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

Acta mathematica sinica, English Series, Series 18 (2002) 1, 1-20. L1→LqPOINCAR´ E INEQUALITIES FOR 0< q < 1IMPLY REPRESENTATION FORMULAS Guozhen Lu(∗) Department of Mathematics Wayne State University Detroit, MI 48202, USA E-mail: [email protected]ayne.edu Carlos P´ erez(∗)(∗∗) Departamento de Matem´aticas Universidad Aut´onoma de Madrid 28049 Madrid, Spain E-mail: carlos.p[email protected] Dedicated to Dick Wheeden on the occasion of his 60th birthday with appreciation and admiration 1991 Mathematics Subject Classification. 46E35, 41A10, 22E25. Key words and phrases. Sobolev spaces, representation formulas, high order derivatives, vector fields, metric spaces, polynomials, doubling measures, Poincar´e inequalities. (*) The first author was supported partly by the U.S. National Science Foundation Grant Nos. DMS96-22996 and DMS99-70352. The second author was supported partly by DGICYT grant PB940192, Spain. Both authors were supported partly by NATO collaborative research grant 972144. (**) The main part of this paper was completed during the second author’s visit at Wright State University, Ohio in June, 1999. He wishes to thank the Department of Mathematics and Statistics at Wright State University for its hospitality and financial support. Typeset by A MS-T EX 1 2 G. LU AND C. P´ EREZ Abstract. Given two doubling measures µand νin a metric space (S, ρ) of homogeneous type and let B0⊂ S be a given ball. It has been a well-known result by now (see [FLW], [FW], [LW1], [LW2]) that the validity of an L1→L1Poincar´e inequality of the following form: ZB |f−fB|dν ≤cr(B)ZB gdµ for all metric balls B⊂B0⊂ S implies a variant of representation formula of fractional intergral type: for ν-a.e. x∈B0, |f(x)−fB0| ≤ CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y) + Cr(B0) µ(B0)ZB0 g(y)dµ(y). One of the main results of this paper shows that an L1to LqPoincar´e inequality for some 0< q < 1, i.e., „ZB |f−fB|qdν«1/q ≤cr(B)ZB gdµ for all metric balls B⊂B0will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak type condition sup λ>0 λν ({x∈B:|f(x)−fB|> λ}) ν(B)≤Cr(B)ZB gdµ also implies the same formula. Analogous theorems related to high order Poincar´e inequalities and Sobolev spaces in metric spaces are also proved. §1. Introduction It is known that L1→L1Poincar´e inequalities are equivalent to the fractional integral estimates in general metric spaces of homogeneous type (see [FLW2], [FW], [LW1-2]). A natural question thus arises: Is the L1→L1Poincar´e inequality the least we need to start in order to derive such representation formulas? In this paper, we study this issue and weaken the hypothesis that an L1→L1Poincar´e inequality has to hold to obtain any kind of representation formulas of fractional type. More precisely, we will show that an L1→LqPoincar´e inequality for some 0 < q < 1 will suffice to derive the pointwise fractional estimates. On the other hand it is well known that the following Kolmogorovs inequality holds for 0 < q < 1, any nonnegative function gand arbitrary measurable set Ewith finite measure (1.1) µ1 µ(E)ZE g(x)qdµ¶1/q ≤cq|g|L1,∞(E,µ). See [GCRdF] p. 485 for instance. We will be using the following notation for the local average Marcinkiewicz quasi-norm |g|L1,∞(E,µ)= sup λ>0 λµ({x∈E:|f(x)|> λ}) µ(E). Hence as an interesting corollary of our main result, we prove that weak type L1,∞→L1 Poincar´e inequality is sufficient to imply fractional representation formulas. L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 3 More precisely, given two doubling measures µand νin a metric space (S, ρ) of homogeneous type and let B0⊂ S be a given ball. It has been a well-known result by now (see [FLW2], [FW], [LW1], [LW2]) that the validity of an L1→L1Poincar´e inequality of the following form: (1.2) ZB |f−fB|dν ≤cr(B)ZB g dµ for all metric balls B⊂B0⊂ S implies a variant of representation formula of fractional integral type: for ν-a.e. x∈B0, |f(x)−fB0| ≤ CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y) + Cr(B0) µ(B0)ZB0 g(y)dµ(y). As usual we use the following notation for the average of fover a ball B,fB= 1 µ(B)RBf dµ Our first main result of this paper demonstrates that an L1to LqPoincar´e inequality for some 0 < q < 1, i.e., µZB |f−fB|qdν¶1/q ≤cr(B)ZB g dµ for all metric balls B⊂B0will suffice to imply the above representation formula. As a by-product of this, we derive our second main result that the weak type condition |f−fB|L1,∞(B,ν)≤C r(B)ZB g dµ also implies the same pointwise estimates. We note that, similar to what was first shown in [LW1] and then in [LW2], the integrals on the right hand side is on the same ball B0, rather than on the enlarged ball (see [FLW2], [FW]). We make emphasis on the fact that the only assumption we make on the measure µis the doubling property. Indeed, it has recently been shown in [LW2] that there is no need to require additional assumptions of reverse doubling of order 1+² or 1 (see [FLW2], [FW]). However, we need to add the second term on the right hand side, which is not harmful at all as far as the Poincar´e type estimates concerned. If we also assume that the measure µis of reverse doubling order 1, then this second term can be dropped (see also [FW] and [LW2]). We mention that the authors in [HK2] derived independently from [LW2] a formula without the second term without the assumption that µis doubling, but with fB0replaced by f1 2B0. It seems that the passage from f1 2B0 to fB0would also result in the second term in the formula. As applications, we provide weaker, but equivalent, definitions of Sobolev spaces of first order in metric spaces than those defined in [H], and further exploited in [FLW2], [FHK] and [LW2]. The implications of L1→LqPoincar´e inequalities of high order to representation formulas also hold and improve those in [LW2]. These also provide us with weaker, and also equivalent, definitions of high order Sobolev spaces in metric spaces defined in [LLW1]. 4 G. LU AND C. P´ EREZ The methods used in this paper are extensions of several techniques adapted from [FLW2], [FW], [LW1], [LW2] and [LLW1]. In particular, we will use similar ideas from [LW2]. However, our case is concerned with the situation q < 1, and there are some subtleties we have to overcome. Some inequalities which hold for q≥1 fail to be true for q < 1. Thus, we have to proceed with caution. We remark in passing that there has been extensive research of proving Lp→Lq Poincar´e inequalities, if a certain type of Lp→LpPoincar´e inequality is already known to exist in the given setting, see [SC], [HK1-2], [BM], [MSC], [GN], [BCSC], [FPW], [MP1-2], [OP]. This is the so-called self-improving property, which can be used to prove Poincar´e inequalities without using representation formula. Thus, by combining with Jerison’s result for Poincar´e inequalities with q=pfor H¨ormander vector fields [J], this argument will recapture the sharp Poincar´e inequalities for H¨ormander vector fields first proved in [L2] (for p > 1) and [FLW1] (for p= 1) by using representation formulas. Direct proofs of representation formulas for H¨ormander vector fields or Grushin vector fields have been given in [F], [FL], [FSe], [L1], [FLW1], [FGW], [CDG], [LM]. Furthermore, it is shown in the papers [FPW], [MP1-2] and [OP] that the SobolevPoincar´e inequalities are special cases of a more general theory that includes, for instance, the classical theorem of John-Nirenberg as well as the Trudinger inequality. The idea there is to replace the expression on the right hand side of (1.2) by a more general “functional” a(B) and to use the Calderon-Zygmund theory, under a certain mild geometric condition on a(see [P] for a survey). We mention that the self-improving property by assuming the initial inequality µZB |f−fB|qdν¶1/q ≤a(B) for some 0 < q < 1 and some quantity a(B) to hold has also been established recently in [FLPW]. To make our paper self-contained, and for the sake of clarity of our presentation, we have decided to treat the case of first order Poincar´e inequalities separately from the ones of high order. The plan of the paper is as follows. In section 2, we prove our results for the first order in general metric spaces. Section 3 contains new definitions of Sobolev spaces of first order in metric spaces. Section 4 deals with the implication of high order Poincar´e inequalities to representation formulas in metric spaces and provides with new definitions of Sobolev spaces of any high order in metric spaces. Acknowledgement This work is an outgrowth of joint work with Bruno Franchi and Richard Wheeden [FLW], [FW], [LW1], [LW2] and [LLW1]. We would like to acknowledge the important contributions they have made in this direction. §2 Representation formulas of first order in metric spaces We begin with the definition of “weak Boman chain domain” defined in [LW2]. Boman chain domains in Euclidean spaces were introduced by Boman in his unpublished work [Bom] and used to prove Poincar´e inequalities on such domains (see [Boj], [Ch], [IN]). Such a notion in metric space seems to be first used in [FGW] and [L2] by slightly modifying the definition in Euclidean spaces. L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 5 Definition 2.1 [LW2]. A domain (i.e., an open connected set) Ωin Sis said to satisfy the Boman chain condition of type σ, M, or to be a member of F(σ, M), if there exist constants σ > 1,M > 0, and a family Fof metric balls B⊂Ωsuch that (1) Ω = SB∈F B (2) PB∈F χσB(x)≤M χΩ(x)for all x∈ S (3) There is a “central ball” B0∈ F such that for each ball B∈ F, there is a positive integer k=k(B)and a chain of balls {Bj}k j=0 for which Bk=Band each BjTBj+1 contains a ball Djwith BjSBj+1 ⊂MDj. (4) B⊂MBjfor all j= 0, . . . , k(B) If we replace the hypothesis that σ > 1by σ= 1, we say that Ωsatisfies the weak Boman chain condition. We do not know if this weaker definition can actually be equivalent to the “Boman chain domain”, where τhas to be taken bigger than 1. It will also be interesting to know if the class of weaker Boman chain domains is strictly larger than the Boman chain domains. We mention that Boman domain is equivalent to John domain as shown independently in [BKL] and [GN]. We now state the following four hypotheses that are modifications of those given in [LW1] and [LW2]. The crucial difference is that we have replaced (H1) there by our L1→LqPoincar´e inequality for some 0 < q < 1, rather than the L1→L1inequality. We note that not all four hypotheses are needed in every theorem. As always, (S, ρ) is a metric space. Let µand νbe doubling measures with respect to metric balls, and let Ω be a domain in S. (H1) fis a function satisfying L1to LqPoincar´e inequality for some 0 < q < 1, i.e., µZB |f−fB|qdν¶1/q ≤cr(B)ZB gdµ for metric balls B⊂Ω. (H2) The measure µin (H1) satisfies a reverse doubling condition of order 1, i.e., there is a constant C > 0 such that if Band ˜ Bare balls with centers in Ω and with B⊂˜ B, then µ(˜ B)≥CÃr(˜ B) r(B)!µ(B). (H3) (S, ρ) has the segment (or geodesic) property that for each pair of points x, y ∈ S, there is a continuous curve γconnecting xand ysuch that ρ(γ(t), γ(s)) = |t−s|. (H4) Ω is a weak Boman chain domain. The main results of this section are improvements of those in [LW2] where L1→L1 Poincar´e inequalities have to be assumed. Remark. Since weak L1implies locally strong L1for 0 < q < 1 as mentioned in the introduction, thus our theorems below still remain to be true if we replace (H1) above by 6 G. LU AND C. P´ EREZ (WH1) fis a function satisfying weak L1to L1Poincar´e inequality, i.e., |f−fB|L1,∞(B,ν)≤C r(B)ZB g dµ for metric balls B⊂Ω. From Kolmogorov’s inequality (1.1) we see that (WH1) implies (H1) for all 0 < q < 1. Theorem 2.2. Let ν, µ be doubling measures on a metric space (S, ρ). Let B0be a ball and suppose that (H1) and (H3) hold with Ω = B0and fB=RBf(y)dν(y). Then for ν-a.e. x∈B0, |f(x)−fB0| ≤CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y) + Cr(B0) µ(B0)ZB0 g(y)dµ(y), where Cdepends only on ν, µ and the constants in (H1). If in addition we impose the reverse doubling condition (H2) in Theorem 2.2, then we have Theorem 2.3. Let ν, µ be doubling measures on a metric space (S, ρ). Let B0be a ball and suppose that (H1), (H2) and (H3) hold with Ω = B0. Then for ν-a.e. x∈B0, |f(x)−fB0| ≤ CZB0 g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y), where Cdepends only on ν, µ and the constants in (H1), (H2). The next theorem is a generalization of Theorem 2.3 to any weak Boman chain domain Ω. Theorem 2.4. Suppose that νand µare doubling measures on a metric space (S, ρ) and that hypotheses (H1)–(H4) hold for a domain Ω⊂ S. Then for ν-a.e. x∈Ω, |f(x)−fB0| ≤ CZΩ g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y), where B0is the central ball in Ω,f, g, ν and µin (H1), and Cdepends only on ν, µ and the constants in (H1), (H2) and (H4). As is well-known, under the segment hypothesis (H3), any metric ball is a Boman chain domain (see [FGW], [L2]), and thus Theorem 2.3 is a special case of Theorem 2.4 . The proof of Theorem 2.2 relies on the construction of the following chain of metric balls given in [LW2], assuming the segment hypothesis (H3). A similar construction was given in [FW], but the following one enables us to select all balls in the chain lying inside entirely the given ball B0. The chain of balls will allow us to prove the representation formulas on the same ball on both sides directly (see [LW2]), rather than using the formula on the enlarged ball to get the corresponding one on the same ball (see [LW1]). A somewhat different chain of finite length is given independently in [HK2]. L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 7 Theorem 2.5 [LW2]. Let (S, ρ)be a metric space in which the segment property (H3) holds. Let B0be a ball in S. Given x∈B0, there exists a chain {Bk}k≥1of balls with the following properties: (1) Bk⊂B0and ρ(Bk, x)→0as k→ ∞. (2) r(B1)≈r(B0)and r(Bk)→0as k→ ∞. (3) If y∈Bk, then ρ(y, x)≈r(Bk). (4) BkTBk−1contains a ball Skwith r(Sk)≈r(Bk)≈r(Bk+1)≈2−kr(B0). (5) If j < k, then Bk⊂cBj. (6) {Bk}k≥1has bounded overlaps, i.e., PkχBk(y)≤cfor all y. The constants of equivalence in (2), (3) and (4) and the constants cin (5) and (6) are independent of x, k, j and B0, but the chain {Bk}depends on x. The following remark is in order. The argument given here is similar to the proof of Theorem A in [LW2]. However, since our case is for q < 1 and then the Minkowski’s inequality fails. Thus, our situation becomes more delicate than the case for q= 1. In particular, we will use the inequality (2.6) µZE (f+g)qdν¶1/q ≤2q"µZE fq¶1/q +µZE gq¶1/q#. However, this inequality does not hold when we have infinitely many terms in the integrand unlike the case for q≥1, namely, we do not have ÃZEÃ∞ X i=1 fi!q dν!1/q ≤C(q) ∞ X i=1 µZE fq idν¶1/q . Therefore we have to proceed with caution, see the estimate for I2below. Proof of Theorem 2.2. We will use Theorem 2.5 to prove Theorem 2.2. Let B0be a ball in Sand suppose that (H1) and the segment property (H3) hold for B0. Given x∈B0, let {Bk}k≥1be a sequence of balls with the properties guaranteed by Theorem 2.5. Then (2.7) |f(x)−fB0| ≤ |f(x)−fB1|+|fB1−fB0|. For the second term on the right in (2.7), we get for ν−a.e. x∈B0that |fB1−fB0|=µZB1 |fB1−fB0|qdν¶1/q ≤CµZB1 |f(y)−fB1|qdν(y)¶1/q +CµZB1 |f(y)−fB0|qdν(y)¶1/q ≤CµZB1 |f(y)−fB1|qdν(y)¶1/q +CµZB0 |f(y)−fB0|qdν(y)¶1/q since ν(B1)≈ν(B0) and νis doubling ≤Cr(B1) µ(B1)ZB1 g dµ +Cr(B0) µ(B0)ZB0 g dµ by the Poincar´e inequality (H1) ≤Cr(B0) µ(B0)ZB0 g dµ 8 G. LU AND C. P´ EREZ since B1⊂B0,r(B1)≈r(B0) and µ(B1)≈µ(B0). Assuming as we may that xis a Lebesgue point for both |f−fB1|qand gwith respect to νand using properties (1)–(3) from Theorem 2.5 and the inequality (2.6), we have for the first term on the right in (2.7) that |f(x)−fB1|= lim k→∞ µZBk |f(y)−fB1|qdν(y)¶1/q ≤2qlim sup k→∞ µZBk |f(y)−fBk|qdν(y)¶1/q + 2qlim sup k→∞ µZBk |fBk−fB1|qdν(y)¶1/q =I1+I2, where I1and I2are defined by the last equality. It is easy to show that I1= 0 for every Lebesgue point xof g. This can be seen by the Poincar´e inequality (H1): I1= lim sup k→∞ µZBk |f(y)−fBk|qdν(y)¶1/q ≤Clim sup k→∞ r(Bk) µ(Bk)ZBk g(y)dµ(y) = 0 ·g(x) = 0. We now estimate I2. By observing that fBj+1 −fBjis a constant function, we have I2= lim sup k→∞ |fBk−fB1| ≤lim sup k→∞ k−1 X j=1 |fBj+1 −fBj| = lim sup k→∞ k−1 X j=1 ÃZSj |fBj+1 −fBj|qdν!1/q ≤2q ∞ X j=1 ÃZSj |fBj+1 −f|qdν!1/q + 2q ∞ X j=1 ÃZSj |fBj−f|qdν!1/q ≤2q ∞ X j=1 ÃZBj+1 |fBj+1 −f|qdν!1/q + 2q ∞ X j=1 ÃZBj |fBj−f|qdν!1/q since Sj⊂Bj∩Bj+1 and ν(Sj)≈ν(Bj)≈ν(Bj+1) by Theorem 2.5. Combining estimates and applying (H1) to the terms of each of the last two sums, we obtain I2≤C ∞ X j=1 r(Bj)ZBj g(y)dµ(y). L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 9 Now, as arguing in [LW2], if y∈Bj, then r(Bj) µ(Bj)≈ρ(x, y) µ(B(y, ρ(x, y))) ≈ρ(x, y) µ(B(x, ρ(x, y))) by part (3) of Theorem 2.5 and the fact that µis a doubling measure. Thus we obtain 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 properties (6) and (1) of Theorem 2.5. This completes the proof of Theorem 2.2 by combining estimates for I1and I2. Proof of Theorem 2.3. By observing that if x, y ∈B0, then ρ(x, y)≤2r(B0) and consequently by (H2), we have r(B0) µ(B0)≤Cρ(x, y) µ(B(x, ρ(x, y))) if x, y ∈B0. Thus, the second term on the right in the conclusion of Theorem 2.2 is bounded by the first term. Proof of Theorem 2.4. Let x∈Ω. By the definition of weak Boman chain domain, we may select B∗with x∈B∗and a chain {Bj}k j=0 connecting B∗=Bkto the central ball B0. We have (2.8) |f(x)−fB0| ≤ |f(x)−fB∗|+|fB∗−fB0|. For the first term on the right side of (2.8), we have by Theorem B that |f(x)−fB∗| ≤ CZB∗ g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y). This holds for ν−a.e. point of B∗, and we may assume it holds for our fixed xby initially excluding from Ω the set of measure zero formed by taking the union of the exceptional sets of measure zero in each Boman ball. Since B∗⊂Ω, we obtain the desired estimate |f(x)−fB∗| ≤ CZΩ g(y)ρ(x, y) µ(B(x, ρ(x, y))) dµ(y). Thus we only need to estimate |fB∗−fB0|.By using the chain {Bj}connecting B0and Bk=B∗, we have |fB∗−fB0| ≤ k X j=1 |fBj−fBj−1|. 16 G. LU AND C. P´ EREZ by part (3) of Theorem 2.5 and the fact that µis a doubling measure. Thus, we get 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 properties (6) and (1) of Theorem 2.5. The proof of Theorem 2.2 now is complete. Remark. We omit the proof of Theorem 4.3 since is similar to that of Theorem 2.3 by using (A2) instead of (H2). Proof of Theorem 4.4. Let x∈Ω. By the definition of weak Boman chain domain, we may select B∗with x∈B∗and a chain {Bj}k j=0 connecting B∗=Bkto the central ball B0. We have (4.6) |f(x)−Pm(B0, f)(x)| ≤|f(x)−Pm(B∗, f)(x)| +|Pm(B∗, f)(x)−Pm(B0, f)(x)|. For the first term on the right side of (4.6), we have by Theorem B that |f(x)−Pm(B∗, f)(x)| ≤ CZB∗ g(y)ρ(x, y)m µ(B(x, ρ(x, y))) dµ(y). This holds for ν−a.e. point of B∗, and we may assume it holds for our fixed xby initially excluding from Ω the set of measure zero formed by taking the union of the exceptional sets of measure zero in each Boman ball. Since B∗⊂Ω, we obtain |f(x)−Pm(B∗, f)(x)| ≤ CZΩ g(y)ρ(x, y)m µ(B(x, ρ(x, y))) dµ(y). We now estimate |Pm(B∗, f)(x)−Pm(B0, f)(x)|.By using the chain {Bj}connecting B0and Bk=B∗and noticing that B∗⊂MBjand x∈B∗, we have |Pm(B∗, f)(x)−Pm(B0, f)(x)| ≤ k X j=1 |Pm(Bj, f)(x)−Pm(Bj−1, f)(x)| ≤ k X j=1 ||Pm(Bj, f)−Pm(Bj−1, f)||L∞ ν(B∗) ≤ k X j=1 ||Pm(Bj, f)−Pm(Bj−1, f)||L∞ ν(MBj). L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 17 If Djis a ball with Dj⊂Bj∩Bj−1⊂MBjand r(Dj)≈r(Bj)≈r(Bj−1), then by (P1) and (P2), the last sum is majorized by C k X j=1 ||Pm(Bj, f)−Pm(Bj−1, f)||L∞ ν(Dj) ≤C k X j=1 ÃZDj |Pm(Bj, f)−Pm(Bj−1, f)|qdν!1/q , which by the inequality (2.6) and doubling is bounded by C2q k X j=1 ÃZDj |Pm(Bj, f)(y)−f(y)|qdν(y)!1/q +C2q k X j=1 ÃZDj |Pm(Bj−1, f)(y)−f(y)|qdν(y)!1/q ≤C2q k X j=1 ÃZBj |Pm(Bj, f)(y)−f(y)|qdν(y)!1/q +C2q k X j=1 ÃZBj−1 |Pm(Bj−1, f)(y)−f(y)|qdν(y)!1/q ≤C k X j=0 Ã1 ν(Bj)ZBj |Pm(Bj, f)(y)−f(y)|qdν(y)!1/q By Poincar´e’s inequality, the last expression above is at most C k X j=0 r(Bj)m µ(Bj)ZBj g(y)dµ(y) =CZΩ   k X j=0 r(Bj)m µ(Bj)χBj(y)   g(y)dµ(y). As shown in [LW2], the sum above in curly brackets is bounded by a fixed multiple of ρ(x, y)m/µ(B(x, ρ(x, y))) for each y∈Ω. Thus, we have completed the proof. By using Theorem (4.2), we will be able to weaken the hypotheses in defining high order Sobolev spaces in metric spaces given in [LLW1] (see also [LLW2]). Definition 4.7. Given a positive integer mand 1< p < ∞, we define the Sobolev class Am,p(Ω) to be the set of functions f∈Lp(Ω) so that for each k= 1,··· , m, there exist rkwith 1≤rk< p and qkwith 0< qk<1, functions gk(x)with 0≤gk∈Lp(Ω), and polynomials Pk(B, f)with (4.8) µZB |f(x)−Pk(B, f)(x)|qkdµ(x)¶1 qk≤r(B)kµZB grk k(x)dµ(x)¶1 rk 18 G. LU AND C. P´ EREZ for every ball B⊂Ω. The polynomials Pk(B, f)are assumed to belong to a linear class which satisfies (P1) and (P2) with constants depending only on k, γ, µ. If f∈Am,p(Ω), we define ||f||Am,p(Ω) =||f||Lp(Ω) + inf {gk} m X k=1 ||gk||Lp(Ω), where the infimum is taken over all sequences such that (4.8) holds for ffor k= 1, . . . , m. It is easy to see that Am,p(Ω) is a linear space. The reason we can impose the Lrknorm rather than the L1norm is because we can show that definition (4.7) is equivalent to the following definitions (4.9) and (4.11) given in [LLW1]. The proof of equivalence follows from our Theorem (4.2) in this section by combining the proofs given in [LLW1]. We shall omit the details here. Definition 4.9 [LLW1]. Given a positive integer mand 1< p < ∞, we define the Sobolev class Bm,p(Ω) to be the set of functions f∈Lp(Ω) so that for each k= 1,· · · , m, there exist functions 0≤gk∈Lp(Ω) and polynomials Pk(B, f)such that (4.10) |f(x)−Pk(B, f)(x)| ≤ ZB ρ(x, y)kgk(y) µ(B(x, ρ(x, y)))dµ(y) + r(B)kZB gk(y)dµ(y) for µ−a.e. x∈Bfor every ball B⊂Ω. The polynomials Pk(B, f)are assumed to belong to a linear class which satisfies (P1) and (P2) with constants depending only on k, γ, µ. If f∈Bm,p(Ω), we define ||f||Bm,p(Ω) =||f||Lp(Ω) + inf {gk} m X k=1 ||gk||Lp(Ω), where the infimum is taken over all sequences such that (4.10) holds for ffor k= 1,··· , m. The class Bm,p(Ω) is clearly a Banach space with norm || · ||Bm,p(Ω). Definition 4.11 [LLW1]. Given a positive integer mand 1< p < ∞, we define the Sobolev class Cm,p(Ω) to be the set of functions f∈Lp(Ω) so that for each k= 1,· · · , m there exist functions 0≤gk∈Lp(Ω) and polynomials Pk(B, f)such that (4.12) |f(x)−Pk(B, f)(x)| ≤ r(B)kgk(x) for µ−a.e. x∈Bfor every metric ball B⊂Ω. The polynomials Pk(B, f)are assumed to belong to a linear class which satisfies (P1) and (P2) with constants depending only on k, γ, µ. If f∈Cm,p(Ω), let ||f||Cm,p(Ω) =||f||Lp(Ω) + inf {gk} m X k=1 ||gk||Lp(Ω). The class Cm,p(Ω) is a Banach space with norm || · ||Cm,p . To show that definitions (4.7), (4.9) and (4.11) are all equivalent, we will need the following theorem. L1→LqPOINCAR´ E INEQUALITIES FOR 0 < q < 1 AND IMPLICATIONS 19 Theorem 4.13. Let 1≤r < ∞,mbe a positive integer, B0⊂Ωbe a fixed ball, and suppose that the segment property (H3) holds for B0. Let fbe a locally integrable function in Ωfor which there exist a function 0≤g∈Lr(Ω) and polynomials Pm(B, f), and 0< q < 1such that the Poincar´e inequality µZB |f(x)−Pm(B, f)(x)|qdµ(x)¶1/q ≤cr(B)mµZB |g(x)|rdµ(x)¶1/r holds for every ball B⊂Ω. The polynomials Pm(B, f)are assumed to belong to a linear class which satisfies (P1) and (P2) with constants depending only on m, γ, µ. Then for µ−a.e. x∈B0, |f(x)−Pm(B0, f)(x)| ≤ Cr(B0)mM(gr)(x)1/r with Cindependent of x. Proof of Theorem 4.13. Let x∈B0. We will use the chain of subballs {Bj}of B0 constructed from Theorem 2.5. The chain depends on x. We may assume without loss of generality that xis a Lebesgue point for both |f−Pm(B0, f)|qand |g|rwith respect to µ. Then by properties (1), (2) and (3) of the chain, |f(x)−Pm(B0, f)(x)|= lim j→∞ ÃZBj |f(y)−Pm(B0, f)(y)|qdµ(y)!1/q ≤lim sup j→∞ 2qÃZBj |f(y)−Pm(Bj, f)(y)|qdµ(y)!1/q + lim sup j→∞ 2qÃZBj |Pm(Bj, f)(y)−Pm(B0, f)(y)|qdµ(y)!1/q =I1+I2. By the Poincar´e inequality, for every Lebesgue point xof |g|r I1≤clim sup j→∞ r(Bj)mÃZBj |g(y)|rdµ(y)!1/r = 0 · |g(x)|= 0 by properties (1), (2) and (3) of the chain. 20 G. LU AND C. P´ EREZ We have for I2 I2≤lim sup j→∞ ||Pm(Bj, f)(y)−Pm(B0, f)(y)||L∞ µ(Bj) ≤lim sup j→∞ j−1 X `=0 ||Pm(B`+1, f)−Pm(B`, f)||L∞ µ(Bj) ≤lim sup j→∞ j−1 X `=0 ||Pm(B`+1, f)−Pm(B`, f)||L∞ µ(cB`)by (5) ≤C ∞ X `=0 ||Pm(B`+1, f)−Pm(B`, f)||L∞ µ(S`)by (4) and (P2) ≤C ∞ X `=0 µZSl |Pm(B`+1, f)−Pm(B`, f)|qdµ¶1/q by (P1) ≤C ∞ X `=0 2qµZB` |Pm(B`, f)(y)−f(y)|qdµ(y)¶1/q +C ∞ X `=0 2qÃZB`+1 |Pm(B`+1, f)(y)−f(y)|qdµ(y)!1/q by (4) ≤C ∞ X `=0 r(B`)mµZB` |g(y)|qdµ(y)¶1/q ≤C ∞ X `=0 r(B`)mM(|g|r)(x)1/r by (3) =C ∞ X `=0 2−`mr(B0)mM(|g|r)(x)1/r by (2) ≤Cr(B0)mM(|g|r)(x)1/r. This completes the proof of Theorem 4.13. By using Theorem 4.13, and arguing similarly as in [LLW1], we will be able to show the equivalence of definitions (4.7), (4.9) and (4.11). We shall omit the details here and refer the reader to [LLW1]. 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