REAL INTERPOLATION OF SOBOLEV SPACES

Transcription

1 REAL INTERPOLATION OF SOBOLEV SPACES NADINE BADR Abstact We pove that W p s a eal ntepolaton space between W p and W p 2 fo p > and p < p < p 2 on some classes of manfolds and geneal metc spaces, whee depends on ou hypotheses Contents Intoducton 2 Pelmnaes 3 2 The doublng popety 3 22 The K-method of eal ntepolaton 4 3 Non-homogeneous Sobolev spaces on Remannan manfolds 5 3 Non-homogeneous Sobolev spaces 5 32 Estmaton of the K-functonal of ntepolaton 5 4 Intepolaton Theoems 3 5 Homogeneous Sobolev spaces on Remannan manfolds 5 6 Sobolev spaces on compact manfolds 8 7 Metc-measue spaces 8 7 Uppe gadents and Poncaé neualty 8 72 Intepolaton of the Sobolev spaces Hp 9 8 Applcatons 2 8 Canot-Caathéodoy spaces 2 82 Weghted Sobolev spaces Le Goups 22 9 Appendx 23 Refeences 24 Intoducton Do the Sobolev spaces W p fom a eal ntepolaton scale fo < p <? The am of the pesent wok s to povde a postve answe fo Sobolev spaces on some metc spaces Let us state hee ou man theoems fo non-homogeneous Sobolev spaces (esp homogeneous Sobolev spaces on Remannan manfolds Theoem Let M be a complete non-compact Remannan manfold satsfyng the local doublng popety (D loc and a local Poncaé neualty (P loc, fo some 2 Mathematcs Subject Classfcaton 46B7, 46M35 Key wods and phases Intepolaton; Sobolev spaces; Poncaé neualty; Doublng popety; Remannan manfolds; Metc-measue spaces

2 < Then fo < p <, W p s a eal ntepolaton space between W and W To pove Theoem, we chaacteze the K-functonal of eal ntepolaton fo nonhomogeneous Sobolev spaces: Theoem 2 Let M be as n Theoem Then thee exsts C > such that fo all f W + W and t > K(f, t, W, W C t ( f (t + f (t ; 2 fo p <, thee s C 2 > such that fo all f Wp and t > K(f, t, W, W C 2 t ( f (t + f (t In the specal case =, we obtan the uppe bound of K n pont 2 f W + W and hence get a tue chaactezaton of K fo evey The poof of ths theoem eles on a Caldeón-Zygmund decomposton fo Sobolev functons (Poposton 35 Above and fom now on, g means ( g see secton 2 fo the defnton of g The eteaton theoem ([6], Chapte 5, Theoem 24 p3 and an mpovement esult fo the exponent of a Poncaé neualty due to Keth-Zhong yeld a moe geneal veson of Theoem Defne = nf { [, [: (P loc holds } Coollay 3 Fo p < p < p 2 wth p >, Wp space between Wp and Wp 2 Moe pecsely whee < θ < such that p = θ p + θ p 2 W p = (W p, W p 2 θ,p s a eal ntepolaton Howeve, f p, we only know that (Wp, Wp 2 θ,p Wp Fo the homogeneous Sobolev spaces, a weak fom of Theoem 2 s avalable Ths esult s pesented n secton 5 The conseuence fo the ntepolaton poblem s stated as follows Theoem 4 Let M be a complete non-compact Remannan manfold satsfyng the global doublng popety (D and a global Poncaé neualty (P fo some < Then, fo < p <, Ẇ p s a eal ntepolaton space between Ẇ and Ẇ Agan, the eteaton theoem mples anothe veson of Theoem 4; see secton 5 below Fo R n and the non-homogeneous Sobolev spaces, ou ntepolaton esult follows fom the leadng wok of Devoe-Schee [4] The method of [4] s based on splne functons Late, smple poofs wee gven by Caldeón-Mlman [9] and Bennett- Shapley [6], based on the Whtney extenson and coveng theoems Snce R n admts (D and (P, we ecove ths esult by ou method Moeove, applyng Theoem 4, we obtan the ntepolaton of the homogeneous Sobolev spaces on R n Notce that ths esult s not coveed by the exstng efeences 2

3 The nteested eade may fnd a wealth of examples of spaces satsfyng doublng and Poncaé neualtes to whch ou esults apply n [], [4], [5], [8], [23] Some comments about the genealty of Theoem - 4 ae n ode Fst of all, completeness of the Remannan manfold s not necessay (see Remak 43 Also, ou technue can be adapted to moe geneal metc-measue spaces, see sectons 7-8 Fnally t s possble to buld examples whee ntepolaton wthout a Poncaé neualty s possble The ueston of the necessty of a Poncaé neualty fo a geneal statement ases Ths s dscussed n the Appendx The ntal motvaton of ths wok was to povde an answe fo the ntepolaton ueston fo Wp Ths poblem was explctly posed n [3], whee the authos ntepolate neualtes of type 2 f p C p f p on Remannan manfolds Let us befly descbe the stuctue of ths pape In secton 2 we evew the notons of a doublng popety as well as the eal K ntepolaton method In sectons 3 to 5, we study n detal the ntepolaton of Sobolev spaces n the case of a complete noncompact Remannan manfold M satsfyng (D and (P (esp (D loc and (P loc We befly menton the case whee M s a compact manfold n secton 6 In secton 7, we explan how ou esults extend to moe geneal metc-measue spaces We apply ths ntepolaton esult to Canot-Caathéodoy spaces, weghted Sobolev spaces and to Le goups n secton 8 Fnally, the Appendx s devoted to an example whee the Poncaé neualty s not necessay to ntepolate Sobolev spaces Acknowledgements I am deeply ndebted to my PhD advso P Ausche, who suggested to study the topc of ths pape, and fo hs constant encouagement and useful advces I would lke to thanks P Koskela fo hs thoough eadng and pocessng of the pape Also I am thankful to P Hajlasz fo hs nteest n ths wok and M Mlman fo communcatng me hs pape wth J Matn [3] Fnally, I am also gateful to G Fexas, wth whom I had nteestng dscussons egadng ths wok 2 Pelmnaes Thoughout ths pape we wll denote by E the chaactestc functon of a set E and E c the complement of E If X s a metc space, Lp wll be the set of eal Lpschtz functons on X and Lp the set of eal, compactly suppoted Lpschtz functons on X Fo a ball B n a metc space, λb denotes the ball co-centeed wth B and wth adus λ tmes that of B Fnally, C wll be a constant that may change fom an neualty to anothe and we wll use u v to say that thee exsts two constants C, C 2 > such that C u v C 2 u 2 The doublng popety By a metc-measue space, we mean a tple (X, d, µ whee (X, d s a metc space and µ a non negatve Boel measue Denote by B(x, the open ball of cente x X and adus > Defnton 2 Let (X, d, µ be a metc-measue space One says that X satsfes the local doublng popety (D loc f thee exst constants >, < C = C( <, 3

4 such that fo all x X, < < we have (D loc µ(b(x, 2 Cµ(B(x, Futhemoe X satsfes a global doublng popety o smply doublng popety (D f one can take = We also say that µ s a locally (esp globally doublng Boel measue Obseve that f X s a metc-measue space satsfyng (D then dam(x < µ(x < ([] Theoem 22 (Maxmal theoem ([] Let (X, d, µ be a metc-measue space satsfyng (D Denote by M the uncenteed Hady-Lttlewood maxmal functon ove open balls of X defned by whee f E := fdµ := E µ(e E µ({x : Mf(x > λ} C λ Mf(x = sup f B B:x B fdµ Then f dµ fo evey λ > ; X 2 Mf Lp C p f Lp, fo < p 22 The K-method of eal ntepolaton The eade can efe to [6], [7] fo detals on the development of ths theoy Hee we only ecall the essentals to be used n the seuel Let A, A be two nomed vecto spaces embedded n a topologcal Hausdoff vecto space V Fo each a A + A and t >, we defne the K-functonal of ntepolaton by K(a, t, A, A = nf ( a A + t a A a=a +a Fo < θ <,, we denote by (A, A θ, the ntepolaton space between A and A : { ( (A, A θ, = a A + A : a θ, = (t θ K(a, t, A, A dt t < } It s an exact ntepolaton space of exponent θ between A and A, see [7], Chapte II Defnton 23 Let f be a measuable functon on a measue space (X, µ deceasng eaangement of f s the functon f defned fo evey t by f (t = nf {λ : µ({x : f(x > λ} t} The maxmal deceasng eaangement of f s the functon f defned fo evey t > by f (t = t t f (sds It s known that (Mf f and µ({x : f(x > f (t} t fo all t > We efe to [6], [7], [8] fo othe popetes of f and f We conclude the pelmnaes by uotng the followng classcal esult ([7] p9: 4 The

5 Theoem 24 Let (X, µ be a measue space whee µ s a totally σ-fnte postve measue Let f L p + L, < p < whee L p = L p (X, dµ We then have ( t p K(f, t, L p, L (f (s p p ds and eualty holds fo p = ; 2 fo < p < p < p, (L p, L p θ,p = L p wth euvalent noms, whee p = θ + θ wth < θ < p p 3 Non-homogeneous Sobolev spaces on Remannan manfolds In ths secton M denotes a complete non-compact Remannan manfold We wte µ fo the Remannan measue on M, fo the Remannan gadent, fo the length on the tangent space (fogettng the subscpt x fo smplcty and p fo the nom on L p (M, µ, p + Ou goal s to pove Theoem 2 3 Non-homogeneous Sobolev spaces Defnton 3 ([2] Let M be a C Remannan manfold of dmenson n Wte Ep fo the vecto space of C functons ϕ such that ϕ and ϕ L p, p < We defne the Sobolev space Wp as the completon of Ep fo the nom ϕ W p = ϕ p + ϕ p We denote W fo the set of all bounded Lpschtz functons on M Poposton 32 ([2], [2] Let M be a complete Remannan manfold Then C and n patcula Lp s dense n W p fo p < Defnton 33 (Poncaé neualty on M We say that a complete Remannan manfold M admts a local Poncaé neualty (P loc fo some < f thee exst constants >, C = C(, > such that, fo evey functon f Lp and evey ball B of M of adus < <, we have (P loc f f B dµ C f dµ B M admts a global Poncaé neualty (P f we can take = n ths defnton Remak 34 By densty of C n W p, we can eplace Lp by C 32 Estmaton of the K-functonal of ntepolaton In the fst step, we pove Theoem 2 n the global case Ths wll help us to undestand the poof of the moe geneal local case 32 The global case Let M be a complete Remannan manfold satsfyng (D and (P, fo some < Befoe we pove Theoem 2, we make a Caldeón- Zygmund decomposton fo Sobolev functons nsped by the one done n [3] To acheve ou ams, we state t fo moe geneal spaces (n [3], the authos only needed the decomposton fo the functons f n C Ths wll be the pncpal tool n the estmaton of the functonal K Poposton 35 (Caldeón-Zygmund lemma fo Sobolev functons Let M be a complete non-compact Remannan manfold satsfyng (D Let < and assume that M satsfes (P Let p <, f Wp and α > Then one can fnd 5 B

6 a collecton of balls (B, functons b W followng popetes hold: and a Lpschtz functon g such that the (3 f = g + b (32 g(x Cα and g(x Cα µ ae x M (33 supp b B, ( b + b dµ Cα µ(b B (34 (35 µ(b Cα p ( f + f p dµ χ B N The constants C and N only depend on, p and on the constants n (D and (P Poof Let f Wp, α > Consde Ω = {x M : M( f + f (x > α } If Ω =, then set g = f, b = fo all so that (32 s satsfed accodng to the Lebesgue dffeentaton theoem Othewse the maxmal theoem Theoem 22 gves us (36 µ(ω Cα p ( f + f ( Cα p f p dµ + < + p p f p dµ In patcula Ω M as µ(m = + Let F be the complement of Ω Snce Ω s an open set dstnct of M, let (B be a Whtney decomposton of Ω ([2] The balls B ae pawse dsjont and thee exst two constants C 2 > C >, dependng only on the metc, such that Ω = B wth B = C B and the balls B have the bounded ovelap popety; 2 = (B = d(x 2, F and x s the cente of B ; 3 each ball B = C 2 B ntesects F (C 2 = 4C woks Fo x Ω, denote I x = { : x B } By the bounded ovelap popety of the balls B, we have that I x N Fxng j I x and usng the popetes of the B s, we easly see that 3 j 3 fo all I x In patcula, B 7B j fo all I x Condton (35 s nothng but the bounded ovelap popety of the B s and (34 follows fom (35 and (36 The doublng popety and the fact that B F yeld (37 ( f + f dµ ( f + f dµ α µ(b Cα µ(b B B Let us now defne the functons b Let (χ be a patton of unty of Ω subodnated to the coveng (B, such that fo all, χ s a Lpschtz functon suppoted n B wth χ C To ths end t s enough to choose χ (x = 6

7 ψ( C d(x, x ( k ψ( C d(x k, x, whee ψ s a smooth functon, ψ = on [, ], k ψ = on [ +C, + [ and ψ We set b 2 = (f f B χ It s clea that supp b B Let us estmate B b dµ and B b dµ We have b dµ = (f f B χ dµ B B C( f dµ + f B dµ B B C f dµ B Cα µ(b We appled Jensen s neualty n the second estmate, and (37 n the last one Snce ((f f B χ = χ f + (f f B χ, the Poncaé neualty (P and (37 yeld ( b dµ C χ f dµ + f f B χ dµ B B B Cα µ(b + C C f dµ B Cα µ(b Theefoe (33 s poved Set g = f b Snce the sum s locally fnte on Ω, g s defned almost eveywhee on M and g = f on F Obseve that g s a locally ntegable functon on M Indeed, let ϕ L wth compact suppot Snce d(x, F fo x supp b, we obtan ( and b ϕ dµ b dµ b f f B dµ = B ( µ(b Cαµ(B χ dµ ( B ( sup d(x, F ϕ(x x M f dµ We used the Hölde neualty, (P and the estmate (37, beng the conjugate of ( Hence b ϕ dµ Cαµ(Ω sup d(x, F ϕ(x Snce f L,loc, we deduce x M that g L,loc (Note that snce b L n ou case, we can say dectly that g L,loc Howeve, fo the homogeneous case secton 5 we need ths obsevaton to conclude that g L,loc It emans to pove (32 Note that χ (x = and χ (x = 7

8 fo all x Ω We have g = f b = f ( χ f (f f B χ = F ( f + f B χ Fom the defnton of F and the Lebesgue dffeentaton theoem, we have that F ( f + f α µ ae We clam that a smla estmate holds fo h = f B χ We have h(x Cα fo all x M Fo ths, note fst that h vanshes on F and s locally fnte on Ω Then fx x Ω and let B j be some Whtney ball contanng x Fo all I x, we have f B f Bj C j α Indeed, snce B 7B j, we get f B f 7Bj f f 7Bj dµ µ(b B C f f 7Bj dµ µ(b j 7B j C j ( f dµ 7B j (38 C j α whee we used Hölde neualty, (D, (P and (37 Analogously f 7Bj f Bj C j α Hence h(x = I x (f B f Bj χ (x C I x f B f Bj CNα Fom these estmates we deduce that g(x Cα µ ae Let us now estmate g We have g = f F + f B χ Snce f F α, stll need to estmate f B χ Note that (39 ( f B C f dµ µ(b B ( (y M( f + f M( f + f (y α whee y B F snce B F The second neualty follows fom the fact that (Mf Mf fo Let x Ω Ineualty (39 and the fact that I x N yeld g(x = I x f B χ 8

9 We conclude that g C α complete I x f B Nα µ ae and the poof of Poposton 35 s theefoe Remak 36 - It s a staghtfowad conseuence of (33 that b W fo all wth b W Cαµ(B 2- Fom the constucton of the functons b, we see that b Wp, wth b W p C f W p It follows that g Wp Hence (g, g satsfes the Poncaé neualty (P p Theoem 32 of [23] assets that fo µ ae x, y M g(x g(y Cd(x, y ((M g p p (x + (M g p p (y Fom Theoem 22 wth p = and the neualty g Cα, we deduce that g has a Lpschtz epesentatve Moeove, the Lpschtz constant s contolled by Cα 3- We also deduce fom ths Caldeón-Zygmund decomposton that g W s fo p s We have ( Ω ( g s + g s dµ s Cαµ(Ω s and ( g s + g s dµ = ( f s + f s dµ F F ( f p f s p + f p f s p dµ F α s p f p W p < Coollay 37 Unde the same hypotheses as n the Caldeón-Zygmund lemma, we have W p W + W s fo p s < Poof of Theoem 2 To pove pat, we begn applyng Theoem 24, pat We have ( t K(f, t, L, L (f (s ds On the othe hand ( t f (s ds ( t = = f(s ds ( t f (t whee n the fst eualty we used the fact that f = ( f and the second follows fom the defnton of f We thus get K(f, t, L, L t ( f (t Moeove, snce the lnea opeato K(f, t, W, W K(f, t, L, L + K( f, t, L, L (I, : W s (M (L s (M; C T M s bounded fo evey s These two ponts yeld the desed neualty We wll now pove pat 2 We teat the case when f W p, p < Let t > We consde the Caldeón-Zygmund decomposton of f of Poposton 35 9

10 wth α = α(t = (M( f + f (t We wte f = b + g = b + g whee (b, g satsfy the popetes of the poposton Fom the bounded ovelap popety of the B s, t follows that fo all b M( b dµ N b dµ B Cα (t µ(b Cα (tµ(ω Smlaly we have b Cα(tµ(Ω Moeove, snce (Mf f and (f + g f + g, we get α(t = (M( f + f (t C ( f (t + f (t Notng that µ(ω t, we deduce that (3 K(f, t, W, W Ct ( f (t + f (t fo all t > and obtan the desed neualty fo f Wp, p < Note that n the specal case whee =, we have the uppe bound of K fo f W Applyng a smla agument to that of [4] Eucldean case we get (3 fo f W + W Hee we wll omt the detals We wee not able to show ths chaactezaton when < snce we could not show ts valdty even fo f W Nevetheless ths theoem s enough to acheve ntepolaton (see the next secton 322 The local case Let M be a complete non-compact Remannan manfold satsfyng a local doublng popety (D loc and a local Poncaé neualty (P loc fo some < Denote by M E the Hady-Lttlewood maxmal opeato elatve to a measuable subset E of M, that s, fo x E and evey locally ntegable functon f on M M E f(x = sup B: x B µ(b E B E f dµ whee B anges ove all open balls of M contanng x and centeed n E We say that a measuable subset E of M has the elatve doublng popety f thee exsts a constant C E such that fo all x E and > we have µ(b(x, 2 E C E µ(b(x, E Ths s euvalent to sayng that the metc-measue space (E, d E, µ E has the doublng popety On such a set M E s of weak type (, and bounded on L p (E, µ, < p

11 Poof of Theoem 2 To fx deas, we assume wthout loss of genealty = 5, = 8 The lowe bound of K s tval (same poof as fo the global case It emans to pove the uppe bound Fo all t >, take α = α(t = (M( f + f (t Consde We have µ(ω t If Ω = M then f dµ + f dµ = Theefoe M Ω = {x M : M( f + f (x > α (t} M f dµ + Ω µ(ω t Ω f (sds + f (sds + f dµ t µ(ω = t ( f (t + f (t f (sds K(f, t, W, W f W Ct ( f (t + f (t Ct ( f (t + f (t f (sds snce We thus obtan the uppe bound n ths case Now assume Ω M Pck a countable set {x j } j J M, such that M = B(x j, 2 and fo all x M, x does not belong to moe than N balls B j := B(x j, Consde a C patton of unty (ϕ j j J subodnated to the balls 2 Bj such that ϕ j, supp ϕ j B j and ϕ j C unfomly wth espect to j Consde f W p, p < Let f j = fϕ j so that f = j J f j We have fo j J, f j L p and f j = f ϕ j + fϕ j L p Hence f j W p (B j The balls B j satsfy the elatve doublng popety wth constant ndependent of the balls B j Ths follows fom the next lemma uoted fom [4] p947 Lemma 38 Let M be a complete Remannan manfold satsfyng (D loc Then the balls B j above, eupped wth the nduced dstance and measue, satsfy the elatve doublng popety (D, wth the doublng constant that may be chosen ndependently of j Moe pecsely, thee exsts C such that fo all j J (3 µ(b(x, 2 B j C µ(b(x, B j x B j, >, and (32 µ(b(x, Cµ(B(x, B j x B j, < 2 Remak 39 Notng that the poof n [4] only used the fact that M s a length space, we obseve that Lemma 38 stll holds fo any length space Recall that a length space X s a metc space such that the dstance between any two ponts x, y X s eual to the nfmum of the lengths of all paths jonng x to y (we mplctly assume that thee s at least one such path Hee a path fom x to y s a contnuous map γ : [, ] X wth γ( = x and γ( = y j J

12 Let us etun to the poof of the theoem Fo any x B j we have M B j( f j + f j (x = ( f µ(b j j + f j dµ B (33 sup B: x B, (B 2 sup B: x B, (B 2 CM( f + f (x B j B C µ(b µ(b j B µ(b whee we used (32 of Lemma 38 Consde now Ω j = { x B j : M B j( f j + f j (x > Cα (t } B ( f + f dµ whee C s the constant n (33 Ω j s an open subset of B j, hence of M, and Ω j Ω M fo all j J Fo the f j s, and fo all t >, we have a Caldeón- Zygmund decomposton smla to the one done n Poposton 35: thee exst b jk, g j suppoted n B j, and balls (B jk k of M, contaned n Ω j, such that (34 f j = g j + k b jk (35 g j (x Cα(t and g j (x Cα(t fo µ ae x M (36 supp b jk B jk, fo ( b jk + b jk dµ Cα (tµ(b jk B jk (37 (38 µ(b jk Cα p (t ( f j + f j p dµ B j k χ Bjk N k wth C and N dependng only on, p and the constants n (D loc and (P loc The poof of ths decomposton wll be the same as n Poposton 35, takng fo all j J a Whtney decomposton (B jk k of Ω j M and usng the doublng popety fo balls whose ad do not exceed 3 < and the Poncaé neualty fo balls whose ad do not exceed 7 < Fo the bounded ovelap popety (38, just note that the adus of evey ball B jk s less than Then apply the same agument as fo the bounded ovelap popety of a Whtney decomposton fo an homogeneous space, usng the doublng popety fo balls wth suffcently small ad By the above decomposton we can wte f = b jk + g j = b + g Let us j J k j J now estmate b W and g W b N N b jk j k Cα (t (µ(b jk j k ( NCα (t µ(ω j N Cα (tµ(ω 2 j

13 We used the bounded ovelap popety of the (Ω j j J s and that of the (B jk k s fo all j J It follows that b Cα(tµ(Ω Smlaly we get b Cα(tµ(Ω Fo g we have g sup x sup x g j (x j J N sup g j (x j J N sup g j j J Cα(t Analogously g Cα(t We conclude that K(f, t, W, W b W + t g W Cα(tµ(Ω + Ct α(t Ct α(t Ct ( f (t + f (t whch completes the poof of Theoem 2 n the case < When = we get the chaactezaton of K fo evey f W + W by applyng agan a smla agument to that of [4] 4 Intepolaton Theoems In ths secton we establsh ou ntepolaton Theoem and some conseuences fo non-homogeneous Sobolev spaces on a complete non-compact Remannan manfold M satsfyng (D loc and (P loc fo some < Fo < p <, we defne the eal ntepolaton space Wp, between W and W by Wp, = (W, W p,p Fom the pevous esults we know that fo f W + W { ( f p,p C t p ( f + f (t p dt t and fo f Wp { ( } f p,p C 2 t p ( f + f p dt p (t t We clam that Wp, = Wp, wth euvalent noms Indeed, f p,p C { } p ( f (t + f (t p dt } p ( C f p + f p ( C f p + f p 3

14 and = C ( f p + f p = C f W p, { } f p,p C 2 ( f (t + f p p (t dt ( C f p + f p ( C f p + f p = C ( f p + f p = C f W p, whee we used that fo l >, f l f l (see [34], Chapte V: Lemma 32 p9 and Theoem 32, p2 Moeove, fom Coollay 37, we have Wp W + W fo < p < Theefoe Wp s a eal ntepolaton space between W and W fo < p < Let us ecall some known facts about Poncaé neualtes wth vayng It s known that (P loc mples (P ploc when p (see [23] Thus f the set of such that (P loc holds s not empty, then t s an nteval unbounded on the ght A ecent esult of Keth and Zhong [28] assets that ths nteval s open n [, + [ Theoem 4 Let (X, d, µ be a complete metc-measue space wth µ locally doublng and admttng a local Poncaé neualty (P loc, fo some < < Then thee exsts ɛ > such that (X, d, µ admts (P ploc fo evey p > ɛ Hee, the defnton of (P loc s that of secton 7 It educes to the one of secton 3 when the metc space s a Remannan manfold Comment on the poof of ths theoem The poof goes as n [28] whee ths theoem s poved fo X satsfyng (D and admttng a global Poncaé neualty (P By usng the same agument and choosng suffcently small ad fo the consdeed balls, (P loc wll gve us (P ( ɛloc fo evey ball of adus less than 2, fo some 2 < mn(,,, beng the constants gven n the defntons of local doublng popety and local Poncaé neualty Defne A M = { [, [: (P loc holds } and M = nf A M When no confuson ases, we wte nstead of M As we mentoned n the ntoducton, ths mpovement of the exponent of a Poncaé neualty togethe wth the eteaton theoem yeld anothe veson of ou ntepolaton esult: Coollay 3 Poof of Coollay 3 Let < θ < such that = θ p p + θ p 2 Case when p > Snce p >, thee exsts A M such that < < p Then = ( θ( p p +θ( p 2 The eteaton theoem [6], Theoem 24 p 3 yelds (W p, W p 2 θ,p = (W p,, W p 2, θ,p = (W, W p,p 4

15 = W p, = W p 2 Case when p Let θ = θ( p p 2 = p The eteaton theoem p appled ths tme only to the second exponent yelds (W p, W p 2 θ,p = (W p, W p 2,p θ,p = (W p, W θ,p = W p,p = W p Theoem 42 Let M and N be two complete non-compact Remannan manfolds satsfyng (D loc Assume that M and N ae well defned Take p p 2,,, 2 Let T be a bounded lnea opeato fom Wp (M to W (N of nom L, =, 2 Then fo evey couple (p, such that p, p > M, > N and (, = ( θ( p p, + θ( p 2, 2, < θ <, T s bounded fom Wp (M to W (N wth nom L CL θ L θ Poof T f W (N C T f (W (N,W 2 (N θ, CL θ L θ f (W p (M,W p 2 (M θ, CL θ L θ f (W p (M,W p 2 (M θ,p CL θ L θ f W p (M We used the fact that K θ, s an exact ntepolaton functo of exponent θ, that Wp (M = (Wp (M, Wp 2 (M θ,p, W (N = (W (N, W 2 (N θ, wth euvalent noms and that (Wp (M, Wp 2 (M θ,p (Wp (M, Wp 2 (M θ, f p Remak 43 Let M be a Remannan manfold, not necessaly complete, satsfyng (D loc Assume that fo some <, a weak local Poncaé neualty holds fo all C functons, that s thee exsts >, C = C(,, λ such that fo all f C and all ball B of adus < we have ( f f B dµ B ( C λb f dµ Then, we obtan the chaactezaton of K as n Theoem 2 and we get by ntepolatng a esult analogous to Theoem 5 Homogeneous Sobolev spaces on Remannan manfolds Defnton 5 Let M be a C Remannan manfold of dmenson n Fo p, we defne E p to be the vecto space of dstbutons ϕ wth ϕ L p, whee ϕ s the dstbutonal gadent of ϕ It s well known that the elements of We eup E p wth the sem nom ϕ E p = ϕ p 5 E p ae n L ploc

16 Defnton 52 We defne the homogeneous Sobolev space Ep/R Remak 53 Fo all ϕ E p, ϕ W p W p as the uotent space = ϕ p, whee ϕ denotes the class of ϕ Poposton 54 ([2] Wp s a Banach space 2 Assume that M satsfes (D and (P fo some < and fo all f Lp, that s thee exsts a constant C > such that fo all f Lp and fo evey ball B of M of adus > we have ( ( (P f f B dµ C f dµ B Then Lp(M Wp s dense n Wp fo p < Poof The poof of tem 2 s mplct n the poof of Theoem 9 n [7] We obtan fo the K-functonal of the homogeneous Sobolev spaces the followng homogeneous fom of Theoem 2, weake n the patcula case =, but agan suffcent fo us to ntepolate Theoem 55 Let M be a complete Remannan manfold satsfyng (D and (P fo some < Let Then thee exsts C such that fo evey F W + W and t > K(F, t, W, W C t f (t whee f E + E and f = F ; 2 fo p <, thee exsts C 2 such that fo evey F W p and t > K(F, t, W, W C 2 t f (t whee f Ep and f = F Befoe we pove Theoem 55, we gve the followng Caldeón-Zygmund decomposton that wll be also n ths case ou pncpal tool to estmate K Poposton 56 (Caldeón-Zygmund lemma fo Sobolev functons Let M be a complete non-compact Remannan manfold satsfyng (D and (P fo some < Let p <, f E p and α > Then thee s a collecton of balls (B, functons b E and a Lpschtz functon g such that the followng popetes hold : B (5 f = g + b (52 g(x C α µ ae (53 supp b B and fo b dµ Cα µ(b B (54 µ(b Cα p f p dµ 6

17 (55 χ B N The constants C and N depend only on, p and the constant n (D Poof The poof goes as n the case of non-homogeneous Sobolev spaces, but takng Ω = {x M : M( f (x > α } as f p s not unde contol We note that n the non-homogeneous case, we used that f L p only to contol g L and b L Remak 57 It s suffcent fo us that the Poncaé neualty holds fo all f E p Coollay 58 Unde the same hypotheses as n the Caldeón-Zygmund lemma, we have Wp W + W fo p < Poof of Theoem 55 The poof of tem s the same as n the non-homogeneous case Let us tun to neualty 2 Fo F W p we take f E p wth f = F Let ( t > and α(t = M( f (t By the Caldeón-Zygmund decomposton wth α = α(t, f can be wtten f = b + g, hence F = b + g, wth b W = b Cα(tµ(Ω and g = g Cα(t Snce fo α = α(t we have µ(ω t, W then we get K(F, t, W, W Ct f (t We can now pove ou ntepolaton esult fo the homogeneous Sobolev spaces Poof of Theoem 4 The poof follows dectly fom Theoem 55 Indeed, tem of Theoem 55 yelds wth F W p ( W, W p,p W p C F p,p, whle tem 2 gves us that Wp ( W, W p,p wth F p,p C F W We conclude that p wth euvalent noms Wp = ( W, W p,p Coollay 59 (The eteaton theoem Let M be a complete non-compact Remannan manfold satsfyng (D and (P fo some < Defne = nf { [, [: (P holds } Then fo p > and p < p < p 2, a eal ntepolaton space between Wp and Wp 2 Applcaton Consde a complete non-compact Remannan manfold M satsfyng (D and (P fo some < 2 Let be the Laplace-Beltam opeato Consde the lnea opeato 2 wth the followng esoluton 2 f = c e t f dt t, f C 7 W p s

18 whee c = π 2 Hee 2 f can be defned fo f Lp as a measuable functon (see [3] In [3], Ausche and Coulhon poved that on such a manfold, we have } µ {x M : 2 f(x > α C α f fo f C, wth [, 2[ In fact one can check that the agument apples to all f Lp E and snce 2 =, 2 can be defned on Lp W by takng uotent whch we keep callng 2 Moeove, Poposton 54 gves us that 2 has a bounded extenson fom W to L, Snce we aleady have then by Coollay 59, we see at once 2 f 2 f 2 (56 2 f p C p f p fo all < p 2 and f W p, wthout usng the agument n [3] 6 Sobolev spaces on compact manfolds Let M be a C compact manfold eupped wth a Remannan metc Then M satsfes the doublng popety (D and the Poncaé neualty (P Theoem 6 Let M be a C compact Remannan manfold Thee exst C, C 2 such that fo all f W + W and all t > we have ( ( ( comp C t f (t + f (t K(f, t, W, W C 2 t f (t + f (t Poof It emans to pove the uppe bound fo K as the lowe bound s tval Indeed, let us consde fo all t > and fo α(t = (M( f + f (t, Ω = {x M; M( f + f (x α(t} If Ω M, we have the Caldeón-Zygmund decomposton as n Poposton 35 wth = and the poof wll be the same as the poof of Theoem 2 n the global case Now f Ω = M, we pove the uppe bound by the same agument used n the poof of Theoem 2 n the local case Thus, n the two cases we obtan the ght hand neualty of ( comp fo all f W + W It follows that Theoem 62 Fo all p < p < p 2, Wp Wp and Wp 2 7 Metc-measue spaces s an ntepolaton space between In ths secton we consde (X, d, µ a metc-measue space wth µ doublng 7 Uppe gadents and Poncaé neualty Defnton 7 (Uppe gadent [26] Let u : X R be a Boel functon We say that a Boel functon g : X [, + ] s an uppe gadent of u f u(γ(b u(γ(a b a g(γ(tdt fo all -Lpschtz cuve γ : [a, b] X Snce evey ectfable cuve admts an ac-length paametzaton that makes the cuve - Lpschtz, the class of -Lpschtz cuves concdes wth the class of ectfable cuves, modulo a paamete change 8

19 Remak 72 If X s a Remannan manfold, u s an uppe gadent of u Lp and u g fo all uppe gadents g of u Defnton 73 Fo evey locally Lpschtz contnuous functon u defned on a open set of X, we defne { lm u(y u(x supy x f x s not solated, Lp u(x = y x d(y,x othewse Remak 74 Lp u s an uppe gadent of u Defnton 75 (Poncaé Ineualty A metc-measue space (X, d, µ admts a weak local Poncaé neualty (P loc fo some <, f thee exst >, λ, C = C(, >, such that fo evey contnuous functon u and uppe gadent g of u, and fo evey ball B of adus < < the followng neualty holds: (P loc ( u u B dµ B ( C λb g dµ If λ =, we say that we have a stong local Poncaé neualty Moeove, X admts a global Poncaé neualty o smply a Poncaé neualty (P f one can take = 72 Intepolaton of the Sobolev spaces H p Befoe defnng the Sobolev spaces H p t s convenent to ecall the followng poposton Poposton 76 (see [22] and [] Theoem 438 Let (X, d, µ be a complete metcmeasue space, wth µ doublng and satsfyng a weak Poncaé neualty (P fo some < < Then thee exst an ntege N, C and a lnea opeato D whch assocates to each locally Lpschtz functon u a measuable functon Du : X R N such that : f u s L-Lpschtz, then Du CL µ ae; 2 f u s locally Lpschtz and constant on a measuable set E X, then Du = µ ae on E; 3 fo locally Lpschtz functons u and v, D(uv = udv + vdu; 4 fo each locally Lpschtz functon u, Lp u Du C Lp u, and hence (u, Du satsfes the weak Poncaé neualty (P We defne now H p = H p(x, d, µ fo p < as the closue of locally Lpschtz functons fo the nom u H p = u p + Du p u p + Lp u p We denote H fo the set of all bounded Lpschtz functons on X Remak 77 Unde the hypotheses of Poposton 76, the unueness of the gadent holds fo evey f H p wth p By unueness of gadent we mean that f u n s a locally Lpschtz seuence such that u n n L p and Du n g L p then g = ae Then D extends to a bounded lnea opeato fom H p to L p In the emanng pat of ths secton, we consde a complete non-compact metcmeasue space (X, d, µ wth µ doublng We also assume that X admts a Poncaé neualty (P fo some < < as defned n Defnton 75 By [27] Theoem 9

20 34, ths s euvalent to say that thee exsts C > such that fo all f Lp and fo evey ball B of X of adus > we have (P f f B dµ C Lp f dµ Defne = nf { ], [: (P holds } B Lemma 78 Unde these hypotheses, and fo < p <, Lp H p s dense n H p Poof See the poof of Theoem 9 n [7] Poposton 79 Caldeón-Zygmund lemma fo Sobolev functons Let (X, d, µ be a complete non-compact metc-measue space wth µ doublng, admttng a Poncaé neualty (P fo some < < Then, the Caldeón-Zygmund decomposton of Poposton 35 stll holds n the pesent stuaton fo f Lp H p, p <, eplacng f by Df Poof The poof s smla, eplacng f by Df, usng that D of Poposton 76 s lnea Snce the χ ae C Lpschtz then Dχ C by tem of Theoem 76 and the b s ae Lpschtz We can see that g s also Lpschtz Moeove, usng the fnte addtvty of D and the popety 2 of Poposton 76, we get the eualty µ ae Dg = Df D( b = Df ( The est of the poof goes as n Poposton 35 B Db Theoem 7 Let (X, d, µ be a complete non-compact metc-measue space wth µ doublng, admttng a Poncaé neualty (P fo some < < Then, thee exst C, C 2 such that fo all f H + H and all t > we have ( met C t ( f (t + Df (t K(f, t, H, H C 2 t ( f (t + Df (t Poof We have ( met fo all f Lp H fom the Caldeón-Zygmund decomposton that we have done Now fo f H, by Lemma 78, f = lm f n n H, wth f n Lpschtz n and f f n H < fo all n Snce fo all n, f n n Lp, thee exst g n, h n such that f n = h n + g n and h n H + t gn H Ct ( f n (t + Dfn (t Theefoe we fnd f g n H + t gn H f f n H + ( h n H + t gn H n + Ct ( f n (t + Dfn (t Lettng n, snce f n n f n L and Df n Df n L, t comes n f n (t f (t and Df n (t Df (t fo all t > Hence ( met n n holds fo f H We pove ( met fo f H +H by the same agument of [4] Theoem 7 (Intepolaton Theoem Let (X, d, µ be a complete non-compact metc-measue space wth µ doublng, admttng a Poncaé neualty (P fo some < < Then, fo < p < p < p 2 2, H p s an ntepolaton space between H p and H p 2 2 We allow p = f = 2

21 Poof Theoem 7 povdes us wth all the tools needed fo ntepolatng, as we dd n the Remannan case In patcula, we get Theoem 7 Remak 72 We wee not able to get ou ntepolaton esult as n the Remmanan case fo p Snce we do not have Poncaé neualty (P p, the unueness of the gadent D does not hold n geneal n H p Remak 73 Othe Sobolev spaces on metc-measue spaces wee ntoduced n the last few yeas, fo nstance M p, N p, C p, P p If X s a complete metc-measue space satsfyng (D and (P fo some < <, t can be shown that fo < p, all the mentoned spaces ae eual to H p wth euvalent noms (see [23] In concluson ou ntepolaton esult caes ove to those Sobolev spaces Remak 74 The pupose of ths emak s to extend ou esults to local assumptons Assume that (X, d, µ s a complete metc-measue space, wth µ locally doublng, and admttng a local Poncaé neualty (P loc fo some < < Snce X s complete and (X, µ satsfes a local doublng condton and a local Poncaé neualty (P loc, then accodng to an obsevaton of Davd and Semmes (see the ntoducton n [], evey ball B(z,, wth < < mn(,, s λ = λ(c(, C( uas-convex, C( and C( beng the constants appeang n the local doublng popety and n the local Poncaé neualty Then, fo < < mn(,, B(z, s λ b-lpschtz to a length space (one can assocate, canoncally, to a λ-uas-convex metc space a length metc space, whch s λ-b-lpschtz to the ognal one Hence, we get a esult smla to the one n Theoem 7 Indeed, the poof goes as that of Theoem 2 n the local case notng that the B j s consdeed thee ae then λ b-lpschtz to a length space wth λ ndependent of j Thus Lemma 38 stll holds (see Remak 39 Theefoe, we get the chaactezaton ( met of K and by ntepolatng, we obtan the coespondance analogue of Theoem 7 8 Applcatons 8 Canot-Caathéodoy spaces An mpotant applcaton of the theoy of Sobolev spaces on metc-measue spaces s to a Canot-Caathéodoy space We efe to [23] fo a suvey on the theoy of Canot-Caathéodoy spaces Let Ω R n be a connected open set, X = (X,, X k a famly of vecto felds defned ( k on Ω, wth eal locally Lpschtz contnuous coeffcents and Xu(x = X j u(x 2 2 We eup Ω wth the Lebesgue measue L n and the Canot-Caathéodoy metc ρ assocated to the X We assume that ρ defnes a dstance Then, the metc space (Ω, ρ s a length space Defnton 8 Let p < We defne Hp,X (Ω as the completon of locally metc 3 Lpschtz functons (euvalently of C functons fo the nom j= f H p,x = f Lp(Ω + Xf Lp(Ω We denote H,X fo the set of bounded metc Lpschtz functon Remak 82 Fo all p, H p,x = W p,x (Ω := {f L p(ω : Xf L p (Ω}, whee Xf s defned n the dstbutonal sense (see fo example [9] Lemma 76 3 that s elatve to the metc ρ of Canot-Caathéodoy 2

22 Adaptng the same method, we obtan the followng ntepolaton theoem fo the H p,x Theoem 83 Consde (Ω, ρ, L n whee Ω s a connected open subset of R n We assume that L n s locally doublng, that the dentty map d : (Ω, ρ (Ω, s an homeomophsm Moeove, we suppose that the space admts a local weak Poncaé neualty (P loc fo some < Then, fo p < p < p 2 wth p >, H p,x s a eal ntepolaton space between H p,x and H p 2,X 82 Weghted Sobolev spaces We efe to [24], [29] fo the defntons used n ths subsecton Let Ω be an open subset of R n eupped wth the Eucldean dstance, w L,loc (R n wth w >, dµ = wdx We assume that µ s -admssble fo some < < (see [25] fo the defnton Ths s euvalent to say, (see [23], that µ s doublng and thee exsts C > such that fo evey ball B R n of adus > and fo evey functon ϕ C (B, (P ϕ ϕ B dµ C ϕ dµ B B wth ϕ B = ϕdµ The A µ(b B weghts, >, satsfy these two condtons (see [25], Chapte 5 Defnton 84 Fo p <, we defne the Sobolev space H p(ω, µ to be the closue of C (Ω fo the nom u H p (Ω,µ = u Lp(µ + u Lp(µ We denote H (Ω, µ fo the set of all bounded Lpschtz functons on Ω Usng ou method, we obtan the followng ntepolaton theoem fo the Sobolev spaces H p(ω, µ: Theoem 85 Let Ω be as n above Then fo < p < p < p 2, H p(ω, µ s a eal ntepolaton space between H p (Ω, µ and H p 2 (Ω, µ As n secton 7, we wee not able to get ou ntepolaton esult fo p snce agan n ths case the unueness of the gadent does not hold fo p Remak 86 Eup Ω wth the Canot-Caathéodoy dstance assocated to a famly of vecto felds wth eal locally Lpschtz contnuous coeffcents nstead of the Eucldean dstance Unde the same hypotheses used n the begnnng of ths secton, just eplacng the balls B by the balls B wth espect to ρ, and by X and assumng that d : (Ω, ρ (Ω, s an homeomophsm, we obtan ou ntepolaton esult As an example we take vectos felds satsfyng a Hömande condton o vectos felds of Gushn type [6] 83 Le Goups In all ths subsecton, we consde G a connected unmodula Le goup eupped wth a Haa measue dµ and a famly of left nvaant vecto felds X,, X k such that the X s satsfy a Hömande condton In ths case the Canot- Caathéodoy metc ρ s s a dstance, and G eupped wth the dstance ρ s complete and defnes the same topology as that of G as a manfold (see [3] page 48 Fom the esults n [2], [32], t s known that G satsfes (D loc Moeove, f G has polynomal gowth t satsfes (D Fom the esults n [33], [35], G admts a local Poncaé 22

23 neualty (P loc If G has polynomal gowth, then t admts a global Poncaé neualty (P Intepolaton of non-homogeneous Sobolev spaces We defne the non-homogeneous Sobolev spaces on a Le goup Wp n the same manne as n secton 3 on a Remannan manfold eplacng by X (see Defnton 3 and Poposton 32 To ntepolate the W p, we dstngush between the polynomal and the exponental gowth cases If G has polynomal gowth, then we ae n the global case If G has exponental gowth, we ae n the local case In the two cases we obtan the followng theoem Theoem 87 Let G be as above Then, fo all p < p < p 2, Wp s an ntepolaton space between Wp and Wp 2, ( = hee Theefoe, we get all the ntepolaton theoems of secton 4 Intepolaton of homogeneous Sobolev spaces Let G be a connected Le goup as befoe We defne the homogeneous Sobolev space Ẇ p n the same manne as n secton 5 on Remannan manfolds eplacng by X Fo these spaces we have the followng ntepolaton theoem Theoem 88 Let G be as above and assume that G has polynomal gowth Then fo p < p < p 2, Ẇ p s a eal ntepolaton space between Ẇ p and Ẇ p 2 9 Appendx In vew of the hypotheses n the pevous ntepolaton esults, a natuel ueston to addess s whethe the popetes (D and (P ae necessay The am of the appendx s to exhbt an example whee at least Poncaé s not needed Consde X = { } (x, x 2,, x n R n ; x x 2 n x 2 n eupped wth the Eucldean metc of R n and wth the Lebesgue measue X conssts of two nfnte closed cones wth a common vetex X satsfes the doublng popety and admts (P n the sense of metc-measue spaces f and only f > n ([23] p7 Denote by Ω the nteo of X Let H p(x be the closue of Lp (X fo the nom f H p (X = f Lp(Ω + f Lp(Ω We defne W p (Ω as the set of all functons f L p (Ω such that f L p (Ω and eup ths space wth the nom f W p (Ω = f H p (X The gadent s always defned n the dstbutonal sense on Ω Usng ou method, t s easy to check that the W p (Ω ntepolate fo all p Also ou ntepolaton esult assets that H p(x s an ntepolaton space between H p (X and H p 2 (X fo p < p < p 2 wth p > n It can be shown that H p(x W p (Ω fo p > n and H p(x = W p (Ω fo p < n Hence H p(x s an ntepolaton space between H p (X and H p 2 (X fo p < p < p 2 < n although the Poncaé neualty does not hold on X fo those p Howeve, we do not know f 23

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