The annular decay property and capacity estimates for thin annuli
aa r X i v : . [ m a t h . A P ] D ec The annular decay property and capacity estimatesfor thin annuli
Anders Bj¨orn
Department of Mathematics, Link¨oping University,SE-581 83 Link¨oping, Sweden ; [email protected] Jana Bj¨orn
Department of Mathematics, Link¨oping University,SE-581 83 Link¨oping, Sweden ; [email protected] Juha Lehrb¨ack
Department of Mathematics and Statistics, University of Jyv¨askyl¨a,P.O. Box 35 ( MaD ) , FI-40014 University of Jyv¨askyl¨a, Finland ; juha.lehrback@jyu.fi Abstract . We obtain upper and lower bounds for the nonlinear variational capacity of thinannuli in weighted R n and in metric spaces, primarily under the assumptions of an annulardecay property and a Poincar´e inequality. In particular, if the measure has the 1-annulardecay property at x and the metric space supports a pointwise 1-Poincar´e inequality at x , then the upper and lower bounds are comparable and we get a two-sided estimate forthin annuli centred at x , which generalizes the known estimate for the usual variationalcapacity in unweighted R n . Most of our estimates are sharp, which we show by supplyingseveral key counterexamples. We also characterize the 1-annular decay property. Key words and phrases : Annular decay property, capacity, doubling measure, metric space,Newtonian space, Poincar´e inequality, Sobolev space, thin annulus, upper gradient, varia-tional capacity, weighted R n .Mathematics Subject Classification (2010): Primary: 31E05; Secondary: 30L99, 31C15,31C45.
1. Introduction
We assume throughout the paper that 1 ≤ p < ∞ and that X = ( X, d, µ ) is ametric space equipped with a metric d and a positive complete Borel measure µ such that 0 < µ ( B ) < ∞ for all balls B ⊂ X . We also let x ∈ X be a fixed butarbitrary point and B r = B ( x , r ) = { x : d ( x, x ) < r } .In this paper, we continue the study of sharp estimates for the variational ca-pacity cap p ( B r , B R ), which we started in Bj¨orn–Bj¨orn–Lehrb¨ack [5]. Therein weconcentrated on the case 0 < r ≤ R , while in the present work we are interestedin the case where the annulus B R \ B r is thin, that is, 0 < R ≤ r < R .Assume for a moment that the measure µ is doubling and that the space X supports a p -Poincar´e inequality. Then it is well known that cap p ( B r , B r ) ≃ µ ( B r ) r − p holds for all 0 < r < diam X . If in addition the exponents 0 < q ≤ q ′ < Anders Bj¨orn, Jana Bj¨orn and Juha Lehrb¨ack ∞ are such that (cid:16) rR (cid:17) q ′ . µ ( B r ) µ ( B R ) . (cid:16) rR (cid:17) q , if 0 < r ≤ R < diam X, (1.1)then, by [5, Theorem 1.1],cap p ( B r , B R ) ≃ ( µ ( B r ) r − p , if p < q,µ ( B R ) R − p , if p > q ′ , (1.2)when 0 < r ≤ R < diam X . However, when r is close to R these estimates areno longer valid; in particular, typically cap p ( B r , B R ) → ∞ when r → R and p > r is close to R , and so it is obviousthat other properties of the space determine the capacities of thin annuli.In (unweighted) R n the following equalities hold for capacities of annuli for all0 < r < R < ∞ (see e.g. [13, p. 35]):cap p ( B r , B R ) = C ( n, p ) | R ( p − n ) / ( p − − r ( p − n ) / ( p − | − p , if p / ∈ { , n } ,C ( n, p ) (cid:0) log Rr (cid:1) − n , if p = n,C ( n, p ) r n − , if p = 1 . When 0 < R ≤ r < R , these yield the estimatecap p ( B r , B R ) ≃ (cid:16) − rR (cid:17) − p m ( B R ) R p , (1.3)where m is the n -dimensional Lebesgue measure.The main goal in this paper is to find general conditions for the space X underwhich estimates similar to (1.3) hold. One such condition is the following measuredecay property, which will play a crucial role in our results. Definition 1.1.
Let η >
0. The measure µ has the η -annular decay ( η -AD ) prop-erty at x ∈ X , if there is a constant C such that for all radii 0 < r < R wehave µ ( B ( x, R ) \ B ( x, r )) ≤ C (cid:16) − rR (cid:17) η µ ( B ( x, R )) . (1.4)If there is a common constant C such that (1.4) holds for all x ∈ X (and allradii 0 < r < R ), then µ has the global η -AD property .For most of the results in this paper it will be enough to require a pointwiseAD property at x , often together with pointwise versions of (reverse) doublingand Poincar´e inequalities. This resembles the situation in [5], where for capacityestimates for nonthin annuli, such as (1.2), it was enough to require doubling (andreverse-doubling) and Poincar´e inequalities to hold pointwise.The global AD property was introduced (under the name volume regularityproperty) in Colding–Minicozzi [8, p. 125] for manifolds and independently byBuckley [7], who called it the annular decay property, for general metric spaces.A variant of the global AD property was already used in David–Journ´e–Semmes [9,p. 41]. Later, the global AD-property has been used by many other authors. Seee.g. Buckley [7] and Routin [19] for more information and applications of the globalAD property. We have not seen any considerations related to the pointwise ADproperty in the literature.If X is a length space and µ is globally doubling, then µ has the global η -ADproperty for some η >
0, see the proof of Lemma 3.3 in [8]. Example 7.2 shows thatthe length space assumption cannot be dropped.The AD property implies the following upper bound for the variational capacity. he annular decay property and capacity estimates for thin annuli 3
Proposition 1.2.
Assume that µ has the η -AD property at x . Then cap p ( B r , B R ) . (cid:16) − rR (cid:17) η − p µ ( B R ) R p , if < r < R. (1.5) If µ has the global η -AD property, then the implicit constant is independent of x . The proof of this result is quite simple, and it is perhaps more interesting thatthere are similar lower bounds and that the estimate is sharp, as we show in Ex-ample 3.3. The sharpness is true even if one assumes that µ has the global η -ADproperty.Lower bounds for capacities are in general considerably more difficult to obtainthan upper bounds. Here we use relatively simple means to obtain lower boundssimilar to the upper bounds, so that we obtain two-sided estimates as in (1.3). Thekey assumption is, as usual, some type of Poincar´e inequality. When both the 1-ADproperty and the 1-Poincar´e inequality are available, our upper and lower boundscoincide, and we obtain the following generalization of (1.3), which is our mainresult. Theorem 1.3.
Assume that X supports a global -Poincar´e inequality and that µ has the global -AD property. Then cap p ( B r , B R ) ≃ (cid:16) − rR (cid:17) − p µ ( B R ) R p , if < R ≤ r < R ≤ diam X . (1.6)As in Proposition 1.2 it is actually enough to require pointwise versions of theassumptions, but then the result is a bit more complicated to formulate; see The-orem 4.3 for the exact statement. Nevertheless, even with global assumptions theparameters are sharp, see Example 7.1. A different type of two-sided estimate forcapacity is obtained in Theorem 4.4.The 1-AD property, which Buckley [7] calls “strong annular decay”, is essential inboth the upper and lower bounds of Theorem 1.3. The 1-AD property is even locallythe best possible AD property, under very mild assumptions, see Proposition 3.4.To further illustrate this useful property, we establish several characterizations ofthe 1-AD property in Section 6. Acknowledgement.
A. B. and J. B. were supported by the Swedish ResearchCouncil. J. L. was supported by the Academy of Finland (grant no. 252108) andthe V¨ais¨al¨a Foundation of the Finnish Academy of Science and Letters. Part of thisresearch was done during several visits of J. L. to Link¨oping University in 2012–15,and one visit of A. B. to the University of Jyv¨askyl¨a in 2015.
2. Preliminaries
In this section we introduce the necessary background notation on metric spaces andin particular on Sobolev spaces and capacities in metric spaces. See the monographsBj¨orn–Bj¨orn [2] and Heinonen–Koskela–Shanmugalingam–Tyson [15] for more ex-tensive treatments of these topics, including proofs of most of the results mentionedin this section.A curve is a continuous mapping from an interval, and a rectifiable curve isa curve with finite length. We will only consider curves which are nonconstant,compact and rectifiable, and thus each curve can be parameterized by its arc length ds . The metric space X is a length space if whenever x, y ∈ X and ε >
0, there isa curve between x and y with length less than (1 + ε ) d ( x, y ).A property is said to hold for p -almost every curve if it fails only for a curvefamily Γ with zero p -modulus, i.e. there exists 0 ≤ ρ ∈ L p ( X ) such that R γ ρ ds = Anders Bj¨orn, Jana Bj¨orn and Juha Lehrb¨ack ∞ for every curve γ ∈ Γ. Following Heinonen–Koskela [14], we introduce uppergradients as follows (they called them very weak gradients).
Definition 2.1.
A Borel function g : X → [0 , ∞ ] is an upper gradient of a function f : X → [ −∞ , ∞ ] if for all curves γ : [0 , l γ ] → X , | f ( γ (0)) − f ( γ ( l γ )) | ≤ Z γ g ds, (2.1)where the left-hand side is considered to be ∞ whenever at least one of the termstherein is infinite. If g : X → [0 , ∞ ] is measurable and (2.1) holds for p -almost everycurve, then g is a p -weak upper gradient of f .The p -weak upper gradients were introduced in Koskela–MacManus [18]. It wasalso shown there that if g ∈ L p ( X ) is a p -weak upper gradient of f , then one canfind a sequence { g j } ∞ j =1 of upper gradients of f such that g j → g in L p ( X ). If f has an upper gradient in L p ( X ), then it has an a.e. unique minimal p -weak uppergradient g f ∈ L p ( X ) in the sense that for every p -weak upper gradient g ∈ L p ( X )of f we have g f ≤ g a.e., see Shanmugalingam [21] and Haj lasz [11]. FollowingShanmugalingam [20], we define a version of Sobolev spaces on the metric measurespace X . Definition 2.2.
For a measurable function f : X → [ −∞ , ∞ ], let k f k N ,p ( X ) = (cid:18)Z X | f | p dµ + inf g Z X g p dµ (cid:19) /p , where the infimum is taken over all upper gradients g of f . The Newtonian space on X is N ,p ( X ) = { f : k f k N ,p ( X ) < ∞} . The quotient space N ,p ( X ) / ∼ , where f ∼ h if and only if k f − h k N ,p ( X ) = 0,is a Banach space and a lattice, see Shanmugalingam [20]. In this paper we assumethat functions in N ,p ( X ) are defined everywhere, not just up to an equivalenceclass in the corresponding function space. This is needed for the definition of uppergradients to make sense. If f, h ∈ N ,p loc ( X ), then g f = g h a.e. in { x ∈ X : f ( x ) = h ( x ) } , in particular g min { f,c } = g f χ { f The Sobolev p -capacity of an arbitrary set E ⊂ X is C p ( E ) = inf u k u k pN ,p ( X ) , where the infimum is taken over all u ∈ N ,p ( X ) such that u ≥ E .The Sobolev capacity is countably subadditive and it is the correct gauge fordistinguishing between two Newtonian functions. If u ∈ N ,p ( X ), then u ∼ v if andonly if they differ only in a set of capacity zero. Moreover, if u, v ∈ N ,p ( X ) and u = v a.e., then u ∼ v . This is the main reason why, unlike in the classical Euclideansetting, we do not need to require the functions admissible in the definition ofcapacity to be 1 in a neighbourhood of E . In (weighted or unweighted) R n , C p isthe usual Sobolev capacity and N ,p ( R n ) and N ,p (Ω) are the refined Sobolev spacesas in Heinonen–Kilpel¨ainen–Martio [13, p. 96], see Bj¨orn–Bj¨orn [2, Theorem 6.7 (ix)and Appendix A.2]. Definition 2.4. The measure µ is doubling at x if there is a constant C > µ ( B ( x, r )) ≤ Cµ ( B ( x, r )) whenever r > . (2.2) he annular decay property and capacity estimates for thin annuli 5 If (2.2) holds with the same constant C > x ∈ X , we say that µ is ( globally ) doubling .We also say that the measure µ is reverse-doubling at x , if there are constants γ, τ > µ ( B ( x, τ r )) ≥ γµ ( B ( x, r )) for all 0 < r ≤ diam X/ τ, and that the measure µ is Ahlfors Q -regular if µ ( B ( x, r )) ≃ r Q for all x ∈ X andall r > µ is doubling at x .If X is connected, or more generally uniformly perfect (see Heinonen [12]), and µ is globally doubling, then µ is also reverse-doubling at every point, with uniformconstants. In the connected case, one can choose τ > γ > x , see e.g. Corollary 3.8 in [2]. If µ is merely doubling at x , thenthe reverse-doubling at x does not follow automatically and has to be imposedseparately whenever needed.The η -AD property at x easily implies that µ is doubling at x . The converseis not true even if X is a length space, as seen by considering m + δ on R , where δ is the Dirac measure at 1, which is doubling at 0, but does not have the η -ADproperty at 0 for any η > 0. (For an absolutely continuous example, consider R equipped with the measure w dx , where w ( x ) = max { , / | x − | (log | x − | ) } .) Definition 2.5. We say that X supports a p -Poincar´e inequality at x if there existconstants C > λ ≥ B = B ( x, r ), all integrablefunctions f on X , and all ( p -weak) upper gradients g of f , Z B | f − f B | dµ ≤ Cr (cid:18)Z λB g p dµ (cid:19) /p , where f B := R B f dµ := R B f dµ/µ ( B ). If C and λ are independent of x , we saythat X supports a ( global ) p -Poincar´e inequality .A nonnegative function w on R n is a p -admissible weight if dµ := w dx isglobally doubling and R n equipped with µ supports a global p -Poincar´e inequality.See Corollary 20.9 in [13] (which is only in the second edition) and Proposition A.17in [2] for why this is equivalent to other definitions in the literature.It is well known that if X supports a global p -Poincar´e inequality, then X isconnected, but in fact even a pointwise p -Poincar´e inequality is sufficient for this. Proposition 2.6. If X supports a p -Poincar´e inequality at x , then X is connectedand C p ( S R ) > for every sphere S R = { x : d ( x, x ) = R } with R < diam X/ . In particular, if µ is globally doubling and X supports a global Poincar´e inequal-ity, then µ is reverse-doubling and τ > Proof. The first part is shown in the same way as in Proposition 4.2 in [2]. For thesecond part, assume that C p ( S R ) = 0. Then 0 is a p -weak upper gradient of χ B R ,as p -almost no curve intersects S R , see [2, Proposition 1.48]. Thus the p -Poincar´einequality is violated for B R . Definition 2.7. Let Ω ⊂ X be open. The variational p -capacity of E ⊂ Ω withrespect to Ω is cap p ( E, Ω) = inf u Z Ω g pu dµ, where the infimum is taken over all u ∈ N ,p ( X ) such that χ E ≤ u ≤ E and u = 0 on X \ E ; we call such functions u admissible for cap p ( E, Ω). Anders Bj¨orn, Jana Bj¨orn and Juha Lehrb¨ack Also the variational capacity is countably subadditive and coincides with theusual variational capacity (see Bj¨orn–Bj¨orn [3, Theorem 5.1] for a proof valid inweighted R n ).Throughout the paper, we write a . b if there is an implicit constant C > a ≤ Cb , where C is independent of the essential parameters involved. Wealso write a & b if b . a , and a ≃ b if a . b . a .Recall that x ∈ X is a fixed but arbitrary point and B r = B ( x , r ). 3. Upper bounds for capacity In this section we prove Proposition 1.2 and show its sharpness. Lemma 3.1. If < r < R , then cap p ( B r , B R ) ≤ µ ( B R \ B r )( R − r ) p . Proof. The function u ( x ) = (cid:18) − dist( x, B r ) R − r (cid:19) + is admissible for cap p ( B r , B R ), and g = ( R − r ) − χ B R \ B r is an upper gradient of u . We thus obtain thatcap p ( B r , B R ) ≤ Z B R g p dµ = µ ( B R \ B r )( R − r ) p . Proof of Proposition . Using the η -AD property at x and Lemma 3.1, we obtainthatcap p ( B r , B R ) ≤ µ ( B R \ B r )( R − r ) p . (cid:18) R − rR (cid:19) η µ ( B R )( R − r ) p = (cid:18) R − rR (cid:19) η − p µ ( B R ) R p . Remark 3.2. Note that if µ has no AD property (in which case we could say that µ has the “0”-AD property), then Lemma 3.1 still givescap p ( B r , B R ) ≤ (cid:16) − rR (cid:17) − p µ ( B R ) R p . This is sharp by Example 3.3 below.Moreover, if µ has local η -AD at x , in the sense that there is some R > < r < R < R as in Proposition 3.4 below, then (1.5) holdswhenever 0 < r < R < R . Similar local versions hold also for our other results.It follows directly from the proof that the constant C from Definition 1.1 canbe used as the implicit constant in (1.5).The following example shows that Proposition 1.2 is sharp. Example 3.3. (This example has been inspired by Example 1.3 in Buckley [7].)Let x = 0, 0 < η < dµ = w dx on R n , n ≥ 1, where w ( x ) = w ( | x | ) and w ( ρ ) = max { , | ρ − | η − } . This is a Muckenhoupt A -weight, by Theorem II.3.4 in Garc´ıa-Cuerva–Rubio deFrancia [10], and it is thus 1-admissible, by Theorem 4 in J. Bj¨orn [6]. It is easilyverified that µ ( B R ) ≃ R n for all R > 0. We also see that µ ( B \ B r ) ≃ (1 − r ) η , if ≤ r ≤ 1. One can check that this is the extreme case showing that the measure he annular decay property and capacity estimates for thin annuli 7 µ has the global η -AD property (and that η is optimal). By Proposition 10.8 inBj¨orn–Bj¨orn–Lehrb¨ack [5], for p > ≤ r < p,w ( B r , B ) ≃ (cid:18)Z r ( w ( ρ ) ρ n − ) / (1 − p ) dρ (cid:19) − p (3.1) ≃ (cid:18)Z r (1 − ρ ) ( η − / (1 − p ) dρ (cid:19) − p ≃ (1 − r ) η − p , which shows that the upper bound in Proposition 1.2 is sharp, with R = 1 fixedand p > d ˜ µ = e w dx and dµ j = w j dx , where e w ( ρ ) := ∞ X j =0 a j w j ( ρ ) , with w j ( ρ ) := w ( q j ρ ) , j = 0 , , ... , for some countable set { q j } ∞ j =0 ⊂ (0 , ∞ ) (e.g. all positive rational numbers) and a j > P ∞ j =0 a j < ∞ . A change of variables shows that µ j ( B R ) = q − nj µ ( B q j R ) ≃ R n and hence ˜ µ ( B R ) ≃ R n . Moreover, for 0 < r ≤ R and x ∈ X ,˜ µ ( B ( x, R ) \ B ( x, r )) = ∞ X j =0 a j µ j ( B ( x, R ) \ B ( x, r ))= ∞ X j =0 a j q − nj µ ( B ( q j x, q j R ) \ B ( q j x, q j r )) . ∞ X j =0 a j q − nj (cid:16) − rR (cid:17) η µ ( B ( q j x, q j R ))= (cid:16) − rR (cid:17) η ˜ µ ( B ( x, R )) , i.e. ˜ µ has the global η -AD property as well. Since e w ≥ a j w j for every j = 0 , , ... ,we also see that η is optimal. Similarly, for every ball B ( x, r ) ⊂ R n , as w is an A -weight, Z B ( x,r ) w j dx = Z B ( q j x,q j r ) w dx . inf B ( q j x,q j r ) w = inf B ( x,r ) w j and summing over all j shows that e w is an A -weight. Finally, using Proposition 10.8in [5] again together with (3.1), we obtain for p > q − j ≤ r < R = q − j ,cap p, e w ( B r , B R ) ≃ q − nj (cid:18)Z Rr (cid:18) ∞ X k =0 a k w ( q k ρ ) (cid:19) / (1 − p ) dρ (cid:19) − p & a j q − nj (cid:18) q − j Z q j r w ( ρ ) / (1 − p ) dρ (cid:19) − p ≃ a j q p − nj cap p,w ( B q j r , B ) ≃ a j (cid:16) − rR (cid:17) η − p ˜ µ ( B R ) R p , and letting r ր R shows that the upper bound in Proposition 1.2 is sharp for all R = q − j .For p = 1 we cannot use Proposition 10.8 in [5]. Instead we do as follows. Let q − j ≤ r < R = q − j , where j = 0 , , ... . Let u be admissible for cap , e w ( B r , B R ), Anders Bj¨orn, Jana Bj¨orn and Juha Lehrb¨ack and let g be an upper gradient of u . We then get, using the unweighted capacitycap ( B r , B R ) and (1.3), Z R n g d ˜ µ ≥ Z S n − Z Rr g ( ρω ) e w ( ρ ) ρ n − dρ dω & a j w j ( r ) Z S n − Z Rr g ( ρω ) ρ n − dρ dω ≥ a j (1 − q j r ) η − cap ( B r , B R ) ≃ a j (cid:16) − rR (cid:17) η − µ ( B R ) R . Taking infimum over all admissible u shows that Proposition 1.2 is sharp also for p = 1.We have the following observation showing that the exponent η = 1 is the largestthat can occur in the AD property, even locally, under a very mild assumption. Proposition 3.4. Let x ∈ X and R > , and assume that µ ( { x } ) = 0 . If (1.4) holds for some η > and all < r < R < R , then η ≤ .Proof. Let 0 < R < R . Using (1.4) we obtain for all integers 1 ≤ k ≤ K , µ ( B R ) = K X i =1 µ ( B iR/K \ B ( i − /RK ) ≤ C K X i =1 (cid:18) − i − i (cid:19) η µ ( B iR/K ) ≤ C k X i =1 i − η µ ( B kR/K ) + C ∞ X i = k +1 i − η µ ( B R ) , (3.2)where C is the constant in (1.4). If η > 1, the series P ∞ i =1 i − η converges and wecan find k such that C P ∞ i = k +1 i − η ≤ . Thus, subtracting the last term in (3.2)from the left-hand side yields µ ( B R ) . µ ( B kR/K ) → , as K → ∞ , which is impossible. Thus η ≤ 4. Lower bounds for capacity We now turn to lower estimates for capacities of thin annuli. The following is ourmain estimate for obtaining the lower bound in Theorem 1.3. As usual for lowerbounds, a key assumption is some sort of a Poincar´e inequality. Theorem 4.1. Assume that ≤ q < p < ∞ , that X supports a q -Poincar´e in-equality at x , and that µ has the η -AD property at x and is reverse-doubling at x with dilation τ > . Then cap p ( B r , B R ) & (cid:16) − rR (cid:17) η ( q − p ) /q µ ( B R ) R p , if < R ≤ r < R ≤ diam X τ . (4.1)If µ has the global η -AD property and supports a global q -Poincar´e inequal-ity, then the implicit constants are independent of x and we may choose τ > X supports global q -Poincar´e inequalities for all q > p . Moreover, Example 7.2 shows that the η -ADproperty cannot be replaced by the assumption that µ is globally doubling or evenAhlfors regular. he annular decay property and capacity estimates for thin annuli 9 Proof. Let u be admissible for cap p ( B r , B R ). Then u = 1 in B r , u = 0 outside B R ,and g u = 0 a.e. outside B R \ B r . The reverse-doubling implies that µ ( B τR \ B R ) & µ ( B R ) from which it follows that | u B τR | < c < 1, and so | u − u B τR | > − c > B R/ . Note that the η -AD property implies that µ is doubling at x . Thus weobtain from the q -Poincar´e inequality at x and H¨older’s inequality that1 . Z B R/ | u − u B τR | dµ . Z B τR | u − u B τR | dµ . R (cid:18)Z B τλR g qu dµ (cid:19) /q . Rµ ( B R ) /q (cid:18)Z B R \ B r g qu dµ (cid:19) /q . Rµ ( B R ) /q µ ( B R \ B r ) /q − /p (cid:18)Z B R \ B r g pu dµ (cid:19) /p . By the η -AD property, µ ( B R \ B r ) . (1 − r/R ) η µ ( B R ). Inserting this into the aboveestimate yields (cid:18)Z B R \ B r g pu dµ (cid:19) /p & µ ( B R ) /q R (cid:16) − rR (cid:17) η (1 /p − /q ) µ ( B R ) /p − /q = µ ( B R ) /p R (cid:16) − rR (cid:17) η ( q − p ) /pq , and (4.1) follows after taking infimum over all admissible u .Theorem 4.1 establishes the lower bound in Theorem 1.3 when p > 1. For p = 1we instead use the following result. In view of Remark 3.2, we can see this as an η = 0 version of Theorem 4.1. Proposition 4.2. Assume that X supports a p -Poincar´e inequality at x , and that µ is doubling at x and reverse-doubling at x with dilation τ > . Then cap p ( B r , B R ) & µ ( B R ) R p , if < R ≤ r < R ≤ diam X τ . (4.2) Moreover, cap p ( B r , B r ) ≃ µ ( B r ) r − p when < r ≤ diam X/ τ .If µ is globally doubling and X supports a global p -Poincar´e inequality, then theimplicit constants are independent of x . Example 7.2 shows that the lower estimate is sharp even under the assumptionsthat µ is globally doubling (or Ahlfors regular) and X supports a global 1-Poincar´einequality. Example 7.1 shows that the p -Poincar´e assumption cannot be weakened,even if it is assumed globally. Example 7.4 shows that the doubling assumptioncannot be dropped (not even if X supports a global 1-Poincar´e inequality and µ is globally reverse-doubling), while Example 7.3 shows that the reverse-doublingassumption cannot be dropped. See also Proposition 7.5. Proof. Let u be admissible for cap p ( B r , B R ). As in the proof of Theorem 4.1 (with q replaced by p ), we get that1 . R (cid:18)Z B τλR g pu dµ (cid:19) /p . Rµ ( B R ) /p (cid:18)Z B R \ B r g pu dµ (cid:19) /p , and (4.2) follows after taking infimum over all admissible u . That cap p ( B r , B r ) ≃ µ ( B r ) r − p follows from this and Lemma 3.1. Theorem 4.3. Assume that X supports a -Poincar´e inequality at x and that µ has the -AD property at x and is reverse-doubling at x with dilation τ > . Then cap p ( B r , B R ) ≃ (cid:16) − rR (cid:17) − p µ ( B R ) R p , if < R ≤ r < R ≤ diam X τ . Even under global assumptions, as in Theorem 1.3, the 1-Poincar´e and 1-ADassumptions cannot be weakened, as shown by Example 7.1. Example 7.3 showsthat the reverse-doubling assumption cannot be dropped, and in particular that it ispossible that µ has the 1-AD property at x and X supports a 1-Poincar´e inequalityat x , but that µ fails to be reverse-doubling at x . Proof. This follows by combining Proposition 1.2 with Theorem 4.1 (for p > 1) andProposition 4.2 (for p = 1). Proof of Theorem . It follows from the global assumptions and Proposition 2.6that X is connected. Hence, X is reverse-doubling at x with τ = . As the im-plicit constants in Theorem 4.3 only depend on the parameters in the assumptions,Theorem 1.3 follows.The following result gives a two-sided estimate of a different form. Theorem 4.4. Assume that µ is globally doubling and that X supports a global p -Poincar´e inequality. Let < R ≤ r < R and δ = R − r . Assume, in addition,that there exists a > such that for every x ∈ B R \ B r there exist x ′ and x ′′ so that B ( x ′ , aδ ) ⊂ B ( x, δ ) ∩ B r and B ( x ′′ , aδ ) ⊂ B ( x, δ ) \ B R . Then cap p ( B r , B R ) ≃ µ ( B R \ B r )( R − r ) p . The balls B ( x ′ , aδ ) in the assumptions of Theorem 4.4 always exist e.g. if X isa length space. The existence of the balls B ( x ′′ , aδ ) is more difficult to guaranteebut there are plenty of spaces where it is true. For example, R n equipped with any p -admissible measure satisfies the assumptions.Observe that the geometric condition is only assumed to hold for the specific r and R under consideration, but whenever the geometric condition is satisfied theimplicit constants in the estimate are independent of r and R . Clearly, the constant2 in B ( x, δ ) is not important and can be replaced by any number ≥ 2. This maybe useful for some spaces containing well distributed holes. The same is true forCorollary 4.5 below.That the geometric assumption cannot be dropped is shown by Example 7.2,while Example 7.1 shows that the Poincar´e assumption cannot be weakened. Ex-amples 7.3 and 7.4 show that the assumption that µ is globally doubling can neitherbe replaced by the assumption that µ is doubling at x , nor by the assumption that µ is globally reverse-doubling, i.e. reverse doubling at every x ∈ X with uniformconstants. Proof. The upper bound follows from Lemma 3.1, so it suffices to prove the lowerbound.Use the Hausdorff maximality principle to find a maximal pairwise disjoint col-lection of balls B ( x j , δ ) with x j ∈ B R \ B r . By maximality, the balls B j = B ( x j , δ )cover B R \ B r . Moreover, since µ is globally doubling, it can be shown that theballs λB j have bounded overlap depending only on λ and the doubling constant of µ . Now, for each j let B ′ j = B ( x ′ j , aδ ) and B ′′ j = B ( x ′′ j , aδ ) as in the assumption ofthe theorem.Let u be admissible for cap p ( B r , B R ). Then u = 1 in B r and u = 0 outside B R .In particular, u = 1 in each B ′ j and u = 0 in each B ′′ j . Since µ is globally doubling, he annular decay property and capacity estimates for thin annuli 11 it follows that u B j ≤ − µ ( B ′′ j ) /µ ( B j ) ≤ c , where c < j . Anapplication of the global p -Poincar´e inequality to B j , and using that g u = 0 a.e.outside B R \ B r , then yields0 < − c ≤ Z B ′ j | u − u B j | dµ . Z B j | u − u B j | dµ . δ (cid:18) µ ( λB j ) Z λB j g pu dµ (cid:19) /p = δ (cid:18) µ ( λB j ) Z λB j ∩ ( B R \ B r ) g pu dµ (cid:19) /p . From this it follows that µ ( λB j ∩ ( B R \ B r )) ≤ µ ( λB j ) . δ p Z λB j ∩ ( B R \ B r ) g pu dµ. Since the balls λB j have bounded overlap and cover B R \ B r , summing over all j gives µ ( B R \ B r ) . δ p Z B R \ B r g pu dµ and taking infimum over all admissible u proves the lower bound.The following corollary partly complements Theorem 4.1 and the lower bound inTheorem 4.3 in the case when the 1-AD property or a 1-Poincar´e inequality are notsatisfied. In particular, if the doubling condition and a 1-Poincar´e inequality holdglobally and p > 1, then the 1-AD condition can be replaced by the geometric con-dition in Theorem 4.4. That the geometric assumption cannot be dropped is shownby Example 7.2, while Example 7.1 shows that the pointwise Poincar´e assumptioncannot be weakened. Corollary 4.5. If the assumptions of Theorem are satisfied and in addition X supports a q -Poincar´e inequality at x for some ≤ q < p , then cap p ( B r , B R ) & (cid:16) − rR (cid:17) q − p µ ( B R ) R p . Proof. This follows directly from Theorem 4.4 and the following Lemma 4.6.The following estimate complements the 1-AD property. In particular, if q = 1then this lower bound, together with the 1-AD property, leads to a sharp two-sidedestimate for µ ( B R \ B r ) when r is close to R . Lemma 4.6. Assume that X supports a q -Poincar´e inequality at x for some ≤ q < ∞ and that µ is doubling at x and reverse-doubling at x . Then µ ( B R \ B r ) & (cid:16) − rR (cid:17) q µ ( B R ) when < R ≤ r < R < diam X τ . Example 7.1 shows that the Poincar´e assumption cannot be weakened, whileExample 7.3 shows that the reverse-doubling condition cannot be omitted. We donot know if the doubling condition can be omitted. Proof. Let u ( x ) = (cid:18) − dist( x, B r ) R − r (cid:19) + . As in the proof of Lemma 3.1, we obtain that Z X g qu dµ ≤ µ ( B R \ B r )( R − r ) q . On the other hand, as in the proof of Theorem 4.1, we get that1 . R (cid:18)Z B τλR g qu dµ (cid:19) /q . Rµ ( B R ) /q µ ( B R \ B r ) /q ( R − r ) , and the claim follows.We can also obtain the following variant of Corollary 4.5. Proposition 4.7. If the assumptions of Theorem are satisfied with p > , thenthere is ≤ q < p such that cap p ( B r , B R ) & (cid:16) − rR (cid:17) q − p µ ( B R ) R p . As seen from the proof below (and those in [1] and [16]) q only depends on p andthe constants in the global doubling condition and the global p -Poincar´e inequality.Moreover, it also follows from the proof that the completion b X of X supports aglobal q -Poincar´e inequality for this q . In fact, it would be enough to require that b X supports a q -Poincar´e inequality at x . Note that Koskela [17, Theorems Aand C] has given counterexamples showing that X may not support any betterPoincar´e inequality than the p -Poincar´e inequality (so Corollary 4.5 is not at ourdisposal). His examples are of the type X = R n \ E , where E ⊂ R n − so theysatisfy the geometric condition in Theorem 4.4 and b X supports a global 1-Poincar´einequality. Proof. Let b X be the completion of X and extend the measure µ so that µ ( b X \ X ) = 0.Then µ is doubling on b X and b X supports a p -Poincar´e inequality, by Proposition 7.1in Aikawa–Shanmugalingam [1]. By Theorem 1.0.1 in Keith–Zhong [16], it followsthat there is 1 ≤ q < p such that b X supports a q -Poincar´e inequality. Now we canapply Lemma 4.6 with respect to b X , and since the estimate of Lemma 4.6 holds forthe measure µ also when restricted to X , this estimate together with Theorem 4.4completes the proof. 5. The blowup of cap p ( B r , B R ) as r → R Theorem 4.1, Corollary 4.5 and Proposition 4.7 all give uniform estimates for theblowup of cap p ( B r , B R ) as r → R . In particular they show thatlim δ → + cap p ( B R − δ , B R ) = lim δ → + cap p ( B R , B R + δ ) = ∞ when the respective assumptions are satisfied.If we are not interested in uniform estimates, but only in the limits above, thenthese can be obtained under considerably weaker assumptions, as we will now show. Proposition 5.1. Assume that ≤ q < p < ∞ and that X supports a q -Poincar´einequality at x . Let R > be such that µ ( X \ B R ) > , which in particular holdsif X \ B R = ∅ . Then lim δ → + cap p ( B R − δ , B R ) = ∞ . (5.1) If in addition µ ( { y : d ( y, x ) = R } ) = 0 , then also lim δ → + cap p ( B R , B R + δ ) = ∞ . (5.2) he annular decay property and capacity estimates for thin annuli 13 If X = B R then cap p ( B R − δ , B R ) = cap p ( B R , B R + δ ) = 0, and thus the condition µ ( X \ B R ) > X supports global q -Poincar´e inequalities for all q > p for neither limit, but we do not know if it is enough to assume that X supportsa p -Poincar´e inequality at x . Moreover, Example 7.2 shows that the assumption µ ( { y : d ( y, x ) = R } ) = 0 cannot be dropped for the limit (5.2) to hold, even if X supports a global 1-Poincar´e inequality. If p = 1 the result fails even if we assumea global 1-Poincar´e inequality, as seen by considering R n or Theorem 1.3. Proof. Assume that 0 < δ < R and let r = R − δ . Let u be admissible forcap p ( B r , B R ). Then, following the ideas in the proof of Theorem 4.1,1 − µ ( B R ) µ ( B R ) ≤ Z B R/ | u − u B R | dµ ≤ µ ( B R ) µ ( B R/ ) Z B R | u − u B R | dµ ≤ CR µ ( B R ) µ ( B R/ ) (cid:18)Z B λR g qu dµ (cid:19) /q = CRµ ( B R ) µ ( B R/ ) µ ( B λR ) /q (cid:18)Z B R \ B r g qu dµ (cid:19) /q ≤ CRµ ( B R ) µ ( B R/ ) µ ( B λR ) /q µ ( B R \ B r ) /q − /p (cid:18)Z B R \ B r g pu dµ (cid:19) /p . Taking infimum over all admissible u shows thatcap p ( B r , B R ) ≥ (cid:18) − µ ( B R ) µ ( B R ) (cid:19) p (cid:18) µ ( B R/ ) µ ( B λR ) /q CRµ ( B R ) (cid:19) p µ ( B R \ B r ) − p/q . (5.3)To see that the first factor in the right-hand side is positive, we note that either X = { y : d ( y, x ) ≤ R } or there is a point y with R < d ( y, x ) < R , as X isconnected by Proposition 2.6. In the former case, µ ( B R \ B R ) = µ ( X \ B R ) > µ ( B R \ B R ) ≥ µ ( B ( y, d ( y, x ) − R )) > 0. Thus the firstfactor in (5.3) is positive, and so is clearly the second one as well. Since the lastfactor tends to ∞ , as δ → + , we see that (5.1) holds.The proof of (5.2) is similar (one can also use (5.3) directly), but in this caseone needs to use that µ ( B R + δ \ B R ) → µ ( { y : d ( y, x ) = R } ) = 0, as δ → + . 6. Characterizations of the 1-AD property Our aim in this section is to characterize the 1-AD property. Theorem 6.1. Let f ( r ) := µ ( B r ) . Then the following are equivalent :(a) µ has the -AD property at x ;(b) f is locally absolutely continuous on (0 , ∞ ) and ρf ′ ( ρ ) . f ( ρ ) for a.e. ρ > f is locally Lipschitz on (0 , ∞ ) and ρf ′ ( ρ ) . f ( ρ ) for a.e. ρ > .If moreover µ is reverse-doubling at x and X supports a -Poincar´e inequalityat x , then also the following condition is equivalent to those above :(d) f is locally Lipschitz on (0 , ∞ ) and ρf ′ ( ρ ) ≃ f ( ρ ) for a.e. ρ with < ρ < diam X/ τ . The assumption of absolute continuity cannot be dropped, as shown by Exam-ple 2.6 in Bj¨orn–Bj¨orn–Lehrb¨ack [5] where X is the usual Cantor ternary set and f is the Cantor staircase function for which f ′ ( ρ ) = 0 ≤ f ( ρ ) /ρ for a.e. ρ > Example 7.1 shows that the 1-Poincar´e assumption (for the last part) cannotbe weakened, even under global assumptions, while Example 7.3 shows that thereverse-doubling assumption cannot be dropped even if X supports a global 1-Poincar´e inequality. Proof. (a) ⇒ (c) By the 1-AD property at x we have for 0 < ε < ρ that f ( ρ ) − f ( ρ − ε ) ε . (cid:18) − ρ − ερ (cid:19) f ( ρ ) ε = f ( ρ ) ρ . (6.1)Since the right-hand side is locally bounded it follows that f is locally Lipschitz on(0 , ∞ ), and thus that f ′ ( ρ ) exists for a.e. ρ > 0. Moreover, by (6.1) we see that f ′ ( ρ ) . f ( ρ ) /ρ whenever f ′ ( ρ ) exists.(c) ⇒ (b) This is trivial.(b) ⇒ (a) Assume that ρf ′ ( ρ ) /f ( ρ ) ≤ M a.e. We have µ ( B R \ B r ) µ ( B R ) = 1 − f ( r ) f ( R ) = 1 − exp( h ( r ) − h ( R )) , (6.2)where h ( ρ ) = log f ( ρ ) is also locally absolutely continuous with h ′ ( ρ ) = f ′ ( ρ ) /f ( ρ ) ≤ M/ρ for a.e. ρ > 0. It follows that h ( r ) − h ( R ) = − Z Rr h ′ ( ρ ) dρ ≥ − M Z Rr dρρ = log (cid:16) rR (cid:17) M . Inserting this into (6.2) yields1 − f ( r ) f ( R ) ≤ − (cid:16) rR (cid:17) M . Finally, Lemma 3.1 from Bj¨orn–Bj¨orn–Gill–Shanmugalingam [4] shows that for t ∈ [0 , { , M } t ≤ − (1 − t ) M ≤ max { , M } t, and applying this with t = 1 − r/R concludes the proof.Thus we have shown that (a)–(c) are equivalent.Now assume that µ is reverse-doubling at x , and that X supports a 1-Poincar´einequality at x .(c) ⇒ (d) Let 0 < ρ < diam X and 0 < ε < ρ . We have already shown that(c) ⇒ (a), so µ has the 1-AD property at x , and in particular µ is doubling at x .Thus Lemma 4.6 (with q = 1) yields f ( ρ ) − f ( ρ − ε ) ε & (cid:18) − ρ − ερ (cid:19) f ( ρ ) ε = f ( ρ ) ρ , showing that f ′ ( ρ ) & f ( ρ ) /ρ whenever f ′ ( ρ ) exists.(d) ⇒ (c) If X is unbounded this is trivial. So assume that X is bounded. If ρ > diam X , then f ( ρ ) = µ ( X ) and f ′ ( ρ ) = 0. As f is locally Lipschitz there is aconstant M such that f ′ ( ρ ) ≤ M for a.e. ρ satisfying diam X < ρ < diam X . Forsuch ρ we have that f ( ρ ) /ρ ≥ f ( B diam X/ ) / diam X and thus ρf ′ ( ρ ) . f ( ρ ) for a.e. ρ > diam X . Together with (d) this yields (c).In R n , the measure of a ball can be obtained by one-dimensional integrationof the surface measures of spheres. To do the same in metric spaces we need thefollowing lemma, which is also useful for verifying the conditions in Theorem 6.1. he annular decay property and capacity estimates for thin annuli 15 Lemma 6.2. Assume that µ is globally doubling and that X supports a global q -Poincar´e inequality for some ≤ q < ∞ . Assume, in addition, that there exists a > such that whenever < r < R ≤ r and δ = R − r , for every x ∈ B R \ B r thereexist x ′ and x ′′ so that B ( x ′ , aδ ) ⊂ B ( x, δ ) ∩ B r and B ( x ′′ , aδ ) ⊂ B ( x, δ ) \ B R .Then the function f ( r ) := µ ( B r ) is locally absolutely continuous on (0 , ∞ ) . The values of the constants a and 2 (in B ( x, δ )) are not important, and bycovering (0 , ∞ ) we can even allow them to be different in different parts. Thus, wecan replace the last assumption by the following condition: for each k ∈ Z thereexist a k > b k ≥ k < r < R ≤ r < k +2 and δ = R − r ,for every x ∈ B R \ B r there exist x ′ and x ′′ so that B ( x ′ , a k δ ) ⊂ B ( x, b k δ ) ∩ B r and B ( x ′′ , a k δ ) ⊂ B ( x, b k δ ) \ B R . Proof. It suffices to show that the measure ν defined on (0 , ∞ ) by ν ( E ) = µ ( { x ∈ X : d ( x , x ) ∈ E } )is absolutely continuous with respect to the Lebesgue measure.Let 0 < r < R ≤ r , δ = R − r and I = ( r, R ). We start by showing that for allmeasurable E ⊂ I , (cid:18) | E || I | (cid:19) q . ν ( E ) ν ( I ) , (6.3)where q is the exponent from the assumed global q -Poincar´e inequality. To this end,set for t > u ( x ) = | E | − Z d ( x ,x )0 χ E ( τ ) dτ, x ∈ X, and note that u = | E | in B r , u = 0 outside B R and g u ( x ) ≤ χ E ( d ( x , x )) a.e.Let the balls B j , B ′ j and B ′′ j be as in the proof of Theorem 4.4. Hence, in thesame way as in the proof of Theorem 4.4, we obtain(1 − c ) | E | . δ (cid:18)Z λB j g qu dµ (cid:19) /q ≤ | I | (cid:18) µ ( { x ∈ λB j : d ( x , x ) ∈ E } ) µ ( λB j ∩ ( B R \ B r )) (cid:19) /q , or equivalently, | E | q µ ( λB j ∩ ( B R \ B r )) . | I | q µ ( { x ∈ λB j : d ( x , x ) ∈ E } ) . Since the balls λB j cover the annulus B R \ B r and have bounded overlap, summingover all j gives (6.3).Now assume for a contradiction that there exists E ⊂ ( r, R ) such that | E | = 0and ν ( E ) > 0. As ν is a Radon measure on (0 , ∞ ), the Lebesgue differentiationtheorem holds with respect to ν , see Remark 1.13 in Heinonen [12]. Thus, thereexists at least one x ∈ E which is a Lebesgue point with respect to ν of the function1 − χ E . Hence, for every ε > 0, there is an interval I ε such that x ∈ I ε ⊂ I and ν ( I ε \ E ) < εν ( I ε ). Applying (6.3) to I ε \ E and I ε in place of E and I gives1 = (cid:18) | I ε \ E || I ε | (cid:19) q . ν ( I ε \ E ) ν ( I ε ) < ε, which is impossible. Thus, the assumption that ν ( E ) > ν is absolutely continuous with respect to the Lebesgue measure onevery interval ( r, R ), and hence on (0 , ∞ ).The following result is now a direct consequence of Theorem 6.1 and Lemma 6.2. Corollary 6.3. Under the assumptions of Lemma , µ has the -AD property at x if and only if the function f ( r ) := µ ( B r ) satisfies ρf ′ ( ρ ) . f ( ρ ) for a.e. ρ > . 7. Counterexamples In this section we provide a number of counterexamples showing that most of ourresults are sharp. The following example shows the sharpness both of the Poincar´eand AD assumptions in Theorem 1.3. It also shows sharpness of the Poincar´eassumptions in several other results. Example 7.1. (Weighted bow-tie) Let X = (cid:8) ( x , ... , x n ) : x + ... + x n ≤ x and − ≤ x ≤ (cid:9) (7.1)as a subset of R n , n ≥ 2, and equip X with the measure dµ = | x | α dx , where α > − n . (Additionally, we can make this example into a length space if we equip X with the inner metric (see [2, Definition 4.41]), which only makes a difference whencalculating distances between the two sides of the origin.) Note that the constant 2in the range of x in (7.1) above was chosen so that we can have R = 1 ≤ diam X below as required in Theorem 1.3.If q ≥ 1, then X supports a global q -Poincar´e inequality if and only if q > n + α or q = 1 ≥ n + α , see Example 5.7 in [2]. Moreover, µ is globally doubling.Let x = ( − , , ... , 0) and η = min { , n + α } . Then for 0 < r < R < diam X we have µ ( B R ) ≃ R n and µ ( B R \ B r ) . (cid:16) − rR (cid:17) η µ ( B R ) , which shows that µ has the η -AD property at x . One can check that this is theextreme case showing that µ has the global η -AD property (and that η is optimal).If 0 < δ < , then µ ( B \ B − δ ) ≃ Z δ ρ α ρ n − dρ ≃ δ n + α , which shows that the Poincar´e assumption in Lemma 4.6 cannot be weakened.Moreover, by Lemma 3.1,cap p ( B − δ , B ) . δ p µ ( B \ B − δ ) ≃ δ n + α − p , which shows that (1.6) fails if n + α > 1, and thus we cannot replace the assumptionof a global 1-Poincar´e inequality in Theorem 1.3 by a global q -Poincar´e inequalityfor any fixed q > 1. Nor can the pointwise q -Poincar´e inequality in Corollary 4.5 bereplaced by assuming that X supports a global q ′ -Poincar´e inequality for any fixed q ′ > q .Conversely, if 0 < δ < and X supports a global p -Poincar´e inequality, i.e. if p > n + α , then a simple reflection argument and [5, Proposition 10.8] imply thatcap p ( B − δ , B ) & cap p ( { } , B (0 , δ )) ≃ δ n + α − p and hence cap p ( B − δ , B ) ≃ δ n + α − p . If η = n + α < 1, then X supports a global 1-Poincar´e inequality and µ has the η -ADproperty at x , but (1.6) fails. Hence we cannot replace the 1-AD assumption inTheorem 1.3 by the η -AD property for any fixed η < 1. In fact, it is only the upperbound in (1.6) that fails. The lower bound therein is still provided by Corollary 4.5.Next, if p = n + α > 1, then X supports a global q -Poincar´e inequality foreach q > p , but not a global p -Poincar´e inequality. Moreover, by Example 5.7in [2], C p ( { } ) = 0 and thus we can test cap p ( B − δ , B ) with u = χ B yieldingcap p ( B − δ , B ) = 0. It also follows from Proposition 2.6 that X does not support he annular decay property and capacity estimates for thin annuli 17 a p -Poincar´e inequality at x . Hence the p -Poincar´e assumption in Proposition 4.2cannot be weakened if p > 1. Moreover, it also follows that it is not enough toassume that X supports global q -Poincar´e inequalities for all q > p in Theorems 4.1and 4.4 when p > 1, as well as for (5.1) in Proposition 5.1 to hold. As in this casewe also have cap p ( B , B δ ) = 0, the same is true for (5.2) in Proposition 5.1.When p = 1 < q we instead choose n and α so that 1 < n + α < q . In particular, X supports a global q -Poincar´e inequality in this case. As above, cap ( B − δ , B ) =0 and X does not support a 1-Poincar´e inequality at x , showing that the Poincar´eassumption in Proposition 4.2 is sharp also for p = 1. It also follows that when p = 1 it is not enough to assume that X supports a global q -Poincar´e inequality forsome fixed q > f ( r ) = µ ( B r ). Let q > n and α sothat 1 < n + α < q . Then µ has the global 1-AD property and X supports a global q -Poincar´e inequality. For ≤ r < R ≤ 1, with R close to r , we see that µ ( B R \ B r ) ≃ (1 − r ) α m ( B R \ B r ) ≃ (1 − r ) α ( R − r )(1 − r ) n − where m is the n -dimensional Lebesgue measure. Hence rf ′ ( r ) = r lim R → r + µ ( B R \ R r ) R − r ≃ r (1 − r ) n + α − r ≃ r n ≃ µ ( B r ) when < r < . Thus condition (d) in Theorem 6.1 fails, which shows that it is not enough to assumethat X supports a global q -Poincar´e inequality for some fixed q > q -Poincar´e inequalities at x for all q > Example 7.2. (This example was introduced by Tessera [22, p. 50] in a differentcontext. See also Routin [19, Section 6] for a more detailed discussion of thisspace.) Let X consist of the intervals (with the natural embedding of R into R )[0 , k − , k ] for even positive k and [ − k , − k − ] for odd positive k , and of thehalf-circles centred at the origin and of radius 2 k lying in the upper half-plane foreven nonnegative k and in the lower half-plane for odd positive k . We equip X with the Euclidean metric d inherited from R and the 1-dimensional Hausdorffmeasure µ . Then X is Ahlfors 1-regular (see Proposition 6.1 in [19]). Moreover, X is bi-Lipschitz equivalent to the half-line [0 , ∞ ) ⊂ R , and hence supports a global1-Poincar´e inequality (by [2, Proposition 4.16]).Let x = 0, k > δ > r = 2 k − δ and R = 2 k + δ . Then µ ( B R \ B r ) ≃ R ≃ µ ( B R ), which shows that µ does not have the η -AD property at x for any η > u which is 1 when | x | < k and 0 when | x | > k ,and decays linearly from 1 to 0 along the half-circle of radius 2 k , it is easy tosee that for the above balls cap p ( B r , B R ) . R − p ≃ µ ( B R ) R − p . Together withProposition 4.2 this shows that cap p ( B r , B R ) ≃ µ ( B R ) R − p , and thus the lowerbound in Proposition 4.2 is sharp. Moreover, the above estimate shows that thegeometric assumption in Theorem 4.4 and Corollary 4.5 cannot be dropped, thatthe η -AD property cannot be replaced by global doubling in Theorem 4.1, and alsothat the assumption µ ( { y : d ( y, x ) = R } ) = 0 cannot be dropped for the limit (5.2)in Proposition 5.1 to hold, even if X supports a global 1-Poincar´e inequality. Example 7.3. Let w be a positive nonincreasing weight function on X = [0 , ∞ ), dµ = w dx and x = 0. Assume that µ ( B ) < ∞ . As w is nonincreasing it iseasy to see that µ is doubling at x . Let f be an integrable function on X with an upper gradient g , and let B = B ( x, r ) ⊂ X be a ball. Then either B = ( a, b ) with0 ≤ a < b or B = [ a, b ) with a = 0 < b . In either case we have Z B | f − f ( a ) | dµ ≤ µ ( B ) Z ba Z ta g ( x ) dx dµ ( t ) = 1 µ ( B ) Z ba Z bx dµ ( t ) g ( x ) dx ≤ µ ( B ) Z ba rw ( x ) g ( x ) dx = r Z B g dµ. It thus follows from Lemma 4.17 in [2] that X supports a global 1-Poincar´e inequal-ity. Moreover, if 0 < R ≤ r < R , then µ ( B R \ B r ) ≤ w ( r )( R − r ) ≤ µ ( B r ) r ( R − r ) ≤ (cid:16) − rR (cid:17) µ ( B R ) . On the other hand, if 0 < r < R , then µ ( B R \ B r ) ≤ µ ( B R ) ≤ (cid:16) − rR (cid:17) µ ( B R ) . Hence µ has the 1-AD property at x .So far we have just assumed that w is nonincreasing, but now assume that w ( x ) = min { , /x } . If R > 2, then by Lemma 3.1,cap p ( B R/ , B R ) . µ ( B R \ B R/ ) R p = log 2 R p , while the right-hand sides (with r = R ) in Theorem 4.1, Proposition 4.2 andTheorem 4.3 are larger than this when R is large enough, since µ ( B R ) → ∞ , as R → ∞ . In particular it follows that µ cannot be reverse-doubling at x (whichalso follows directly from µ ( B R \ B R/ ) = log 2) and that the reverse-doublingassumption in Theorem 4.1, Proposition 4.2 and Theorem 4.3 cannot be dropped.Moreover, as µ ( B R \ B R/ ) = log 2 and µ ( B R ) → ∞ as R → ∞ , the inequalityin Lemma 4.6 fails in this case, showing that the reverse-doubling assumption inLemma 4.6 cannot be dropped either.Write f ( r ) = µ ( B r ), as in Theorem 6.1. Then f ( r ) = 1 + log r for r ≥ 1. For ρ > ρf ′ ( ρ ) = 1 f ( ρ ), so condition (d) in Theorem 6.1 fails. Thus thereverse-doubling assumption in (the last part of) Theorem 6.1 cannot be dropped.Finally, if we instead let w ( r ) = e − r , then µ ( B R ) = 1 − e − R . By Lemma 3.1 wehave for R > p ( B R/ , B R ) ≤ cap p ( B R/ , B R ) . µ ( B R \ B R/ ) R p . e − R/ R p . As µ ( B R \ B R/ ) ≃ e − R/ for R > 1, this shows that the estimate in Theorem 4.4fails in this case. Thus the global doubling assumption therein cannot be replacedby assuming that µ is doubling at x . Note that the geometric assumption inTheorem 4.4 is satisfied in this case. Example 7.4. Let this time w be a positive nondecreasing weight function on X = [0 , ∞ ), dµ = w dx and x = 0. As in Example 7.3 we get that X supports aglobal 1-Poincar´e inequality (estimate using the right end point of the ball insteadof the left end point).Let B be a ball with right end point b . Then µ (2 B \ B ) ≥ µ ( { x ∈ B : x >b } ) ≥ µ ( B ), as w is nondecreasing. Hence µ is globally reverse-doubling, i.e.reverse-doubling at every x ∈ X with uniform constants. he annular decay property and capacity estimates for thin annuli 19 Now let w ( x ) = ( e − /x /x , ≤ x ≤ , e − , x ≥ , which is a continuous nondecreasing function such that µ ( B R ) = e − /R when 0 Assume that X supports a -Poincar´e inequality at x and that µ is reverse-doubling at x with dilation τ > . Then cap ( B r , B R ) & µ ( B r ) r , if < R ≤ r < R ≤ diam X τ . (7.2)Example 7.1 shows that the 1-Poincar´e assumption cannot be weakened, evenif it is assumed globally. Example 7.3 shows that the reverse-doubling assumptioncannot be dropped. Proof. Let u be admissible for cap ( B r , B R ). As in the proof of Theorem 4.1, weget that 1 . Z B r | u − u B τR | dµ ≤ µ ( B τR ) µ ( B r ) Z B τR | u − u B τR | dµ . R µ ( B τR ) µ ( B r ) Z B τλR g u dµ . rµ ( B r ) Z B R \ B r g u dµ, and (7.2) follows after taking infimum over all admissible u . References H. Aikawa and N. Shanmugalingam, Carleson-type estimates for p -harmonic functions and the conformal Martin boundary of John domains inmetric measure spaces, Michigan Math. J. (2005), 165–188.2. A. Bj¨orn and J. Bj¨orn, Nonlinear Potential Theory on Metric Spaces, EMSTracts in Mathematics , European Math. Soc., Z¨urich, 2011.3. A. Bj¨orn and J. Bj¨orn, The variational capacity with respect to nonopensets in metric spaces, Potential Anal. (2014), 57–80.4. A. Bj¨orn, J. Bj¨orn, J. Gill and N. Shanmugalingam, Geometricanalysis on Cantor sets and trees, to appear in J. Reine Angew. Math. , arXiv:1302.3887 . A. Bj¨orn, J. Bj¨orn and J. Lehrb¨ack, Sharp capacity estimates for annuliin weighted R n and in metric spaces, Preprint, , arXiv:1312.1668 . J. Bj¨orn, Poincar´e inequalities for powers and products of admissible weights, Ann. Acad. Sci. Fenn. Math. (2001), 175–188.7. S. Buckley, Is the maximal function of a Lipschitz function continuous?, Ann.Acad. Sci. Fenn. Math. (1999), 519–528.8. T. H. Colding and W. P. Minicozzi II, Liouville theorems for harmonicsections and applications, Comm. Pure Appl. Math. (1998), 113–138.9. G. David, J.-L. Journ´e and S. Semmes, Op´erateurs de Calder´on–Zygmund,fonctions para-accr´etives et interpolation, Rev. Mat. Iberoam. :4 (1985), 1–56.10. J. Garc´ıa-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalitiesand Related Topics, North-Holland, Amsterdam, 1985.11. P. Haj lasz, Sobolev spaces on metric-measure spaces, in Heat Kernels andAnalysis on Manifolds , Graphs and Metric Spaces ( Paris , ), Contemp.Math. , pp. 173–218, Amer. Math. Soc., Providence, RI, 2003.12. J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York,2001.13. J. Heinonen, T. Kilpel¨ainen and O. Martio, Nonlinear Potential Theoryof Degenerate Elliptic Equations, J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces withcontrolled geometry, Acta Math. (1998), 1–61.15. J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, SobolevSpaces on Metric Measure Spaces, New Math. Monographs , Cambridge Univ.Press, Cambridge, 2015.16. S. Keith and X. Zhong, The Poincar´e inequality is an open ended condition, Ann. of Math. (2008), 575–599.17. P. Koskela, Removable sets for Sobolev spaces, Ark. Mat. (1999), 291–304.18. P. Koskela and P. MacManus, Quasiconformal mappings and Sobolevspaces, Studia Math. (1998), 1–17.19. E. Routin, Distribution of points and Hardy type inequalities in spaces ofhomogeneous type, J. Fourier Anal. Appl. (2013), 877–909.20. N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces tometric measure spaces, Rev. Mat. Iberoam. (2000), 243–279.21. N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. (2001), 1021–1050.22. R. Tessera, Volume of spheres in doubling metric measured spaces and ingroups of polynomial growth, Bull. Soc. Math. France135