Subelliptic Poincaré inequalities: the case p < 1

Publicacions Matem`atiques, Vol 39 (1995), 313–334. SUBELLIPTIC POINCAR´ E INEQUALITIES: THE CASE p<1 S. Buckley, P. Koskela and G. Lu1 Abstract We obtain (weighted) Poincar´e type inequalities for vector fields satisfying the H¨ormander condition for p<1 under some assumptions on the subelliptic gradient of the function. Such inequalities hold on Boman domains associated with the underlying CarnotCarath´eodory metric. In particular, they remain true for solutions to certain classes of subelliptic equations. Our results complement the earlier results in these directions for p≥1. 1. Introduction One of the main purposes of this paper is to derive a Poincar´e-type inequality of the form (1.1)   1 |B| B |f(x)−fB|qdx  1 q ≤cr    1 |B| B   j |Xj,∇f(x)|2  p 2 dx   1 p in Euclidean space RNfor 0 <p<1 and certain values q>p, where {Xj}m j=1 is a collection of smooth vector fields which satisfy the H¨ormander condition (see [H]) provided that fis a suitable function whose subelliptic gradient satisfies a weak reverse H¨older condition (defined below). Here, Bdenotes any suitably restricted ball of radius r relative to a metric which is naturally associated with {Xj}(as in, for example, [FP] and [NSW]), fBis some constant (we may assume fB=|B|−1Bf(x)dx if f∈L1(B)), and cis a constant which depends 1The first author was partially supported by NSF Grant DMS–9207715. The second author was partially supported by NSF Grant DMS–9305742 and by the Academy of Finland. The third author was partially supported by NSF Grant DMS–9315963. 314 S. Buckley, P. Koskela, G. Lu on the reverse H¨older constant of the subelliptic gradient of fbut is otherwise independent of f, and is also independent of B. More generally, we shall prove one-weight and two-weight versions of this result, and also results for more general domains. Inequality (1.1) was derived in [J] for q=pand 1 ≤p<∞, and this result was improved by the third author in the case p>1in[L2] where (1.1) is proved for 1 <p<Q,q=pQ/(Q−p), and Q(≥N) denotes the homogeneous dimension of RNassociated with {Xj}(see below for the definition). Recently the limiting case p= 1 and q=Q Q−1was proved in [FLW] by establishing a new representation formula that improves the previous one proved in [L1]; this inequality was then applied to the relative isoperimetric inequality. In the cases p=Qand p>Q,f was shown to be in BMO and H¨older classes, respectively, when |Xf| is assumed to be in Lp loc (see [L3], [L4]); embedding theorems on the Campanato-Morrey spaces and from the Morrey spaces to BMO and Lipschitz spaces were also shown. Some related inequalities have been studied in [BM], [HK], and [MS]. There have been very few Poincar´e-type results for p<1, mainly because there are easy counterexamples even in the case when {Xi}N i=1 are the constant vector fields in the coordinate directions, i.e., Xi=∂ ∂xi, 1≤i≤N(see [BK]). However, in this particular case, it has recently been proved that fsatisfies a Poincar´e inequality if the gradient of f satisfies a weak reverse H¨older condition [BK]. We show in this paper that a similar result is true in the setting of vector fields of the above type and also in a weighted context. In the unweighted case, we will show that (1.1) holds for such functions fif 0 <p<1, p<q<∞, and pand qare related by a natural balance condition involving the local doubling order of Lebesgue measure for metric balls (see [FLW] for p≥1). This balance condition (introduced earlier in [CW]) can actually be shown to be necessary and sufficient for the validity of the Sobolev-Poincar´e inequality. To state our theorems, we need to introduce some notation and definitions. Let Ω be a domain in RN, and let {Xj}m j=1 be a collection of C∞real vector fields defined in a neighbourhood of the closure ΩofΩ. For a multi-index α=(i1,... ,i k), the commutator [Xi1,[Xi2,... ,[Xik−1,X ik]] ...] of length k=|α|will be denoted by Xα. Throughout this paper we assume that the vector fields satisfy H¨ormander’s condition: there exists some positive integer ssuch that {Xα}|α|≤sspan RNat each point of Ω. For the sake of brevity, we shall refer to such a family of vector fields as a H¨ormander family. With any H¨ormander family, we can associate a metric as follows. 315 First let us say that γ:[a, b]→Ωisanadmissible curve if it is Lipschitz and there exist functions ci(t), a≤t≤b, satisfying m i=1 ci(t)2≤1 and γ(t)=m i=1 ci(t)Xi(γ(t)) for almost every t∈[a, b]. A natural metric on Ω associated with X1,... ,X mis (ξ,η) = inf{b≥0:∃an admissible curve γ:[0,b]→Ω such that γ(0) = ξ, and γ(b)=η}. Such a metric is often called a Carnot-Carath´eodory metric. It follows from the work of Busemann [Bu, p. 25] that any two points in Ω can be joined by a geodesic (a rectifiable path whose -length equals the -distance between its end-points). We assume that any geodesic is canonically parametrised by the -arclength of its initial segment. The (open) metric ball with centre xand radius rwill be denoted B(x, r)={y:(x, y)<r}.is locally equivalent to the various other metrics defined in [NSW], and generates the same topology as the Euclidean metric. It is shown there that Lebesgue measure is locally doubling on small balls: if K⊂⊂ Ω and δ>0 is sufficiently small, then (1.2) |B(x, 2r)|≤C|B(x, r)|,x∈K, 0<r<δ. Thus (Ω,) is (locally) a homogeneous space in the sense of CoifmanWeiss. If Bis a metric ball and t>0, r(B) and rBboth denote the radius of B,tB denotes the “t-dilate” of B(the concentric ball with radius t·rB), and zBdenotes the centre of B. In proofs, Cdenotes any constant whose exact value is unimportant for the purposes of the proof. By the Rothschild-Stein lifting theorem (see [RS]), the vector fields {Xi}m i=1 on Ω ⊂RNcan be lifted to vector fields {˜ Xi}m i=1 in ˜ Ω=Ω× T⊂RN×RM−N, where Tis the unit ball in RM−N, by adding extra variables so that the resulting vector fields are free, i.e., the only linear relation between the commutators of order less than or equal to sat each point of ˜ Ω are the antisymmetric and Jacobi’s identity. Let G(m, s) be the free Lie algebra of steps with mgenerators, that is the quotient of the free Lie algebra with mgenerators by the ideal generated by the commutators of order at least s+ 1. Then {Xα}|α|≤sare free if and only if N= dim G(m, s). We define the homogeneous dimension of Ω to be Q≡s j=1 jmj, where mjis the number of linearly independent commutators of length jfor the lifted vector fields. A weight function w(x) on an open subset Eof Ω is a nonnegative function on Ewhich is locally integrable with respect to Lebesgue measure and not everywhere zero. Given 0 <α<1, we say that a positive 316 S. Buckley, P. Koskela, G. Lu Borel measure µon an open subset EofΩisanα-strong doubling measure (or simply a strong doubling measure) on Eif there exists Cµ>0 such that µ(2B∩E)≤Cµµ(B∩E) for all metric balls Bfor which αB ⊂E.Ifdµ(x)=w(x)dx for some weight w, we say that wis an α-strong doubling weight.If1≤p<∞, we say that a weight wis in the class (local)-Ap(E)≡Ap(E, , dx) if there is some constant Cw≥1 such that   1 |B| B wdx    1 |B| B w−1/(p−1) dx  p−1 ≤Cw,when 1 <p<∞, 1 |B|B wdx≤Cwess inf Bw, when p=1, for all metric balls 2B⊂E. Since Lebesgue measure is doubling with respect to metric balls, we can develop the usual theory of these Muckenhoupt Apclasses as in [Ca], at least for balls B=B(x, r) with 0 <r<r 0 and xbelonging to a compact subset of E. In particular, it follows easily from the above definition that Apweights are (local) doubling weights: µ(2B)≤Cµµ(B) whenever 4B⊂E. These weights are not necessarily strongly doubling. We should note that our definition is more “local” than Calder´on’s (since we are only assuming the defining inequality for balls Bsuch that 2B⊂E), but Calder´on’s proofs can be easily adjusted to handle this variant; our more local definition allows us to handle certain important classes of weights that would not belong to a more restricted class (for instance, positive powers of distance to the boundary). We now introduce the notion of functions satisfying a weak reverse H¨older condition. Given a doubling measure µon an open subset E of Ω, we say that a non-negative function won Eis in WRHp(E,µ)if w≡ 0, w∈Lp loc(E,µ), and if there is a constant Cw,µ >0 such that 1 µ(B)B w(x)pdµ1 p ≤Cw,µ 1 µ(B)σB w(x)qdµ1 q for all B:σB⊂E. Here q,σ, and σare parameters satisfying 0 <q<p,1<σ≤σ. These definitions are independent of the choice of q,σ,σ, as we shall show in Lemma 1.4 (whose proof is adapted from the proof for Lebesgue measure and Euclidean space in [IN]). When dµ(x)=v(x)dx, we shall abuse notation as before and write WRHp(E,v) in place of WRHp(E,µ); in the particular case v= 1, we simply write WRHp(E). For a convenient choice of q,σ,σ, the smallest constant Cw,µ for which the above defining 317 inequality remains valid is called the WRHp(E,µ) constant of w. Since we use this constant only as an upper bound on the variability of the gradient, it follows from Lemma 1.4 below that the exact choices of q,σ, σ, are unimportant for our purposes. Before stating Lemma 1.4, let us first state the following Whitney decomposition result of Coifman and Weiss (see [CoWe, Theorem III.1.3]). Lemma 1.3. If Eis a proper open subset of a homogeneous space (S, d, µ), then there exists a family Fof disjoint metric balls Band constants 1<K 1<K 2<K 3,Msuch that (a) E=B∈F K1B. (b) B∈F χK2B(x)≤Mχ E(x)for all x∈S. (c) K3Bintersects Ecfor every B∈F. Note that by examining the proof of Lemma 1.3 in [CoWe], it is easily verified that the constants K1,K2/K1, and K3/K2can be chosen arbitrarily and independently, provided that they exceed certain lower bounds. For the first and last of these constants, this is essentially trivial, while increasing K2/K1corresponds to using smaller balls in the proof of this lemma. Lemma 1.4. Let Gbe an open subset of a homogeneous space (S, d, µ) and let F(G)be the set of metric balls contained in G. Suppose that for some 0<q<pand non-negative f∈Lp loc(µ), there are constants A>1 and 1<σ 0≤σ 0such that (1.5) − B fpdµ1/p ≤A− σ0B fqdµ1/q ∀B:σ 0B∈F(G). Then for any 0<r<qand 1<σ≤σ<σ  0, there exists a constant A>1such that (1.6) − B fpdµ1/p ≤A− σB frdµ1/r ∀B:σB∈F(G). In fact, we can choose A=C0As/(σ−1)C0, where s=(r−1−p−1)/(q−1− p−1)and C0is a sufficiently large constant independent of f,A,σ, and σ. Proof: Without loss of generality, we assume that σ=σ<σ 0. Let E⊂Gbe any metric ball. For simplicity, we normalise the metric and the measure so that rE= 1 and µ(E)=1. LetW∞be the set of all 318 S. Buckley, P. Koskela, G. Lu metric balls in the Whitney decomposition of Ewith constants K1,K2, K3chosen so that K2is larger than 2K4≡2σ 0K1(so that we can choose Bin (1.5) to be the K1-dilate of any Whitney ball). For all k≥0, let Wkbe the set of all Whitney balls of radius greater than 2−k, and let Ek=B∈WkK1B. Since µis doubling, there exists 0 <b<1 such that µ(B)≥bkfor all B∈Wk, and hence there are at most Mb−k balls in Wk. Letting t=p(q−r)/q(p−r), we see that 0 <t<1 and q−1=tp−1+(1−t)r−1. Now (1.5) and H¨older’s inequality imply that − K1B fpdµ1/p ≤A− K4B fpdµt/p− K4B frdµ(1−t)/r and so K1B fpdµ ≤Ap− E frdµ(1−t)p/r K4B fpdµt µ(B)(1−t)(1−p/r). Now µ(B)(1−t)(1−p/r)≤b−k 1for all B∈Wk, where b1=b(1−t)(p−r)/p. Thus Ek fpdµ ≤ B∈WkK1B fpdµ ≤Apb−k 1− E frdµ(1−t)p/r  B∈WkK4B fpdµt . If x∈K4Bfor some B∈Wk, then d(x, Ec)≥K42−k(since K2>2K4). On the other hand, if x∈K1B, for some B∈W∞, then d(x, Ec)≤ d(x, zB)+d(zB,Ec)≤(K1+K3)rB. It follows that if we fix an integer m>log2[(K1+K3)/K4], then Q∈WkK4B⊂Ek+m, and so writing b2=b·b1, we get (1.7) Ek fpdµ ≤MApb−k 2− E frdµ(1−t)p/r Ek+m fpdµt . Iterating (1.7) we see that Ek fpdµ ≤MAp− E frdµ(1−t)p/rαl b−γl 2Ek+ml fpdµtl where αl=l−1 j=0 tjand γl=l−1 j=0(k+mj)tj. Letting l→∞, we see that αl→(1 −t)−1and γl→(k(1 −t)+mt)/(1 −t)2and so (1.8) − Ek fpdµ1/p ≤CA1/(1−t)b−k/p(1−t) 2− E frdµ1/r . 319 If we choose kto be the least integer larger than log2[(K1+K3)σ/(σ−1)], then σ−1E⊂Ek, and so (1.6) follows for all admissible Bby choosing E=σB in (1.8). The last statement of the theorem follows since 1/(1 − t)=sand b−k/p(1−t) 2≤C(σ−1)−C0for sufficiently large C0. It was shown in [FLW] that, given a compact subset Kof Ω and a ball B=B(x, r), r<r 0,x∈K, there exist positive constants γand c, depending on Kand r0(or on Bif we wish), so that (1.9) |J|≤cr(J) r(I)Nγ |I| for all balls I,Jwith I⊂J⊂B. We shall call γthe doubling order of Lebesgue measure for B. We always have N≤Nγ ≤Q, where Qis the homogeneous dimension defined previously. We can of course choose Nγ =Q, but smaller values may arise for particular vector fields, and these values may vary with B(x, r). If E⊂Ωisopenandf∈C1(E), we write Xjf(x)=Xj(x),∇f(x),j=1,... ,m, and |Xf(x)|2= m  j=1 |Xjf(x)|2, where ∇fis the usual gradient of fand ,is the usual inner product on RN. We now state the unweighted version of our main Poincar´e estimate, which essentially generalises the main result of [BK] and extends the main result of [L2]. Theorem 1.10. Let Kbe a compact subset of Ω. There exists r0>0 depending on K,Ωand {Xj}such that if E=B(x, r)is a ball with x∈Kand 0<r<r 0, and if 1/q =1/p −1/(Nγ),0<p<1, where γ is defined by (1.9) for E, then there exists a constant fEsuch that 1 |E|E |f(x)−fE|qdx1 q ≤C0r1 |E|E |Xf(x)|pdx1 p for any f∈C1(E)provided that |Xf|∈WRH1(E). The constant C0 depends on p,K,Ω,{Xj}, the WRH1(E)-constant, and the constants 320 S. Buckley, P. Koskela, G. Lu cand γin (1.9). We may choose fE=|E|−1Ef(x)dx for any compactly contained sub-ball Eof E, (in which case the constant C0also depends on the choice of E). This result and its proof is a hybrid of the main unweighted results of [FLW] and [L2] with the main result of [BK]. As mentioned earlier, we may always choose Nγ =Q. Note that if we assume Theorem 1.10 for a particular choice of the subball E, and use also the corresponding Poincar´e inequality for p= 1 (as in [FLW]), a standard argument gives Theorem 1.10 for all valid choices of E. Accordingly we shall prove this theorem only when Eis the “central” ball in an appropriate Whitney decomposition of E. Similarly, if f∈L1(E), we readily see that fEcan be chosen to be the average of fover all of E(as it is usually defined when p≥1). Similar comments apply to the weighted results below which, for simplicity, we state only for a particular choice of fE. Given 0 <p<1, p<q<∞, and a metric ball E⊂Ω, we shall be interested in weights w1,w2on Efor which the following balance condition holds: (1.11) r(I) r(J)w2(I) w2(J)1 q ≤cw1(I) w1(J)1 p for all metric balls I,Jwith I⊂J⊂E. Note that in the case of Lebesgue measure (w1=w2= 1), (1.11) reduces to (1.9) if 1/q =1/p − 1/(Nγ). A balance condition of type (1.11) was introduced previously in [CW] to study weighted Poincar´e inequalities (see also [FGuW], [FLW], and [L1]). We shall use notation such as (1.11)p0,q0when we wish to refer to (1.11) with parameters p=p0and q=q0. We now state a weighted Poincar´e inequality for 0 <p<1, p<q, which complements the case 1 ≤p<qconsidered in [FLW]. It also generalises Theorem 1.10, since Lebesgue measure is doubling on small balls centred in K. Theorem 1.12. Let Kbe a compact subset of Ω. There exists r0>0 depending on K,Ωand {Xj}such that if E=B(x, r)is a ball with x∈Kand 0<r<r 0,if0<p<1,p<q<∞, and if w1,w2are weights satisfying the balance condition (1.11) for E, with w1∈A1(E) and w2α-strongly doubling on Efor sufficiently small α=α(Ω) >0, 321 then (1.13) 1 w2(E)E |f(x)−fE|qw2(x)dx1 q ≤C0r1 w1(E)E |Xf(x)|pw1(x)dx1 p for any f∈C1(E), with fE=w2(1 2E)−11 2Ef(x)w2(x)dx, provided |Xf|∈WRH1(E,w1). The constant C0depends only on p,K,Ω,{Xj}, the WRH1(E,w1)-constant, and the constants in the conditions imposed on w1and w2. For the necessity of (1.11) see Section 2. In the particular case when w≡w1=w2is α-strongly doubling (for α<1/11, say), we claim that (1.11) is always satisfied for some q−1=p−1−δwhere δ∈(0,1/p)is dependent only on the strong doubling constant. This is clearly true if rI>r J/10, so we assume rI<r J/10. We may also assume that (zJ,z I)>r J/10. We claim that I⊂J1⊂Jfor some metric ball J1for which rJ1= rJ/10. To construct J1, let g:[0,s]→J,s<r J, be the geodesic curve for which g(0) = zJand g(s)=zI. Let Btbe the ball of radius rJ/10 −rI, and centre g(t), and let t0= inf{0<t≤s:zI∈Bt}. Since gis a geodesic, it readily follows that I⊂J1⊂Jif rJ1=rJ/10 and zJ1=g(t0). Clearly, w(J1)/w(J)>β>0 for some 0 <β<1 dependent only on the strong doubling constant. Continuing this process, we can create a finite nested sequence of metric balls Jiof radius rJ/10iwhose final member Jmcontains Ibut has radius not more than 10rI. It follows easily that w(I)/w(J)>Cβ mif rI/rJ<10−m, establishing our claim. The following one-weighted corollary of Theorem 1.12 now follows. Corollary 1.14. Let Kbe a compact subset of Ω. Then there exists r0depending on K,Ωand {Xj}such that if E=B(x, r)is a ball with x∈Kand 0<r<r 0,ifw∈A1(E, , dx)is α-strongly doubling on E for sufficiently small α=α(Ω) >0, and if q≤q0,0<p≤1, where p<q 0=q0(p, w), then 1 w(E)E |f(x)−fE|qw(x)dx1 q ≤C0r1 w(E)E |Xf(x)|pw(x)dx1 p for any f∈C1(E)provided that |Xf|∈WRH1(E,w). The constant C0 depends on p,K,Ω,{Xj},w, the WRH1(E)-constant, and the constants cand γin (1.9). Also, we may take fE=w(1 2E)−11 2Ef(x)w(x)dx. 328 S. Buckley, P. Koskela, G. Lu instances of “E”to“B∗” in (2.9) above to get a proof of the following Poincar´e inequality: f−fDLq w2(D)≤C0XfLp w1(D), where fDis the w2-average of fover B∗, and C0now depends also on D. (2) As for the case p≥1, some sort of balance condition is needed to prove a two-weighted inequality like (1.13) (assuming w1,w2are as in Theorem 1.12). Let us show that (1.11) must hold for all I⊂J=E. It suffices by strong doubling to prove this for rI<r J/100, 8I⊂J. There is a bump function fwhich equals 1 on I, is supported on 2I, and such that |Xf|≤C/rI(see [L1, Lemma 7.12]). It is not clear that |Xf|∈WRH1(E), but Lemma 2.1 of [BK] can be adapted to show that g=M(|Xf|2)1/2belongs to WRH1(4I),where M(|Xf|2) is the restricted maximal function M(|Xf|2)(x) = sup B=B(x,r) 2B⊂4I 1 w2(B)B |Xf|2w2dx. Extending gto be zero on E\4I, it is clear that g∈WRH1(E) (since it has compact support in 3I) and that |Xf|≤g. The proof of Theorem 1.12 and the Lp-boundedness of the maximal function then show that (1.13) holds for f. By the doubling property of w2, we see that for t∈{0,1},w2({x∈E|f(x)=t})≥Cw2(I) and so, no matter what choice of constant fEwe make, the left hand side of (1.13) is at least C(w2(I)/w2(E))1/q. However, the right-hand side of (1.13) is at most Cr(E) r(I)w1(I) w1(E)1/p , and (1.11) readily follows. Thus, if we have a weighted Poincar´e inequality on Eand on all of its sub-balls (with a uniform constant C0), (1.11) holds for all I⊂J⊂E. (3) One can also prove some results for more general domains, for example metric versions of the John-αdomains defined in [BK]. We shall not state or prove such results here as the statements lack elegance and we suspect this method does not give sharp results for such domains. Suffice it to say that one can prove such results in certain cases where it is possible to modify the above proof as in [BK] to avoid the use of Lemma 2.3. In particular this can be done if q<1. 329 Finally, let us give an application of the above to solutions of a class of quasilinear subelliptic equations. We show that the conclusion of Theorem 1.12 holds for these solutions with fEreplaced by f(x), where x is an arbitrary point of 1 2E. In order to deduce this from Theorem 1.12 we have to verify two facts: a reverse H¨older inequality for |Xu|when u is a solution, and an estimate for |u(x)−uE|,where uEis the average of uin 1 2E. The first of these is a consequence of a Caccioppoli type inequality and the second essentially follows from a mean-value inequality established in [L4]. Results of this type have appeared in [Z], [BK] (with Xi=∂ ∂xi), and in [L5] for H¨ormander families. In particular we shall show here that Theorem 5.14 of [BK] for 0 <p<1 remains true for H¨ormander vector fields, thereby extending the result of [L5] to the case 0 <p<1. We consider quasilinear second order subelliptic partial differential equations of the form (2.10) m  j=1 X∗ jAj(x, u, X1u, X2u,... ,X mu)+B(x, u, X1u, X2u, . . . , Xmu)=0, under certain structral assumptions. Harnack inequalities for weak solutions, subsolutions, and supersolutions of (2.10) have been established in [L4]. We let x,ηdenote vectors in RNand Rmrespectively, and Xu = (X1u,... ,X mu). A(x, u, η)=(A1(x, u, η),... ,A m(x, u, η)) and B(x, u, η) are, respectively, vector and scalar measurable functions defined on Ω ×R×Rm, where Ω is a domain in RN. For all M<∞and for all (x, u, η)∈Ω×(−M,M)×RN, we assume the structure of (2.10) satisfies the following inequalities: (2.11) |A(x, u, η)|≤a0|η|s−1+(a1(x)|u|)s−1, η·A(x, u, η)≥|η|s−(a2(x)|u|)s |B(x, u, η)|≤b0|η|s+b1(x)|η|s−1+(b2(x))s|u|s−1 where s>1, a0,b0are constants, ai(x), bi(x) are nonnegative measurable functions; s,a0,b0,ai(x), bi(x) may possibly depend on M. All the coefficients ai(x), bi(x) are assumed to be in certain subspaces of Lt loc(Ω), where t= max(s, Q) (see [L4] for the details). We now define the notion of a solution to the equation (2.10). First, for a domain D⊂Ω,the Sobolev class W1,s(D)(W1,s 0(D)), is defined as the closure of C1-smooth (and compactly supported) functions in the norm ||u|| =||u||s,D +||Xu||s,D. Here ||·|| s,D is the usual Ls-norm in 330 S. Buckley, P. Koskela, G. Lu D. Integration by parts shows that each u∈W1,s(D) has a unique subgradient Xu ∈Ls(D). We say uis a weak solution of (2.10) in Ω if ubelongs to W1,s loc (Ω) and Ω {Xφ·A(x, u, Xu)−φB(x, u, Xu)}dx =0 for all bounded φ(x)∈W1,s 0(Ω). Given ai(x), bi(x)∈LQ loc(Ω), u(x)∈L∞ loc, a standard approximation argument shows that, if the above equation holds for all φ(x)∈C1 0(Ω), then it still holds for all φ(x) given in the definition. As an application of Theorem 1.10 we obtain the following result. We note that the boundedness assumption on ucan be dropped if b0=0 (and (2.11) is true for all M>0 with parameters independent of M). Theorem 2.12. Let Kbe a compact subset of Ω. There exists r0such that whenever E=B(x, r),x∈K, and 0<r<r 0, the following holds. Let 0<p<1and 1/q =1/p −1/(Nγ), where γis defined by (1.9) for E(we can always choose Nγ =Q). Let u∈W1,s loc (E),|u|≤M,bea weak solution of (2.10). Then there is a constant Cdepending on the structure conditions (2.11), p,s,q,||u||s, 1 2E,Ω, and b0Msuch that for any point x0∈1 2E 1 |E|E |u−u(x0)|q1 q ≤Cr1 |E|E |Xu|p1 p , where ρ(E)is the radius of E. In order to deduce Theorem 2.12 from Theorem 1.10 we have to verify that Xu satisfies a weak reverse H¨older inequality and then replace the average of uover 1 2Eby u(x0). We first establish a weak reverse H¨older inequality. Caccioppoli Estimate. If u∈W1,s(E) is a solution of (2.10), then for any metric ball Bsuch that 2B⊂E,anyξ∈C∞ 0(2B), and any constant c,wehave 2B ξs|Xu|s≤C2B (|ξ|s+|Xξ|s)|u−c|s. This is a special case of formula (3.26) in [L4] derived by setting q= 1 and replacing uby u−c(note that u−cis a solution to an 331 equation also satisfying structural conditions (2.11)). In particular, if we select a cut-off function relative to the Carnot-Carath´eodory metric (see Lemma (7.12) in [L1]), we can choose 0 ≤ξ≤1, ξ∈C∞ 0(2B), ξ=1onBand |Xξ|≤Cr−1 B.Thus B |Xu|s≤Crs B2B |u−c|s, and so by using the Sobolev-Poincar´e inequality on 2B(see [L2]) to control the right-hand side of the above inequality we see that |Xu|∈ WRHs(E). Therefore, the solutions satisfy the hypothesis of Theorem 1.10. Let B=1 2E, and let uBbe the average value of uon B. For simplicity, we normalise so that rB=1. Now |u(x)−u(x0)|≤|u(x)−uB|+|uB−u(x0)|, so it suffices to control the q-integrals of the two right-hand side terms. For the first term, we use Theorem 1.10. By H¨older’s inequality, and Theorem 2.13 below, the second term can be bounded the Ls(B)-norm of |u−uB|. Using the classical Poincar´e inequality, this can then be bounded by the Ls-norm of |Xu|, which is comparable with the Lp-norm of |Xu|by Lemma 1.4, since we have shown that |Xu|∈WRHs(E). 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Str¨ omberg and A. Torchinsky, Weights, sharp maximal functions and Hardy spaces, Bull. Amer. Math. Soc. 3(80), 1053–1056. 334 S. Buckley, P. Koskela, G. Lu [Z] W. P. Ziemer, A Poincar´e-type inequality for solutions of elliptic differential equations, Proc. Amer. Math. Soc. 97 (1986), 286–90. S. Buckley: Department of Mathematics St. Patrick’s College Maynooth, Co. Kildare IRELAND e-mail: sbuc[email protected]y.ie P. Koskela: Department of Mathematics University of Jyv¨askyl¨a P.O. Box 35 FIN-40351 Jyv¨askyl¨a FINLAND e-mail: pkosk[email protected].fi G. Lu: Department of Mathematics Wright State University Dayton, Ohio 45435 U.S.A. e-mail: [email protected] Rebut el 16 de Gener de 1995