Generalized Poincaré inequalities: Sharp Self Improving Properties

Transcription

1 Internatonal mathematcs research notces, 2 998, 0 6. Generalzed Poncaré nequaltes: Sharp Self Improvng Propertes Paul MacManus and Carlos Pérez Introducton The man purpose of ths paper s both to mprove the man result n [FPW] and to provde a drect proof that avods the use of the dyadc sets for spaces of homogeneous type. oth proofs are based on a classcal good λ nequalty (cf. [G]). Let (S, d, µ) be a space of homogeneous type, and a : [0, ) be a functonal defned on the famly of all balls n S. Recall that d denotes a pseudo-metrc on S and that µ s a measure that s doublng wth respect to d. We want to show that f a locally ntegrable functon f satsfes f f dµ C a() () for all balls, then one can deduce hgher L p ntegrablty of f. As usual, f denotes the µ average of f over. Ths type of nequalty s refered to as a (weak) Poncaré nequalty. The standard example of a functon satsfyng () s any f W,p loc (RD ). In ths case ( /p a() = r f dx) p, Supported by grant ERFMICT of the TMR programme of the European Unon. Ths research was carred out durng a stay at the Unversdad Autónoma de Madrd. Research partally supported by DGICYT grant P94092, Span.

2 where r s the radus of the ball. The Sobolev Embeddng Theorem says that f p < D, then any functon n W,p loc les n Lr loc for r = Dp/(D p). Saloff-Coste ([S-C]) showed that f S s a smooth manfold equpped wth the metrc and gradent assocated to a certan sub-ellptc operator, and f ( ) /2 ( /2 f f 2 dµ C r f dµ) 2 (2) µ(2) 2 holds for all smooth functons f and for all balls, then t follows that W,2 loc les n Lr loc for r = 2D/(D 2) (D s the doublng dmenson of µ). Haj lasz and Koskela ([HaK2], also [HaK]) obtaned a much more general verson of ths result by showng that f S s a metrc space and f satsfes () wth ( /p a() = C r g dx) p (3) µ(τ) τ for some g L p loc (S) and some τ, then f les n weak Lr loc (S) for r = Dp/(D p). Franch, Pérez, and Wheeden ([FPW]) proved that ths same self-mprovng phenomenon holds for general S and for a very wde class of functonals, descrbed below. Ther results, however, only partally contan those of Haj lasz and Koskela as the class of functonals they consdered only contans (3) for τ =. At frst glance, these results ( [HaK2], [FPW]) appear to be weaker than ether the Embeddng Theorem or the theorem of Saloff-Coste gven that they only show that f belongs to weak L r loc and not to Lr loc. However, t turns out that the weak result always mples the strong result when () holds unformly for a large famly of dfferentable functons, see secton 2. Ths s precsely the stuaton consdered by Saloff-Coste and n the Embeddng Theorem. For a dscusson of the numerous settngs and functonals for whch () holds and for more about the hstory of the problem see [FPW] and [Ha]. Trudnger s Inequalty (the case p = D of the Embeddng Theorem) can also be vewed as a self-mprovement of a 2

3 famly of Poncaré nequaltes. Ths case s treated n [HaK2] and n [MP]. The paper [HaK2] shows that many of the classcal theorems about Sobolev functons actually hold n much more general metrc spaces for functons satsfyng () wth the functonal (3). The man theorem n [FPW] showed that f w s an A weght and the par a and w satsfy the condton D r defned below, then any functon satsfyng () les n weak L r loc (S, dw). Defnton. Let 0 < r < and w be a weght. We say that a satsfes the weghted D r condton f there exsts a fnte constant c such that for each ball and any famly { } of parwse dsjont sub-balls of, a( ) r w( ) c r a() r w(). (4) Here w() = w dµ. The model example of a functonal satsfyng D r s the fractonal average ( ) /p ν() a() = r() α (5) where α 0, 0 < p and ν s a nonnegatve measure. When αp < D ths functonal satsfes D r wth r = Dp/(D αp). In partcular, the functonals (3) satsfy D r for r = Dp/(D p) when τ =. Ths s no longer true when τ >. In ths case they only satsfy D r for any r < Dp/(D p). Let us show that the functonal (5) really does satsfy the stated D r condton. Recall { } r(2 ) D that µ( 2 ) c r( ) µ( ) whenever and 2 are balls wth 2, and note that αr = D( r ) and that p r. Then p { } a( ) r r( ) D r µ( ) = µ( ) p ν( ) r p {c r()d } r p ν( ) r p and consequently a( ) r µ( ) } r {c r()d p ν( ) r p 3

4 } r {c r()d p [ r p ν( )] C { r() D } r p ν() r p = C a() r. efore statng the paper s man result, we ntroduce the notaton g L r, (,w) = sup λ>0 ( w({x : g(x) > λ}) λ w() for the normalzed weak L r norm, where w(e) denotes w dµ for any measurable set E. E We denote the smallest constant c for whch (4) holds by a. We always have a. ) /r Theorem.2 Let 0 be a ball n S and let δ > 0 be gven. Set 0 = ( + δ)k 0. Suppose that the functonal a satsfes the weghted D r condton (4) for some 0 < r < and some w A (µ). If f s a locally ntegrable functon on 0 for whch there exst constants τ and f a > 0 such that for all balls wth 0 f f dµ f a a(τ), (6) then there exsts a constant C ndependent of f and 0 such that f f 0 L r, (0,w) C a f a a(τ 0 ). (7) Ths extends the results of [FPW], by allowng the factor τ, and of [HaK2] by allowng the weght w and much more general functonals than (3). The constant ( + δ)k s far better than the correspondng constant obtaned n [FPW] and s probably the best possble wthn the general context of space of homogeneous type. We wll show n 5 that n general the weak L r norm n nequalty (7) cannot be replaced by the strong L r norm. As mentoned earler, the strong nequalty can be obtaned n certan crcumstances. We dscuss ths n the next secton. We do, however, have the followng corollary: 4

5 Corollary.3 Under the same hypothess as Theorem.2, f 0 < p < r, then there exsts a constant C = C(p) ndependent of f and 0 such that ( ) /p f f 0 p wdµ C a f w( 0 ) a a(τ 0 ). (8) 0 Ths s a consequence of the nequalty ( /p h dν) p c p,r h ν(e), L r, (E,ν) E whch holds n any measure space of fnte measure and whenever n p les between 0 and r. In many stuatons t s possble to replace τ 0 by 0 n both Theorem.2 and Corollary.3. Ths s the case, for example, when the ball 0 satsfes the oman chan condton. The pont s that the nequalty (7) actually holds for all sub-balls of 0 and that the oman chan condton allows us to terate ths nequalty along chans of sub-balls of 0. We refer the nterested reader to Remark 2.6 and the last part of Secton 5 n [FPW]. Other types of chan condtons yeld the same sort of results, see [HaK], for example. Let us compare our stuaton wth that n the Campanato-Morrey spaces. These spaces are defned as those functons on R D satsfyng () wth the functonal a() = δ, where δ s postve. It s well known (and easly verfed) that ths condton s equvalent to requrng that f dµ C δ (9) for all balls. The same property holds replacng the L norm by the L p norm on the left hand sde of these nequaltes. Ths means that the cancellaton does not play any role n these spaces. These spaces do not have the self-mprovng propertes descrbed n the prevous corollary. Indeed, an example of Pccnn ([P]) shows that for any postve δ there s a functon f n the correspondng Campanato-Morrey space whch does not belong to any L p loc (RD ) for any p >. 5

6 Ths example allows us to show that the condton D r s sharp n the followng weak sense. Defne D r,ɛ to be the condton obtaned by rasng w() and w( ) to the power of + ɛ n (4). Then Theorem.2 fals to hold for r > when D r s replaced by D r,ɛ for any postve ɛ. Let S be R D, f be the functon of Pccnn correspondng to δ = ɛ/r, and a() = ɛ/r. Ths functonal satsfes D r,ɛ wth the weght beng Lebesgue measure, and so the hypotheses of the theorem hold wth D r beng replaced by D r,ɛ and wth the nequalty (6) holdng for any ball n S. If the concluson of the theorem were to hold, then, by the corollary, f would belong to L p loc (RD ) for some p >. Ths contradcton proves that the theorem fals n ths case. We are ndebted to F. Leonett for brngng ths example of Pccnn to our attenton. The proof of our theorem s a modfcaton of the classcal proof that bounds the norm of the maxmal functon by the norm of the sharp maxmal functon, see [J], for example. Assocated to the famly of balls = { : x 0 and r δr 0 } (0) are the functons M f(x) = sup :x f dµ and M # f(x) = sup :x f f dµ The heart of the proof, Proposton 4., conssts of provng a good λ nequalty relatng these two functons. 2 Poncaré type nequaltes The purpose of ths secton s to show that we can use our man result to establsh sharp Poncaré nequaltes for dfferental operators. The results here follow from Theorem.2 n 6

7 the same way that the correspondng results n [FPW] follow from the man theorem there. We refer to [FPW] for detals. We now consder a functonal b(, f) of two varables of the form b : F (0, ), where F s an appropate set of functons contaned n L loc (S). In applcatons, the man examples of F are the Lpschtz class, or Sobolev classes, although our results are not restrcted to these classcal spaces. Typcal examples of b are those assocated wth Poncaré nequaltes, namely b(, f) = b X (, f) = r()α Xf dν, () where X s a dfferental operator wth X = 0,.e., wth no zero order term. In partcular, n Eucldean space, X could be m or some other approprate combnaton of partal dervatves. The man property we need s a certan stablty property under truncatons. Ths dea was orgnally ntroduced n [LN] and exploted n [SW], [FGaW], [FLW] and [CLSC]. Gven a nonnegatve functon g, the truncaton τ λ (g) s defned by τ λ (g) = mn{g, 2 λ} mn{g, λ} = 0 f g(x) λ g(x) λ f λ < g(x) 2 λ λ f g(x) > 2 λ. We shall assume that the class F has the followng propertes: f F f + λ, λf F for λ R. f F f F 7

8 f F τ λ ( f ) F for λ 0 We also assume that the followng natural relatonshps between the functonal b and F hold: b(, f) = b(, f + λ) for all f F and λ R b(, f ) b(, f) for all f F There exst r and a constant C such that for any nonnegatve f F, any ball and any sequence λ k of the form {λ k = 2 k λ}, k =, 2,..., λ > 0, we have b(, τ λk (f)) r c b(, f) r. (2) k= Observe, for example, that for the functonal b = b X defned n (), we have b(, τ λk (f)) = r()α X(τ λk (f)) dν = r()α {x :λ k <f(x) λ k+ } and then (2) readly follows snce the domans of ntegraton are dsjont. Xf dν, We assume that b and F have all the propertes lsted above and that b also satsfes the followng condton (a weghted D r condton whch s unform n f for f F) for some r and some w A (µ): b(, f) r w( ) C r b(, f) r w() (3) for all f F, every ball and every famly { } of parwse dsjont sub-balls of. Theorem 2. Let 0 be a ball and let δ > 0. Set 0 = ( + δ)k 0. Suppose that the functonal b and the class F satsfy all of the condtons above (ncludng (3)). If there are constants C and τ, both at least, for whch 8

9 for every f F and 0, then f f dµ c b(τ, f), (4) wth C ndependent of 0 and f F. ( ) /r f f 0 r wdµ C b(τ w( 0 ) 0, f) (5) 0 Corollary 2.2 Let µ and ν be doublng measures, p 0, τ, and X be a dfferental operator for whch ( /p0 f f dµ C r() Xf p 0 dν) (6) ν(τ) τ for all balls and all Lpschtz functons f. Let p 0 p q < and let (w,v) be a par of weghts such that w A (µ), v A p/p0 (ν), and the followng balance condton holds (cf. [ChW]): r( ) r() ( ) /q wdµ C wdµ ( ) /p vdν (7) vdν for all balls, such that. Then ( ( /q f f wdµ) q C r() w() v(τ 0 ) wth C ndependent of f and. τ b 0 Xf p vdν ) /p (8) As remarked after Theorem.2, the concluson of Theorem 2. s vald wth τ 0 replaced by 0 on the rghthand sde of these nequaltes whenever 0 satsfes the oman chan condton. 9

10 If we let X be a collecton of smooth vector felds satsfyng the Hörmander condton on, say, a ball 0, then by Jerson s theorem [Je] we have (6) wth p 0 = τ = and µ = ν = d x. Ths allows us to deduce the man results of [FLW] wthout makng use of the representaton formula obtaned n that paper. For nstance, there s a constant c such that ( ) /Q f f Q dx c r() Xf(x) dx (9) for each ball 0, where Q s the homogeneous dmenson of the vector felds and s the Lebesgue measure of. Ths unweghted estmate s also derved n [GN] by dfferent means. A smlar nequalty can be establshed for non-smooth vector felds X = X λ of Grushn type snce a correspondng Poncaré nequalty (6) was derved n [FGuW] by means of a representaton formula. Recall that the balls n both of these cases are not Eucldean balls, but rather are defned n terms of the metrc assocated wth the vector felds. We ntally obtan these results wth the larger ball η, η > on the rght sde, but we may then take η = by usng the oman chan condton as n [FGuW] and [FLW]. 3 Defntons A quasmetrc d on a set S s a functon d : S S [0, ) satsfyng () d(x, y) = 0 f and only f x = y () d(x, y) = d(y, x) for all x, y () There exsts a fnte constant K such that d(x, y) K(d(x, z) + d(z, y)) for all x, y, z. Gven x S and r > 0, we let (x, r) = {y S : d(x, y) < r} and refer to (x, r) as the ball wth center x and radus r. If a ball s gven, then x denotes the center of, whle 0

11 the radus wll be denoted ether by r or r(). A space of homogeneous type (S, d, µ) s a set S together wth a quasmetrc d and a nonnegatve orel measure µ on S such that µ((x, r)) s fnte for all x S and r > 0, and the doublng condton 0 < µ((x, 2r)) C µ((x, r)), (20) holds for all x S and r > 0. The doublng assumpton (20) s global n nature,.e., t s assumed to hold for all x S and all r > 0. In many mportant cases, however, doublng s a local property, lmted to ponts x n compact sets and to small values of r. In such cases, our man results hold locally. As usual, we say that w s a weght f w s a nonnegatve locally ntegrable functon. For a measurable set E, we denote w(e) = w(y) dµ(y). E We also recall that the weght w s a A (µ) weght f there are postve constants C and α such that w(e) C ( ) α µ(e) w() (2) for every ball and every measurable set E. We denote by D = log C the doublng order of µ, where C s the smallest constant n (20). y teratng (20), we then have µ( ) c µ for every par, of balls such that. ( ) D r() (22) r( )

12 4 Proof of Theorem.2 We fx δ > 0 and a ball 0. We assume that f, a, and w, are as n statement of the theorem. Recall that 0 denotes the ball 0 = ( + δ)k 0. The constants K, D, and c µ wll be refered to as the geometrc data, and any constant that depends only on these wll be called a geometrc constant. Constants denoted by c, c 0,... wll be geometrc constants. All other constants wll depend (at most) on the geometrc data, the A constants of w, τ, and δ. Recall that we have to prove the nequalty f f 0 L r, (0,w) c a f a a(τ 0 ). Observe that wthout loss of generalty we may assume that f a = and that f b0 = 0. Now, f f 0 L r, (0,w) f L r, ( 0,w) + f 0 and usng (6), the doublng property of µ and the fact that f b0 = 0 we fnd that Therefore, f 0 f dµ µ( 0 ) 0 c µ( 0 ) b 0 f dµ c a(τ 0 ). f f 0 L r, (0,w) f L r, ( 0,w) + c a(τ 0 ), and we are left wth showng that for all λ > 0 w({x 0 : f(x) > λ}) C a r λ r a(τ 0 ) r w( 0 ). (23) A key object n our proof s the famly of balls = { : x 0 and r δ r 0 }. (24) 2

13 Ths famly has the followng propertes: 0 (25) τ τ 0 (26) The maxmal functon and the sharp maxmal functon assocated to of a locally ntegrable functon g are, respectvely, M g(x) = sup :x g dµ and M # g(x) = sup :x g g dµ, where we understand that the supremum s zero f x s not contaned n any element of. oth of these functons are zero outsde 0. The Lebesgue dfferentaton theorem and the defnton of the bass mply that g(x) M g(x) for almost every x 0. We defne for λ > 0 the set Ω λ = {x S : M f(x) > λ}. Ths set s contaned n 0. As f(x) M f(x) a.e., the proof of (23) s reduced to showng that the same weak-type estmate holds for Ω λ : For notatonal convenence we ntroduce Notce that w(ω λ ) C a r λ r a(τ 0 ) r w( 0 ) (27) Av() = f dµ and Osc() = f f dµ M f(x) = sup Av() and that M # f(x) = sup Osc() :x :x Snce the average of f over 0 s zero we have Osc( 0 ) = Av( 0 ). We also have, usng (6), that Av( 0 ) a(τ 0 ) and Osc() a(τ). It turns out that t wll be more useful for us to use a slght modfcaton of M # f(x) nstead of M # f(x) tself. We set M # +f(x) = max{m # f(x), Osc( 0 )} (28) 3

14 Ths s smply the sharp maxmal functon of f wth respect to the famly of balls { 0 }. The next lemma contans a key nequalty of good λ type relatng M and M # +f. We defne for λ > 0 the set Σ λ = {x S : M # +f(x) > λ}. Proposton 4. There exst geometrc constants N and ɛ 0 > 0, and a constant C such that for all λ > 0 and all 0 < ɛ < ɛ 0, w(ω Nλ ) C ɛ α w(ω λ ) + Cw(Σ ɛλ ) (29) where α s the A exponent of w, as n (2). From ths good λ nequalty we mmedately obtan, n the usual way, sup λ>0 λ r w(ω λ ) C sup λ>0 λ r w(σ λ ) (30) Combnng ths nequalty and Lemma 4.2 below we obtan (27). Ths means that the proof of the theorem has been reduced to that of Proposton 4.. Lemma 4.2 There exsts a postve constant C such that for all λ > 0 w(σ λ ) C a r a(τ 0 ) r λ r w( 0 ), Proof: The proof s by a standard coverng lemma. If x s n the set Σ λ then there s a ball contanng x and for whch λ < Osc() a(τ). y defnton of M # +f, wll be an element of or just 0. In any case, (we use property (26) here), the ball x = τ whch stll contans x, s a sub-ball of τ 0, and λ < a( x ). Pck a Vtal type cover of Σ λ by balls {P } { x }. Then the balls {P } are parwse dsjont sub-balls of τ 0 and Σ λ cp. Therefore, usng the condton D r and doublng of w, w(σ λ ) w(cp ) C w(p ) C λ r a(p ) r w(p ) C a r λ r a(τ 0 ) r w( 0 ) We now proceed wth the proof of the proposton. The followng lemma, although easy, s nevertheless mportant. 4

15 ( Av( Lemma 4.3 Let such that Av() > λ. Then r() c ) /D 0 ) 0 r( 0 ). λ Indeed, snce 0 we have λ < Av() µ( ( 0 ) Av( r( 0 ) c ) D 0 ) µ Av( r() 0 ) c D 0 ( ) D r(0 ) Av( r() 0 ), and the lemma follows. ( ) D We frst observe that (29) holds for small λ s. Set M = K 2 (2K + ) 2 c0 M, γ =, δ and ɛ 0 = γ. Suppose that λ < γ Av( 0 ). Then ɛλ < Av( 0 ), and because M # +f(x) Osc( 0 ) = Av( 0 ) for every x 0 we have Σ ɛλ = 0. ut Ω Nλ 0, and (29) follows. Henceforth, we wll be assumng that λ γ Av( 0 ). Observe that for such λ s we have by Lemma 4.3 that whenever such that Av() > λ. r() δ M r( 0), Lemma 4.4 Suppose that Ω λ s not empty. Then there exsts a countable famly { } of parwse dsjont balls n such that ) Ω λ, where = K(2K + ). ) For each, r( ) δ M r( 0). ) Av( ) > λ for all. v) If σ and σ 2, then Av(σ ) λ. 5

16 Proof: The proof s based on a stoppng tme or Calderón Zygmund decomposton combned wth a Vtal type lemma. For x Ω λ there exsts a ball wth x such that Av( ) > λ. Let R = R x be defned by R = sup{r() :, s a multple of and Av() > λ} Lemma 4.3 and the range of λ mply that R δ M r( 0). There exsts a ball x whch s a multple of, whose radus satsfes R/2 < r( x ) R, and for whch λ < Av( x ). Ths ball satsfes ), ), and v). Part ) mples that x Ω λ x Ω λ, whle we obvously have Ω λ x Ω λ x. Thus these two nclusons are actually equaltes. Pckng a Vtal type sub-cover of { x } x Ωλ satsfyng ). gves us a famly of parwse dsjont balls { } { x } x Ωλ Fx N = c µ (2K) D >, where c µ s the constant appearng n (22). We have Ω N λ Ω λ, hence w(ω N λ ) = w(ω Nλ Ω λ ) w({x : M f(x) > Nλ}). (3) Set = M. Lemma 4.4 ) mples that. For each fxed, we clam that {x : M f(x) > Nλ} = {x : M (fχ )(x) > Nλ}. (32) The drecton s clear. Suppose now that x les n the set on the left. Then x and there exsts a ball contanng x such that Nλ < f dµ. (33) Lemma 4.3 along wth our choce of γ mples that r() δ M r( 0). In fact, the ball must satsfy r() r( ). Suppose not. Then r() > r( ) and so = (x, 2Kr ). Hence, f dµ µ( ) µ( f dµ c µ (2K) D ) µ( f dµ ) 6

17 Observe that s n, snce the center s n 0 and ts radus s smaller than δ r( 0 ). Now property v) of Lemma 4.4 mples that and as a result µ( ) f dµ λ f dµ c µ (2K) D λ Nλ Ths contradcts (33), and so proves that r() r( ). Ths nequalty n turn mples that K(2K + ) =, and therefore Nλ < Ths yelds the drecton n (32). Now, f dµ = f χ dµ M (fχ )(x), f(x) f(x) f + f f(x) f + Av( ) f(x) f + λ In the last nequalty we have used Lemma 4.4 v) and the fact that s a multple (of at least 2) of that les n. We now fnd, usng (32), that {x : M f(x) > Nλ} {x : M ((f f Let us denote ths last set by E. Recallng (3), we see that )χ )(x) > (N )λ} w(ω N λ ) w(e ). The next stage s to splt the ndex set n two. Say that I f Osc( ) ɛ λ and that II f Osc( ) > ɛ λ. Then w(ω N λ ) I w(e ) + II w(e ) = I + II. 7

18 For I, we use the fact that M s of weak type (, ). We frst control the unweghted measure of E : µ(e ) C λ S f f χ dµ = C µ( ) λ f f µ( dµ ) = C λ Osc( ) µ( ) C ɛ µ( ). Then because w A (µ), we obtan for each I that w(e ) C ɛ α w( ). Consequently, I C ɛ α I w( ) C ɛ α I ( ) w( ) = C ɛ α w C ɛ α w(ω λ ) I The last nequalty s a consequence of Lemma 4.4 ). To estmate II, note that f II and x, then M # +f(x) Osc( ) > ɛλ. Ths means that s contaned n Σ ɛλ. Therefore we have II II w( ) C II w( ) = C w ( ) C w(σ ɛλ ) Combnng these estmates for I and II gves the good λ estmate (29). Ths concludes the II proof of the Theorem. 5 The strong estmate s false n general The pont of ths secton s to show that the L r, weak norm nequalty (7) cannot be replaced n general by the strong L r norm. Consder the subsets of R d gven by S k = k {0} where k s the closed ball centered at (2 k, 0,, 0) and of radus 2 k /0. Each par of the S k have precsely one pont n common, namely, the orgn. Denote the Eucldean metrc restrcted to S k by ρ k. Now set 8

19 S = k=0s k. We can use the ρ k to defne a metrc on S by passng through the orgn. Set ρ k (a, b) ρ(a, b) = ρ k (a, 0) + ρ j (0, b) f a, b k f a k, b j, and k j It s easy to see that ρ s a metrc and that t s equvalent to the usual Eucldean metrc on S. There s an explct formula for ρ: b a f a, b k for some k ρ(a, b) = a + b otherwse If we now equp S wth ths metrc and Lebesgue measure we get a S.H.T. Furthermore, t s compact and d-regular. Ths last means that there are constants c and c 2 such that c r d λ((x, r)) c 2 r d for each x and r. Let r >, and consder the functon f = k=0 2kd/r χ k and defne a functonal a by a() = δ 0() = /r f 0 /r 0 otherwse It s obvous that a D r, n fact ths functonal s a specal case of our model example defned n (5). Observe that f s essentally x d/r restrcted to S. We want to show that f satsfes f f dµ C a(), (34) for every ball. There are two cases to consder: 0 and 0 /. In the frst case (34) holds because = (0, c r ) and so f f 2 f c f c r /r c r /r 9

20 Suppose now that 0 /. The nature of the metrc ρ mples that ntersects only one of the balls k. Ths means that f s constant on and so (34) holds agan. The functon f does not belong to L r loc. Therefore we cannot replace the weak Lr, norm n nequalty (7) by the strong L r norm. References [CLSC] D. akry, T. Coulhon, M. Ledoux and L. Saloff Coste, Sobolev nequaltes n dsguse, Indana U. Math. J. 44 (995), [G] D. L. urkholder and R. F. Gundy, Extrapolaton and nterpolaton of quaslnear operators on martngales, Acta Math. 24 (970), [ChW] S. Chanllo and R. L. Wheeden, Weghted Poncaré and Sobolev nequaltes and estmates for weghted Peano maxmal functons, Amer. J. Math. 07 (985), [FGaW]. Franch, S. Gallot and R. L. Wheeden, Sobolev and sopermetrc nequaltes for degenerate metrcs, Math. Ann. 300 (994), [FGuW]. Franch, C. E. Gutérrez and R. L. Wheeden, Weghted Sobolev Poncaré nequaltes for Grusn type operators, Comm. P. D. E. 9 (994), [GN] N. Garofalo and D. Nheu, Isopermetrc and Sobolev nequaltes for Carnot Carathéodory Spaces and the Exstence of Mnmal Surfaces, Comm. n Pure and Appled Math. 69 (996),

21 [FLW]. Franch, G. Lu and R. L. Wheeden, Representaton formulas and weghted Poncaré nequaltes for Hörmander vector felds, Ann. Inst. Fourer 45 (995), [FPW]. Franch, C. Pérez and R. L. Wheeden, Self Improvng Propertes of John Nrenberg and Poncaré Inequaltes on Spaces of Homogeneous Type, to appear Journal of Functonal Analyss. [GCRdF] J. Garca-Cuerva and J. L. Rubo de Franca, Weghted Norm Inequaltes and Related Topcs, North Holland Math. Studes 6, North Holland, Amsterdam 985. [Ha] P. Haj lasz, Geometrc Approach to Sobolev Spaces and adly Degenerated Ellptc Equatons, Internatonal Centre for Theoretcal Physcs, Publcaton 390 (995) [HaK] P. Haj lasz and P. Koskela, Sobolev meets Poncaré, C. R. Acad. Sc. Pars, 320 (995), [HaK2] P. Haj lasz and P. Koskela, Sobolev met Poncaré, preprnt [Je] D. Jerson, The Poncaré nequalty for vector felds satsfyng Hörmander s condton, Duke Math. J. 53 (986), [J] J. L. Journé, Calderon-Zygmund operators, pseudo dffrental operators and the Cauchy ntegral of Calderon, Lecture Notes n Math. 994 (983) Sprnger-Verlag. [LN] R. Long and F. Ne, Weghted Sobolev nequalty and egenvalue estmates of Schrödnger operators, Lect. Notes n Math. 494 (990), 3 4. [MP] P. MacManus and C. Pérez, Trudnger Inequaltes wthout Dervatves, preprnt 2

22 [P] Pccnn L. C., Incluson tra spaz d Morrey, oll. Un. Mat. Ital. (4) 2 (969), [S-C] L. Saloff-Coste, A Note on Poncaré, Sobolev, and Harnack Inequaltes, Inter. Mat. Res. Notces 2 (992), [SW] E. T. Sawyer and R. L. Wheeden, Weghted nequaltes for fractonal ntegrals on Eucldean and homogeneous spaces, Amer. J. Math. 4 (992), Paul MacManus Department of Mathematcs Unversty of Ednburgh Ednburgh EH9 3JZ Scotland e-mal: pmm@@maths.ed.ac.uk Carlos Pérez Departmento de Matemátcas Unversdad Autónoma de Madrd Madrd Span e-mal: carlos.perez@@uam.es 22