 3 C n 2 x m 2 + dx, y 2 n 2 K C f x m, fy dy C f x m+n 2

10 46 XUAN THINH DUONG, JI LI, AND ADAM SIKORA where we use he facs ha n > 2, x > and ha in his case, dx, y x Similarly, m 2 I 32 C fy dy C f m 2 + dx, y 2 m 2 x m Case 2: By Theorem A Poin exp f 3 x exp K K m 2 cdx, y2 fy dy Hence i is bounded by CMfx Combining he esimaes of he wo cases, we obain I 3 C f L M λ The esimaes of I, I 2 and I 3 ogeher imply 7 We now urn o he esimae of 8 Similarly o he proof of 7, we have { x R n \K : sup exp fx > λ } { x R n \K : sup exp f x > λ } + { x R n \K : sup exp f 2 x > λ } + { x R n \K : sup exp f 3 x > λ } =: II + II 2 + II 3 We noe ha he esimae of II is similar o ha of I 2, while he esimae of II 2 is similar o ha of I Moreover, he esimae of II 3 is similar o ha of I 3 Therefore we can verify ha 8 holds Finally, we urn o he esimae of 9 We have { x K : sup exp fx > λ } { x K : sup exp f x > λ } + { x K : sup exp f 2 x > λ } + { x K : sup exp f 3 x > λ } =: III + III 2 + III 3 Also, we poin ou ha he esimae of III is similar o ha of I 3 and ha he esimae of III 2 is similar o ha of II 3 Concerning he erm III 3, we firs noe ha in his case x K We have exp f 3 x C K m 2 exp cdx, y2 fy dy I is easy o see ha he righ-hand side of he above inequaliy is bounded by CMfx Thus, we have III 3 C f λ

11 BOUNDEDNESS OF MAXIMAL FUNCTIONS 47 Hence, we can see ha 9 holds Now 7, 8 and 9 ogeher imply ha 6 holds, ie, M is of weak ype, Nex, noe ha he semigroup exp is submarkovian so M is bounded on L M This ogeher wih 6, implies ha M is bounded on L p M for all < p < The proof of Theorem 3 is complee References X T Duong and A McInosh, Singular inegral operaors wih non-smooh kernels on irregular domains, Rev Ma Iberoamericana 5 999, A Grigor yan and L Saloff-Cose, Hea kernel on manifolds wih ends, Ann Ins Fourier Grenoble, no5, , F Nazarov, S Treil and A Volberg, Cauchy inegral and Calderón-Zygmund operaors on nonhomogeneous spaces, Inerna Mah Res Noices, Vol 5, 997, p F Nazarov, S Treil and A Volberg, Weak ype esimaes and Colar inequaliies for Calderón-Zygmund operaors on nonhomogeneous spaces, Inerna Mah Res Noices, Vol 9, 998, p F Nazarov, S Treil and A Volberg, The T b- heorem on non-homogeneous spaces, Aca Mah, Vol 90, 2003, No 2, p J Maeu, P Maila, A Nicolau, J Orobig, BMO for non doubling measures, Duke Mah J, , EM Sein, Harmonic analysis: Real variable mehods, orhogonaliy and oscillaory inegrals, Princeon Univ Press, Princeon, NJ, X Tolsa, BMO, H, and Calderón-Zygmund operaors for non doubling measures, Mah Ann , X Tolsa, A proof of he weak, inequaliy for singular inegrals wih non doubling measures based on a Calderón-Zygmund decomposiion, Publ Ma , Xuan Thinh Duong, Deparmen of Mahemaics, Macquarie Universiy, NSW 209 Ausralia address: xuanduong@mqeduau Ji Li, Deparmen of Mahemaics, Sun Ya-sen Universiy, Guangzhou, 50275, China address: liji6@mailsysueducn Adam Sikora, Deparmen of Mahemaics, Macquarie Universiy, NSW 209 Ausralia address: adamsikora@mqeduau