L1 → Lq Poincaré inequalities for 0 < q < 1 imply representation formulas

Ac a ma hema ica sinica, English Se ies, Se ies 18 (2002) 1, 1-20. L1→LqPOINCAR´ E INEQUALITIES FOR 0< q < 1IMPLY REPRESENTATION FORMULAS Guozhen Lu(∗) Depa men o Ma hema ics Wayne S a e Uni e si y De oi , MI 48202, USA E-mail: [email p o ec ed]ayne.edu Ca los P´ e ez(∗)(∗∗) Depa amen o de Ma em´a icas Uni e sidad Au ´onoma de Mad id 28049 Mad id, Spain E-mail: ca los.p[email p o ec ed] Dedica ed o Dick Wheeden on he occasion o his 60 h bi hday wi h app ecia ion and admi a ion 1991 Ma hema ics Subjec Classi ica ion. 46E35, 41A10, 22E25. Key wo ds and ph ases. Sobole spaces, ep esen a ion o mulas, high o de de i a i es, ec o ields, me ic spaces, polynomials, doubling measu es, Poinca ´e inequali ies. (*) The i s au ho was suppo ed pa ly by he U.S. Na ional Science Founda ion G an Nos. DMS96-22996 and DMS99-70352. The second au ho was suppo ed pa ly by DGICYT g an PB940192, Spain. Bo h au ho s we e suppo ed pa ly by NATO collabo a i e esea ch g an 972144. (**) The main pa o his pape was comple ed du ing he second au ho ’s isi a W igh S a e Uni e si y, Ohio in June, 1999. He wishes o hank he Depa men o Ma hema ics and S a is ics a W igh S a e Uni e si y o i s hospi ali y and inancial suppo . Typese by A MS-T EX 1 2 G. LU AND C. P´ EREZ Abs ac . Gi en wo doubling measu es µand νin a me ic space (S, ρ) o homogeneous ype and le B0⊂ S be a gi en ball. I has been a well-known esul by now (see [FLW], [FW], [LW1], [LW2]) ha he alidi y o an L1→L1Poinca ´e inequali y o he ollowing o m: ZB | − B|dν ≤c (B)ZB gdµ o all me ic balls B⊂B0⊂ S implies a a ian o ep esen a ion o mula o ac ional in e g al ype: o ν-a.e. x∈B0, | (x)− B0| ≤ CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y) + C (B0) µ(B0)ZB0 g(y)dµ(y). One o he main esul s o his pape shows ha an L1 o LqPoinca ´e inequali y o some 0< q < 1, i.e., „ZB | − B|qdν«1/q ≤c (B)ZB gdµ o all me ic balls B⊂B0will su ice o imply he abo e ep esen a ion o mula. As an immedia e co olla y, we can show ha he weak ype condi ion sup λ>0 λν ({x∈B:| (x)− B|> λ}) ν(B)≤C (B)ZB gdµ also implies he same o mula. Analogous heo ems ela ed o high o de Poinca ´e inequali ies and Sobole spaces in me ic spaces a e also p o ed. §1. In oduc ion I is known ha L1→L1Poinca ´e inequali ies a e equi alen o he ac ional in eg al es ima es in gene al me ic spaces o homogeneous ype (see [FLW2], [FW], [LW1-2]). A na u al ques ion hus a ises: Is he L1→L1Poinca ´e inequali y he leas we need o s a in o de o de i e such ep esen a ion o mulas? In his pape , we s udy his issue and weaken he hypo hesis ha an L1→L1Poinca ´e inequali y has o hold o ob ain any kind o ep esen a ion o mulas o ac ional ype. Mo e p ecisely, we will show ha an L1→LqPoinca ´e inequali y o some 0 < q < 1 will su ice o de i e he poin wise ac ional es ima es. On he o he hand i is well known ha he ollowing Kolmogo o s inequali y holds o 0 < q < 1, any nonnega i e unc ion gand a bi a y measu able se Ewi h ini e measu e (1.1) µ1 µ(E)ZE g(x)qdµ¶1/q ≤cq|g|L1,∞(E,µ). See [GCRdF] p. 485 o ins ance. We will be using he ollowing no a ion o he local a e age Ma cinkiewicz quasi-no m |g|L1,∞(E,µ)= sup λ>0 λµ({x∈E:| (x)|> λ}) µ(E). Hence as an in e es ing co olla y o ou main esul , we p o e ha weak ype L1,∞→L1 Poinca ´e inequali y is su icien o imply ac ional ep esen a ion o mulas. L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 3 Mo e p ecisely, gi en wo doubling measu es µand νin a me ic space (S, ρ) o homogeneous ype and le B0⊂ S be a gi en ball. I has been a well-known esul by now (see [FLW2], [FW], [LW1], [LW2]) ha he alidi y o an L1→L1Poinca ´e inequali y o he ollowing o m: (1.2) ZB | − B|dν ≤c (B)ZB g dµ o all me ic balls B⊂B0⊂ S implies a a ian o ep esen a ion o mula o ac ional in eg al ype: o ν-a.e. x∈B0, | (x)− B0| ≤ CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y) + C (B0) µ(B0)ZB0 g(y)dµ(y). As usual we use he ollowing no a ion o he a e age o o e a ball B, B= 1 µ(B)RB dµ Ou i s main esul o his pape demons a es ha an L1 o LqPoinca ´e inequali y o some 0 < q < 1, i.e., µZB | − B|qdν¶1/q ≤c (B)ZB g dµ o all me ic balls B⊂B0will su ice o imply he abo e ep esen a ion o mula. As a by-p oduc o his, we de i e ou second main esul ha he weak ype condi ion | − B|L1,∞(B,ν)≤C (B)ZB g dµ also implies he same poin wise es ima es. We no e ha , simila o wha was i s shown in [LW1] and hen in [LW2], he in eg als on he igh hand side is on he same ball B0, a he han on he enla ged ball (see [FLW2], [FW]). We make emphasis on he ac ha he only assump ion we make on he measu e µis he doubling p ope y. Indeed, i has ecen ly been shown in [LW2] ha he e is no need o equi e addi ional assump ions o e e se doubling o o de 1+² o 1 (see [FLW2], [FW]). Howe e , we need o add he second e m on he igh hand side, which is no ha m ul a all as a as he Poinca ´e ype es ima es conce ned. I we also assume ha he measu e µis o e e se doubling o de 1, hen his second e m can be d opped (see also [FW] and [LW2]). We men ion ha he au ho s in [HK2] de i ed independen ly om [LW2] a o mula wi hou he second e m wi hou he assump ion ha µis doubling, bu wi h B0 eplaced by 1 2B0. I seems ha he passage om 1 2B0 o B0would also esul in he second e m in he o mula. As applica ions, we p o ide weake , bu equi alen , de ini ions o Sobole spaces o i s o de in me ic spaces han hose de ined in [H], and u he exploi ed in [FLW2], [FHK] and [LW2]. The implica ions o L1→LqPoinca ´e inequali ies o high o de o ep esen a ion o mulas also hold and imp o e hose in [LW2]. These also p o ide us wi h weake , and also equi alen , de ini ions o high o de Sobole spaces in me ic spaces de ined in [LLW1]. 4 G. LU AND C. P´ EREZ The me hods used in his pape a e ex ensions o se e al echniques adap ed om [FLW2], [FW], [LW1], [LW2] and [LLW1]. In pa icula , we will use simila ideas om [LW2]. Howe e , ou case is conce ned wi h he si ua ion q < 1, and he e a e some sub le ies we ha e o o e come. Some inequali ies which hold o q≥1 ail o be ue o q < 1. Thus, we ha e o p oceed wi h cau ion. We ema k in passing ha he e has been ex ensi e esea ch o p o ing Lp→Lq Poinca ´e inequali ies, i a ce ain ype o Lp→LpPoinca ´e inequali y is al eady known o exis in he gi en se ing, see [SC], [HK1-2], [BM], [MSC], [GN], [BCSC], [FPW], [MP1-2], [OP]. This is he so-called sel -imp o ing p ope y, which can be used o p o e Poinca ´e inequali ies wi hou using ep esen a ion o mula. Thus, by combining wi h Je ison’s esul o Poinca ´e inequali ies wi h q=p o H¨o mande ec o ields [J], his a gumen will ecap u e he sha p Poinca ´e inequali ies o H¨o mande ec o ields i s p o ed in [L2] ( o p > 1) and [FLW1] ( o p= 1) by using ep esen a ion o mulas. Di ec p oo s o ep esen a ion o mulas o H¨o mande ec o ields o G ushin ec o ields ha e been gi en in [F], [FL], [FSe], [L1], [FLW1], [FGW], [CDG], [LM]. Fu he mo e, i is shown in he pape s [FPW], [MP1-2] and [OP] ha he Sobole - Poinca ´e inequali ies a e special cases o a mo e gene al heo y ha includes, o in- s ance, he classical heo em o John-Ni enbe g as well as he T udinge inequali y. The idea he e is o eplace he exp ession on he igh hand side o (1.2) by a mo e gen- e al “ unc ional” a(B) and o use he Calde on-Zygmund heo y, unde a ce ain mild geome ic condi ion on a(see [P] o a su ey). We men ion ha he sel -imp o ing p ope y by assuming he ini ial inequali y µZB | − B|qdν¶1/q ≤a(B) o some 0 < q < 1 and some quan i y a(B) o hold has also been es ablished ecen ly in [FLPW]. To make ou pape sel -con ained, and o he sake o cla i y o ou p esen a ion, we ha e decided o ea he case o i s o de Poinca ´e inequali ies sepa a ely om he ones o high o de . The plan o he pape is as ollows. In sec ion 2, we p o e ou esul s o he i s o de in gene al me ic spaces. Sec ion 3 con ains new de ini ions o Sobole spaces o i s o de in me ic spaces. Sec ion 4 deals wi h he implica ion o high o de Poinca ´e inequali ies o ep esen a ion o mulas in me ic spaces and p o ides wi h new de ini ions o Sobole spaces o any high o de in me ic spaces. Acknowledgemen This wo k is an ou g ow h o join wo k wi h B uno F anchi and Richa d Wheeden [FLW], [FW], [LW1], [LW2] and [LLW1]. We would like o acknowledge he impo an con ibu ions hey ha e made in his di ec ion. §2 Rep esen a ion o mulas o i s o de in me ic spaces We begin wi h he de ini ion o “weak Boman chain domain” de ined in [LW2]. Boman chain domains in Euclidean spaces we e in oduced by Boman in his unpublished wo k [Bom] and used o p o e Poinca ´e inequali ies on such domains (see [Boj], [Ch], [IN]). Such a no ion in me ic space seems o be i s used in [FGW] and [L2] by sligh ly modi ying he de ini ion in Euclidean spaces. L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 5 De ini ion 2.1 [LW2]. A domain (i.e., an open connec ed se ) Ωin Sis said o sa is y he Boman chain condi ion o ype σ, M, o o be a membe o F(σ, M), i he e exis cons an s σ > 1,M > 0, and a amily Fo me ic balls B⊂Ωsuch ha (1) Ω = SB∈F B (2) PB∈F χσB(x)≤M χΩ(x) o all x∈ S (3) The e is a “cen al ball” B0∈ F such ha o each ball B∈ F, he e is a posi i e in ege k=k(B)and a chain o balls {Bj}k j=0 o which Bk=Band each BjTBj+1 con ains a ball Djwi h BjSBj+1 ⊂MDj. (4) B⊂MBj o all j= 0, . . . , k(B) I we eplace he hypo hesis ha σ > 1by σ= 1, we say ha Ωsa is ies he weak Boman chain condi ion. We do no know i his weake de ini ion can ac ually be equi alen o he “Boman chain domain”, whe e τhas o be aken bigge han 1. I will also be in e es ing o know i he class o weake Boman chain domains is s ic ly la ge han he Boman chain domains. We men ion ha Boman domain is equi alen o John domain as shown independen ly in [BKL] and [GN]. We now s a e he ollowing ou hypo heses ha a e modi ica ions o hose gi en in [LW1] and [LW2]. The c ucial di e ence is ha we ha e eplaced (H1) he e by ou L1→LqPoinca ´e inequali y o some 0 < q < 1, a he han he L1→L1inequali y. We no e ha no all ou hypo heses a e needed in e e y heo em. As always, (S, ρ) is a me ic space. Le µand νbe doubling measu es wi h espec o me ic balls, and le Ω be a domain in S. (H1) is a unc ion sa is ying L1 o LqPoinca ´e inequali y o some 0 < q < 1, i.e., µZB | − B|qdν¶1/q ≤c (B)ZB gdµ o me ic balls B⊂Ω. (H2) The measu e µin (H1) sa is ies a e e se doubling condi ion o o de 1, i.e., he e is a cons an C > 0 such ha i Band ˜ Ba e balls wi h cen e s in Ω and wi h B⊂˜ B, hen µ(˜ B)≥Cà (˜ B) (B)!µ(B). (H3) (S, ρ) has he segmen (o geodesic) p ope y ha o each pai o poin s x, y ∈ S, he e is a con inuous cu e γconnec ing xand ysuch ha ρ(γ( ), γ(s)) = | −s|. (H4) Ω is a weak Boman chain domain. The main esul s o his sec ion a e imp o emen s o hose in [LW2] whe e L1→L1 Poinca ´e inequali ies ha e o be assumed. Rema k. Since weak L1implies locally s ong L1 o 0 < q < 1 as men ioned in he in oduc ion, hus ou heo ems below s ill emain o be ue i we eplace (H1) abo e by 6 G. LU AND C. P´ EREZ (WH1) is a unc ion sa is ying weak L1 o L1Poinca ´e inequali y, i.e., | − B|L1,∞(B,ν)≤C (B)ZB g dµ o me ic balls B⊂Ω. F om Kolmogo o ’s inequali y (1.1) we see ha (WH1) implies (H1) o all 0 < q < 1. Theo em 2.2. Le ν, µ be doubling measu es on a me ic space (S, ρ). Le B0be a ball and suppose ha (H1) and (H3) hold wi h Ω = B0and B=RB (y)dν(y). Then o ν-a.e. x∈B0, | (x)− B0| ≤CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y) + C (B0) µ(B0)ZB0 g(y)dµ(y), whe e Cdepends only on ν, µ and he cons an s in (H1). I in addi ion we impose he e e se doubling condi ion (H2) in Theo em 2.2, hen we ha e Theo em 2.3. Le ν, µ be doubling measu es on a me ic space (S, ρ). Le B0be a ball and suppose ha (H1), (H2) and (H3) hold wi h Ω = B0. Then o ν-a.e. x∈B0, | (x)− B0| ≤ CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y), whe e Cdepends only on ν, µ and he cons an s in (H1), (H2). The nex heo em is a gene aliza ion o Theo em 2.3 o any weak Boman chain domain Ω. Theo em 2.4. Suppose ha νand µa e doubling measu es on a me ic space (S, ρ) and ha hypo heses (H1)–(H4) hold o a domain Ω⊂ S. Then o ν-a.e. x∈Ω, | (x)− B0| ≤ CZΩ g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y), whe e B0is he cen al ball in Ω, , g, ν and µin (H1), and Cdepends only on ν, µ and he cons an s in (H1), (H2) and (H4). As is well-known, unde he segmen hypo hesis (H3), any me ic ball is a Boman chain domain (see [FGW], [L2]), and hus Theo em 2.3 is a special case o Theo em 2.4 . The p oo o Theo em 2.2 elies on he cons uc ion o he ollowing chain o me ic balls gi en in [LW2], assuming he segmen hypo hesis (H3). A simila cons uc ion was gi en in [FW], bu he ollowing one enables us o selec all balls in he chain lying inside en i ely he gi en ball B0. The chain o balls will allow us o p o e he ep esen a ion o mulas on he same ball on bo h sides di ec ly (see [LW2]), a he han using he o mula on he enla ged ball o ge he co esponding one on he same ball (see [LW1]). A somewha di e en chain o ini e leng h is gi en independen ly in [HK2]. L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 7 Theo em 2.5 [LW2]. Le (S, ρ)be a me ic space in which he segmen p ope y (H3) holds. Le B0be a ball in S. Gi en x∈B0, he e exis s a chain {Bk}k≥1o balls wi h he ollowing p ope ies: (1) Bk⊂B0and ρ(Bk, x)→0as k→ ∞. (2) (B1)≈ (B0)and (Bk)→0as k→ ∞. (3) I y∈Bk, hen ρ(y, x)≈ (Bk). (4) BkTBk−1con ains a ball Skwi h (Sk)≈ (Bk)≈ (Bk+1)≈2−k (B0). (5) I j < k, hen Bk⊂cBj. (6) {Bk}k≥1has bounded o e laps, i.e., PkχBk(y)≤c o all y. The cons an s o equi alence in (2), (3) and (4) and he cons an s cin (5) and (6) a e independen o x, k, j and B0, bu he chain {Bk}depends on x. The ollowing ema k is in o de . The a gumen gi en he e is simila o he p oo o Theo em A in [LW2]. Howe e , since ou case is o q < 1 and hen he Minkowski’s inequali y ails. Thus, ou si ua ion becomes mo e delica e han he case o q= 1. In pa icula , we will use he inequali y (2.6) µZE ( +g)qdν¶1/q ≤2q"µZE q¶1/q +µZE gq¶1/q#. Howe e , his inequali y does no hold when we ha e in ini ely many e ms in he in eg and unlike he case o q≥1, namely, we do no ha e ÃZEÃ∞ X i=1 i!q dν!1/q ≤C(q) ∞ X i=1 µZE q idν¶1/q . The e o e we ha e o p oceed wi h cau ion, see he es ima e o I2below. P oo o Theo em 2.2. We will use Theo em 2.5 o p o e Theo em 2.2. Le B0be a ball in Sand suppose ha (H1) and he segmen p ope y (H3) hold o B0. Gi en x∈B0, le {Bk}k≥1be a sequence o balls wi h he p ope ies gua an eed by Theo em 2.5. Then (2.7) | (x)− B0| ≤ | (x)− B1|+| B1− B0|. Fo he second e m on he igh in (2.7), we ge o ν−a.e. x∈B0 ha | B1− B0|=µZB1 | B1− B0|qdν¶1/q ≤CµZB1 | (y)− B1|qdν(y)¶1/q +CµZB1 | (y)− B0|qdν(y)¶1/q ≤CµZB1 | (y)− B1|qdν(y)¶1/q +CµZB0 | (y)− B0|qdν(y)¶1/q since ν(B1)≈ν(B0) and νis doubling ≤C (B1) µ(B1)ZB1 g dµ +C (B0) µ(B0)ZB0 g dµ by he Poinca ´e inequali y (H1) ≤C (B0) µ(B0)ZB0 g dµ 8 G. LU AND C. P´ EREZ since B1⊂B0, (B1)≈ (B0) and µ(B1)≈µ(B0). Assuming as we may ha xis a Lebesgue poin o bo h | − B1|qand gwi h espec o νand using p ope ies (1)–(3) om Theo em 2.5 and he inequali y (2.6), we ha e o he i s e m on he igh in (2.7) ha | (x)− B1|= lim k→∞ µZBk | (y)− B1|qdν(y)¶1/q ≤2qlim sup k→∞ µZBk | (y)− Bk|qdν(y)¶1/q + 2qlim sup k→∞ µZBk | Bk− B1|qdν(y)¶1/q =I1+I2, whe e I1and I2a e de ined by he las equali y. I is easy o show ha I1= 0 o e e y Lebesgue poin xo g. This can be seen by he Poinca ´e inequali y (H1): I1= lim sup k→∞ µZBk | (y)− Bk|qdν(y)¶1/q ≤Clim sup k→∞ (Bk) µ(Bk)ZBk g(y)dµ(y) = 0 ·g(x) = 0. We now es ima e I2. By obse ing ha Bj+1 − Bjis a cons an unc ion, we ha e I2= lim sup k→∞ | Bk− B1| ≤lim sup k→∞ k−1 X j=1 | Bj+1 − Bj| = lim sup k→∞ k−1 X j=1 ÃZSj | Bj+1 − Bj|qdν!1/q ≤2q ∞ X j=1 ÃZSj | Bj+1 − |qdν!1/q + 2q ∞ X j=1 ÃZSj | Bj− |qdν!1/q ≤2q ∞ X j=1 ÃZBj+1 | Bj+1 − |qdν!1/q + 2q ∞ X j=1 ÃZBj | Bj− |qdν!1/q since Sj⊂Bj∩Bj+1 and ν(Sj)≈ν(Bj)≈ν(Bj+1) by Theo em 2.5. Combining es ima es and applying (H1) o he e ms o each o he las wo sums, we ob ain I2≤C ∞ X j=1 (Bj)ZBj g(y)dµ(y). L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 9 Now, as a guing in [LW2], i y∈Bj, hen (Bj) µ(Bj)≈ρ(x, y) µ(B(y, ρ(x, y))) ≈ρ(x, y) µ(B(x, ρ(x, y))) by pa (3) o Theo em 2.5 and he ac ha µis a doubling measu e. Thus we ob ain I2≤C ∞ X j=1 ZBj g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y) ≤CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y) by p ope ies (6) and (1) o Theo em 2.5. This comple es he p oo o Theo em 2.2 by combining es ima es o I1and I2. P oo o Theo em 2.3. By obse ing ha i x, y ∈B0, hen ρ(x, y)≤2 (B0) and consequen ly by (H2), we ha e (B0) µ(B0)≤Cρ(x, y) µ(B(x, ρ(x, y))) i x, y ∈B0. Thus, he second e m on he igh in he conclusion o Theo em 2.2 is bounded by he i s e m. P oo o Theo em 2.4. Le x∈Ω. By he de ini ion o weak Boman chain domain, we may selec B∗wi h x∈B∗and a chain {Bj}k j=0 connec ing B∗=Bk o he cen al ball B0. We ha e (2.8) | (x)− B0| ≤ | (x)− B∗|+| B∗− B0|. Fo he i s e m on he igh side o (2.8), we ha e by Theo em B ha | (x)− B∗| ≤ CZB∗ g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y). This holds o ν−a.e. poin o B∗, and we may assume i holds o ou ixed xby ini ially excluding om Ω he se o measu e ze o o med by aking he union o he excep ional se s o measu e ze o in each Boman ball. Since B∗⊂Ω, we ob ain he desi ed es ima e | (x)− B∗| ≤ CZΩ g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y). Thus we only need o es ima e | B∗− B0|.By using he chain {Bj}connec ing B0and Bk=B∗, we ha e | B∗− B0| ≤ k X j=1 | Bj− Bj−1|. 16 G. LU AND C. P´ EREZ by pa (3) o Theo em 2.5 and he ac ha µis a doubling measu e. Thus, we ge I2≤C ∞ X j=1 ZBj g(y)ρ(x, y)m µ(B(x, ρ(x, y))) dµ(y) ≤CZB0 g(y)ρ(x, y)m µ(B(x, ρ(x, y))) dµ(y) by p ope ies (6) and (1) o Theo em 2.5. The p oo o Theo em 2.2 now is comple e. Rema k. We omi he p oo o Theo em 4.3 since is simila o ha o Theo em 2.3 by using (A2) ins ead o (H2). P oo o Theo em 4.4. Le x∈Ω. By he de ini ion o weak Boman chain domain, we may selec B∗wi h x∈B∗and a chain {Bj}k j=0 connec ing B∗=Bk o he cen al ball B0. We ha e (4.6) | (x)−Pm(B0, )(x)| ≤| (x)−Pm(B∗, )(x)| +|Pm(B∗, )(x)−Pm(B0, )(x)|. Fo he i s e m on he igh side o (4.6), we ha e by Theo em B ha | (x)−Pm(B∗, )(x)| ≤ CZB∗ g(y)ρ(x, y)m µ(B(x, ρ(x, y))) dµ(y). This holds o ν−a.e. poin o B∗, and we may assume i holds o ou ixed xby ini ially excluding om Ω he se o measu e ze o o med by aking he union o he excep ional se s o measu e ze o in each Boman ball. Since B∗⊂Ω, we ob ain | (x)−Pm(B∗, )(x)| ≤ CZΩ g(y)ρ(x, y)m µ(B(x, ρ(x, y))) dµ(y). We now es ima e |Pm(B∗, )(x)−Pm(B0, )(x)|.By using he chain {Bj}connec ing B0and Bk=B∗and no icing ha B∗⊂MBjand x∈B∗, we ha e |Pm(B∗, )(x)−Pm(B0, )(x)| ≤ k X j=1 |Pm(Bj, )(x)−Pm(Bj−1, )(x)| ≤ k X j=1 ||Pm(Bj, )−Pm(Bj−1, )||L∞ ν(B∗) ≤ k X j=1 ||Pm(Bj, )−Pm(Bj−1, )||L∞ ν(MBj). L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 17 I Djis a ball wi h Dj⊂Bj∩Bj−1⊂MBjand (Dj)≈ (Bj)≈ (Bj−1), hen by (P1) and (P2), he las sum is majo ized by C k X j=1 ||Pm(Bj, )−Pm(Bj−1, )||L∞ ν(Dj) ≤C k X j=1 ÃZDj |Pm(Bj, )−Pm(Bj−1, )|qdν!1/q , which by he inequali y (2.6) and doubling is bounded by C2q k X j=1 ÃZDj |Pm(Bj, )(y)− (y)|qdν(y)!1/q +C2q k X j=1 ÃZDj |Pm(Bj−1, )(y)− (y)|qdν(y)!1/q ≤C2q k X j=1 ÃZBj |Pm(Bj, )(y)− (y)|qdν(y)!1/q +C2q k X j=1 ÃZBj−1 |Pm(Bj−1, )(y)− (y)|qdν(y)!1/q ≤C k X j=0 Ã1 ν(Bj)ZBj |Pm(Bj, )(y)− (y)|qdν(y)!1/q By Poinca ´e’s inequali y, he las exp ession abo e is a mos C k X j=0 (Bj)m µ(Bj)ZBj g(y)dµ(y) =CZΩ   k X j=0 (Bj)m µ(Bj)χBj(y)   g(y)dµ(y). As shown in [LW2], he sum abo e in cu ly b acke s is bounded by a ixed mul iple o ρ(x, y)m/µ(B(x, ρ(x, y))) o each y∈Ω. Thus, we ha e comple ed he p oo . By using Theo em (4.2), we will be able o weaken he hypo heses in de ining high o de Sobole spaces in me ic spaces gi en in [LLW1] (see also [LLW2]). De ini ion 4.7. Gi en a posi i e in ege mand 1< p < ∞, we de ine he Sobole class Am,p(Ω) o be he se o unc ions ∈Lp(Ω) so ha o each k= 1,··· , m, he e exis kwi h 1≤ k< p and qkwi h 0< qk<1, unc ions gk(x)wi h 0≤gk∈Lp(Ω), and polynomials Pk(B, )wi h (4.8) µZB | (x)−Pk(B, )(x)|qkdµ(x)¶1 qk≤ (B)kµZB g k k(x)dµ(x)¶1 k 18 G. LU AND C. P´ EREZ o e e y ball B⊂Ω. The polynomials Pk(B, )a e assumed o belong o a linea class which sa is ies (P1) and (P2) wi h cons an s depending only on k, γ, µ. I ∈Am,p(Ω), we de ine || ||Am,p(Ω) =|| ||Lp(Ω) + in {gk} m X k=1 ||gk||Lp(Ω), whe e he in imum is aken o e all sequences such ha (4.8) holds o o k= 1, . . . , m. I is easy o see ha Am,p(Ω) is a linea space. The eason we can impose he L kno m a he han he L1no m is because we can show ha de ini ion (4.7) is equi alen o he ollowing de ini ions (4.9) and (4.11) gi en in [LLW1]. The p oo o equi alence ollows om ou Theo em (4.2) in his sec ion by combining he p oo s gi en in [LLW1]. We shall omi he de ails he e. De ini ion 4.9 [LLW1]. Gi en a posi i e in ege mand 1< p < ∞, we de ine he Sobole class Bm,p(Ω) o be he se o unc ions ∈Lp(Ω) so ha o each k= 1,· · · , m, he e exis unc ions 0≤gk∈Lp(Ω) and polynomials Pk(B, )such ha (4.10) | (x)−Pk(B, )(x)| ≤ ZB ρ(x, y)kgk(y) µ(B(x, ρ(x, y)))dµ(y) + (B)kZB gk(y)dµ(y) o µ−a.e. x∈B o e e y ball B⊂Ω. The polynomials Pk(B, )a e assumed o belong o a linea class which sa is ies (P1) and (P2) wi h cons an s depending only on k, γ, µ. I ∈Bm,p(Ω), we de ine || ||Bm,p(Ω) =|| ||Lp(Ω) + in {gk} m X k=1 ||gk||Lp(Ω), whe e he in imum is aken o e all sequences such ha (4.10) holds o o k= 1,··· , m. The class Bm,p(Ω) is clea ly a Banach space wi h no m || · ||Bm,p(Ω). De ini ion 4.11 [LLW1]. Gi en a posi i e in ege mand 1< p < ∞, we de ine he Sobole class Cm,p(Ω) o be he se o unc ions ∈Lp(Ω) so ha o each k= 1,· · · , m he e exis unc ions 0≤gk∈Lp(Ω) and polynomials Pk(B, )such ha (4.12) | (x)−Pk(B, )(x)| ≤ (B)kgk(x) o µ−a.e. x∈B o e e y me ic ball B⊂Ω. The polynomials Pk(B, )a e assumed o belong o a linea class which sa is ies (P1) and (P2) wi h cons an s depending only on k, γ, µ. I ∈Cm,p(Ω), le || ||Cm,p(Ω) =|| ||Lp(Ω) + in {gk} m X k=1 ||gk||Lp(Ω). The class Cm,p(Ω) is a Banach space wi h no m || · ||Cm,p . To show ha de ini ions (4.7), (4.9) and (4.11) a e all equi alen , we will need he ollowing heo em. L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 19 Theo em 4.13. Le 1≤ < ∞,mbe a posi i e in ege , B0⊂Ωbe a ixed ball, and suppose ha he segmen p ope y (H3) holds o B0. Le be a locally in eg able unc ion in Ω o which he e exis a unc ion 0≤g∈L (Ω) and polynomials Pm(B, ), and 0< q < 1such ha he Poinca ´e inequali y µZB | (x)−Pm(B, )(x)|qdµ(x)¶1/q ≤c (B)mµZB |g(x)| dµ(x)¶1/ holds o e e y ball B⊂Ω. The polynomials Pm(B, )a e assumed o belong o a linea class which sa is ies (P1) and (P2) wi h cons an s depending only on m, γ, µ. Then o µ−a.e. x∈B0, | (x)−Pm(B0, )(x)| ≤ C (B0)mM(g )(x)1/ wi h Cindependen o x. P oo o Theo em 4.13. Le x∈B0. We will use he chain o subballs {Bj}o B0 cons uc ed om Theo em 2.5. The chain depends on x. We may assume wi hou loss o gene ali y ha xis a Lebesgue poin o bo h | −Pm(B0, )|qand |g| wi h espec o µ. Then by p ope ies (1), (2) and (3) o he chain, | (x)−Pm(B0, )(x)|= lim j→∞ ÃZBj | (y)−Pm(B0, )(y)|qdµ(y)!1/q ≤lim sup j→∞ 2qÃZBj | (y)−Pm(Bj, )(y)|qdµ(y)!1/q + lim sup j→∞ 2qÃZBj |Pm(Bj, )(y)−Pm(B0, )(y)|qdµ(y)!1/q =I1+I2. By he Poinca ´e inequali y, o e e y Lebesgue poin xo |g| I1≤clim sup j→∞ (Bj)mÃZBj |g(y)| dµ(y)!1/ = 0 · |g(x)|= 0 by p ope ies (1), (2) and (3) o he chain. 20 G. LU AND C. P´ EREZ We ha e o I2 I2≤lim sup j→∞ ||Pm(Bj, )(y)−Pm(B0, )(y)||L∞ µ(Bj) ≤lim sup j→∞ j−1 X `=0 ||Pm(B`+1, )−Pm(B`, )||L∞ µ(Bj) ≤lim sup j→∞ j−1 X `=0 ||Pm(B`+1, )−Pm(B`, )||L∞ µ(cB`)by (5) ≤C ∞ X `=0 ||Pm(B`+1, )−Pm(B`, )||L∞ µ(S`)by (4) and (P2) ≤C ∞ X `=0 µZSl |Pm(B`+1, )−Pm(B`, )|qdµ¶1/q by (P1) ≤C ∞ X `=0 2qµZB` |Pm(B`, )(y)− (y)|qdµ(y)¶1/q +C ∞ X `=0 2qÃZB`+1 |Pm(B`+1, )(y)− (y)|qdµ(y)!1/q by (4) ≤C ∞ X `=0 (B`)mµZB` |g(y)|qdµ(y)¶1/q ≤C ∞ X `=0 (B`)mM(|g| )(x)1/ by (3) =C ∞ X `=0 2−`m (B0)mM(|g| )(x)1/ by (2) ≤C (B0)mM(|g| )(x)1/ . This comple es he p oo o Theo em 4.13. By using Theo em 4.13, and a guing simila ly as in [LLW1], we will be able o show he equi alence o de ini ions (4.7), (4.9) and (4.11). We shall omi he de ails he e and e e he eade o [LLW1]. 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