The annular decay property and capacity estimates for thin annuli

We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted $\mathbf{R}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar\'e inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincar\'e inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted $\mathbf{R}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also characterize the $1$-annular decay property.


Introduction
We assume throughout the paper that 1 ≤ p < ∞ and that X = (X, d, µ) is a metric 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 0 ∈ X be a fixed but arbitrary point and B r = B(x 0 , r) = {x : d(x, x 0 ) < r}.
In this paper, we continue the study of sharp estimates for the variational capacity cap p (B r , B R ), which we started in Björn-Björn-Lehrbäck [3]. Therein we concentrated on the case 0 < 2r ≤ R, while in the present work we are interested in the case where the annulus B R \ B r is thin, that is, 0 < 1 2 R ≤ r < R. Assume for a moment that the measure µ is doubling and that the space X supports a p-Poincaré inequality. Then it is well known that cap p (B r , B 2r ) µ(B r )r −p holds for all 0 < r < 1 8 diam X. If in addition the exponents 0 < q ≤ q < ∞ are such that if p > q , (1.2) when 0 < 2r ≤ R < 1 4 diam X. However, when r is close to R these estimates are no longer valid; in particular, typically cap p (B r , B R ) → ∞ when r → R and p > 1. Moreover, the difference in the growth bounds in (1.1) does not play any role when r is close to R, and so it is obvious that other properties of the space determine the capacities of thin annuli.
In (unweighted) R n the following equalities hold for capacities of annuli for all 0 < r < R < ∞ (see e.g. [11, p. 35 if p = n, C(n, p)r n−1 , if p = 1.
When 0 < 1 2 R ≤ r < R, these yield the estimate where m is the n-dimensional Lebesgue measure. The main goal in this paper is to find general conditions for the space X under which estimates similar to (1.3) hold. One such condition is the following measure decay property, which will play a crucial role in our results. Definition 1.1. Let 0 < η ≤ 1. The measure µ has the η-annular decay (η-AD) property at x ∈ X, if there is a constant C such that for all radii 0 < r < R we have µ(B(x, R) \ B(x, r)) ≤ C 1 − r R η µ(B(x, R)). (1.4) If there is a common constant C such that (1.4) holds for all x ∈ X (and all radii 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 pointwise AD property at x 0 , often together with pointwise versions of (reverse) doubling and Poincaré inequalities. This resembles the situation in [3], where for capacity estimates for nonthin annuli, such as (1.2), it was enough to require doubling (and reverse-doubling) and Poincaré inequalities to hold pointwise.
The global AD property was introduced (under the name volume regularity property) in p. 125] for manifolds and independently by Buckley [5], who called it the annular decay property, for general metric spaces. A variant of the global AD property was already used in David-Journé-Semmes [7, p. 41]. Later, the global AD-property has been used by many other authors. See e.g. Buckley [5] and Routin [15] for more information and applications of the global AD property. We have not seen any considerations related to the pointwise AD property in the literature.
If X is a length space and µ is globally doubling, then µ has the global η-AD property for some η > 0, see the proof of Lemma 3.3 in [6]. The AD property implies the following upper bound for the variational capacity. Proposition 1.2. Assume that µ has the η-AD property at x 0 . Then If µ has the global η-AD property, then the implicit constant is independent of x 0 .
The proof of this result is quite simple, and it is perhaps more interesting that there are similar lower bounds and that the estimate is sharp, as we show in Example 3.3. The sharpness is true even if one assumes that µ has the global η-AD property.
Lower bounds for capacities are in general considerably more difficult to obtain than upper bounds. Here we use relatively simple means to obtain lower bounds similar to the upper bounds, so that we obtain two-sided estimates as in (1.3). The key assumption is, as usual, some type of Poincaré inequality. When both the 1-AD property and the 1-Poincaré inequality are available, our upper and lower bounds coincide, and we obtain the following generalization of (1.3), which is our main result. Theorem 1.3. Assume that X supports a global 1-Poincaré inequality and that µ has the global 1-AD property. Then As in Proposition 1.2 it is actually enough to require pointwise versions of the assumptions, but then the result is a bit more complicated to formulate; see Theorem 4.3 for the exact statement. Nevertheless, even with global assumptions the parameters are sharp, see Example 6.1.
The 1-AD property, which is the best possible annular decay and which Buckley [5] calls "strong annular decay", is essential in both the upper and lower bounds of Theorem 1.3. To further illustrate this useful property, we establish several characterizations of the 1-AD property in Section 5.

Preliminaries
In this section we introduce the necessary background notation on metric spaces and in particular on Sobolev spaces and capacities in metric spaces. See the monographs Björn-Björn [1] and Heinonen-Koskela-Shanmugalingam-Tyson [13] for more extensive treatments of these topics, including proofs of most of the results mentioned in this section.
A curve is a continuous mapping from an interval, and a rectifiable curve is a 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 is a 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 curve family Γ with zero p-modulus, i.e. there exists 0 ≤ ρ ∈ L p (X) such that γ ρ ds = ∞ for every curve γ ∈ Γ. Following Heinonen-Koskela [12], we introduce upper gradients as follows (they called them very weak gradients).
where the left-hand side is considered to be ∞ whenever at least one of the terms therein is infinite. If g : X → [0, ∞] is measurable and (2.1) holds for p-almost every curve, then g is a p-weak upper gradient of f .
The p-weak upper gradients were introduced in Koskela-MacManus [14]. It was also shown there that if g ∈ L p (X) is a p-weak upper gradient of f , then one can find 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 upper gradient 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 [17] and Haj lasz [9]. Following Shanmugalingam [16], we define a version of Sobolev spaces on the metric measure space X.
, is a Banach space and a lattice, see Shanmugalingam [16]. In this paper we assume that functions in N 1,p (X) are defined everywhere, not just up to an equivalence class in the corresponding function space. This is needed for the definition of upper gradients to make sense. If f, h ∈ N 1,p loc (X), then where the infimum is taken over all u ∈ N 1,p (X) such that u ≥ 1 on E.
The Sobolev capacity is countably subadditive and it is the correct gauge for distinguishing between two Newtonian functions. If u ∈ N 1,p (X), then u ∼ v if and only if they differ only in a set of capacity zero. Moreover, if u, v ∈ N 1,p (X) and u = v a.e., then u ∼ v. This is the main reason why, unlike in the classical Euclidean setting, we do not need to require the functions admissible in the definition of capacity to be 1 in a neighbourhood of E. In (weighted or unweighted) R n , C p is the usual Sobolev capacity and N 1,p (R n ) and N 1,p (Ω) are the refined Sobolev spaces as in Heinonen-Kilpeläinen-Martio [11, p. 96], see Björn-Björn [1, Theorem 6.7 (ix) and Appendix A.2].
If (2.2) holds with the same constant C > 0 for all 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 > 1 such that The global doubling condition is often assumed in the metric space literature, but for many of our estimates it will be enough to assume that µ is doubling at x. If X is connected, or more generally uniformly perfect (see Heinonen [10]), and µ is globally doubling, then µ is also reverse-doubling at every point, with uniform constants. In the connected case, one can choose τ > 1 arbitrarily and find γ > 1 independent of x, see e.g. Corollary 3.8 in [1]. If µ is merely doubling at x, then the reverse-doubling at x does not follow automatically and has to be imposed separately whenever needed.
The η-AD property at x 0 easily implies that µ is doubling at x 0 . The converse is not true even if X is a length space, as seen by considering m + δ 1 on R, where δ 1 is the Dirac measure at 1, which is doubling at 0, but does not have the η-AD property at 0 for any η > 0. (For an absolutely continuous example, consider R equipped with the measure w dx, where w( Definition 2.5. We say that X supports a p-Poincaré inequality at x if there exist constants C > 0 and λ ≥ 1 such that for all balls B = B(x, r), all integrable functions f on X, and all (p-weak) upper gradients g of f , It is well known that if X supports a global p-Poincaré inequality, then X is connected, but in fact even a pointwise p-Poincaré inequality is sufficient for this. Proposition 2.6. If X supports a p-Poincaré inequality at x 0 , then X is connected and In particular, if µ is globally doubling and X supports a global Poincaré inequality, then µ is reverse-doubling and τ > 1 can be chosen arbitrarily.
Proof. The first part is shown in the same way as in Proposition 4.2 in [1]. For the second 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 [1,Proposition 1.48]. Thus the p-Poincaré inequality is violated for B 2R .
where the infimum is taken over all u ∈ N 1,p (X) such that χ E ≤ u ≤ 1 in Ω and u = 0 on X \ Ω; we call such functions u admissible for cap p (E, Ω).
Also the variational capacity is countably subadditive and coincides with the usual variational capacity (see Björn-Björn [2, Theorem 5.1] for a proof valid in weighted R n ).
Throughout the paper, we write a b if there is an implicit constant C > 0 such that a ≤ Cb, where C is independent of the essential parameters involved. We  also write a b if b a, and a b if a b a. Recall that x 0 ∈ X is a fixed but arbitrary point and B r = B(x 0 , r).

Upper bounds for capacity
In this section we prove Proposition 1.2 and show its sharpness.
Proof. The function is admissible for cap p (B r , B R ), and g = (R − r) −1 χ B R \Br is an upper gradient of u. We thus obtain that Proof of Proposition 1.2. Using the η-AD property at x 0 and Lemma 3.1, we obtain that 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 gives This is sharp by Example 3.3 below. Moreover, if µ has local η-AD at x 0 , in the sense that there is some R 0 > 0 such that (1.4) holds for all 0 < r < R < R 0 , then also (1.5) holds whenever 0 < r < R < R 0 . Similar local versions hold for our other results, as well.
It follows directly from the proof that the constant C from Definition 1.1 can be used as the implicit constant in (1.5).
The following example shows that Proposition 1.2 is sharp.  [5].) Let x 0 = 0, 0 < η < 1 and dµ = w dx on R n , n ≥ 1, where w(x) = w(|x|) and This is a Muckenhoupt A 1 -weight, by Theorem II.3.4 in García-Cuerva-Rubio de Francia [8], and thus µ = w dx is globally doubling and supports a global 1-Poincaré inequality, by Theorem 4 in J. Björn [4]. It is easily verified that µ(B R ) R n for all R > 0. We also see that µ(B 1 \ B r ) (1 − r) η , if 1 2 ≤ r ≤ 1. One can check that this is the extreme case showing that the measure µ has the global η-AD property (and that η is optimal). By Proposition 10.8 in Björn-Björn-Lehrbäck [3], for p > 1 and 1 2 ≤ r < 1, which shows that the upper bound in Proposition 1.2 is sharp when p > 1.
For p = 1 we cannot use Proposition 10.8 in [3]. Instead we observe that when 1 2 ≤ r < 1 then which shows the sharpness also for p = 1.

Lower bounds for capacity
We now turn to lower estimates for capacities of thin annuli. The following is our main estimate for obtaining the lower bound in Theorem 1.3. As usual for lower bounds, a key assumption is some sort of a Poincaré inequality.
Theorem 4.1. Assume that 1 ≤ q < p < ∞, that X supports a q-Poincaré inequality at x 0 , and that µ has the η-AD property at x 0 and is reverse-doubling at x 0 with dilation τ > 1. Then If µ has the global η-AD property and supports a global q-Poincaré inequality, then the implicit constant is independent of x 0 and we may choose τ > 1 arbitrarily, see the proof of Theorem 1.3 below. Example 6.2 shows that the reverse-doubling assumption cannot be dropped, while Example 6.1 shows that it is not enough to assume that X supports global q-Poincaré inequalities for all q > p.
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 τ R \ B R ) µ(B R ) from which it follows that |u B τ R | < c < 1, and so |u − u B τ R | > 1 − c > 0 in B R/2 . Note that the η-AD property implies that µ is doubling at x 0 . Thus we obtain from the q-Poincaré inequality at x 0 and Hölder's inequality that By the η-AD property, µ(B R \B r ) (1−r/R) η µ(B R ). Inserting this into the above estimate yields   Assume that X supports a p-Poincaré inequality at x 0 , and that µ is doubling at x 0 and reverse-doubling at x 0 with dilation τ > 1. Then Moreover, cap p (B r , B 2r ) µ(B r )r −p when 0 < r ≤ diam X/4τ . If µ is globally doubling and X supports a global p-Poincaré inequality, then the implicit constants are independent of x 0 . Example 6.1 shows that the p-Poincaré assumption cannot be weakened, even if it is assumed globally, while Example 6.2 shows that the reverse-doubling assumption cannot be dropped.
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 that , and (4.2) follows after taking infimum over all admissible u. That cap p (B r , B 2r ) µ(B r )r −p follows from this and Lemma 3.1.

Theorem 4.3.
Assume that X supports a 1-Poincaré inequality at x 0 and that µ has the 1-AD property at x 0 and is reverse-doubling at x 0 with dilation τ > 1. Then Even under global assumptions, as in Theorem 1.3, the 1-Poincaré and 1-AD assumptions cannot be weakened, as shown by Example 6.1. Example 6.2 shows that the reverse-doubling assumption cannot be dropped, and in particular that it is possible that µ has the 1-AD property at x 0 and X supports a 1-Poincaré inequality at x 0 , but that µ fails to be reverse-doubling at x 0 .
Proof of Theorem 1.3. It follows from the global assumptions and Proposition 2.6 that X is connected. Hence, X is reverse-doubling at x 0 with τ = 3 2 . As the implicit constants in Theorem 4.3 only depend on the parameters in the assumptions, Theorem 1.3 follows.

Characterizations of the 1-AD property
Our aim in this section is to prove the following characterizations of the 1-AD property.
If moreover µ is reverse-doubling at x 0 and X supports a 1-Poincaré inequality at x 0 , 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 0 < ρ < diam X/2τ .
The assumption of absolute continuity cannot be dropped, as shown by Example 2.6 in Björn-Björn-Lehrbäck [3] 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 > 0. Example 6.1 shows that the 1-Poincaré assumption (for the last part) cannot be weakened, even under global assumptions, while Example 6.2 shows that the reverse-doubling assumption cannot be dropped even if X supports a global 1-Poincaré inequality.
Proof. (a) ⇒ (c) By the 1-AD property at x 0 we have for 0 < ε < ρ that 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 (5.1) we see that and Thus we have shown that (a)-(c) are equivalent.
Now assume that µ is reverse-doubling at x 0 , and that X supports a 1-Poincaré inequality at x 0 .
The following estimate completes the proof of Theorem 5.1. It also complements the 1-AD property. In particular, if q = 1 then this lower bound, together with the 1-AD property, leads to a sharp two-sided estimate for the measures of thin annuli.
Lemma 5.2. Assume that X supports a q-Poincaré inequality at x 0 for some 1 ≤ q < ∞ and that µ is doubling at x 0 and reverse-doubling at x 0 . Then Example 6.1 shows that the Poincaré assumption cannot be weakened, while Example 6.2 shows that the reverse-doubling condition cannot be omitted. We do not know if the doubling condition can be omitted.

Proof. Let
As in the proof of Lemma 3.1, we obtain that Hence, as in the proof of Theorem 4.1, we get that from which the claim follows.

Counterexamples
In this section we provide counterexamples showing that most of our results are sharp. The following example shows the sharpness of the Poincaré and the AD assumptions in Theorem 1.3 and in several other results.
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 [1,Definition 4.41]), which only makes a difference when calculating distances between the two sides of the origin.) Note that the constant 2 in the range of x 1 in (6.1) above was chosen so that we can have R = 1 ≤ 1 3 diam X below as required in Theorem 1.3.
If η = n + α < 1, then X supports a global 1-Poincaré inequality and µ has the η-AD property at x 0 , but (1.6) fails. Hence we cannot replace the 1-AD assumption in Theorem 1.3 by the η-AD property for any fixed η < 1. Next, if p = n + α > 1, then X supports a global q-Poincaré inequality for each q > p, but not a global p-Poincaré inequality. Moreover, by Example 5.7 in [1], C p ({0}) = 0 and thus we can test cap p (B 1−δ , B 1 ) with u = χ B1 yielding cap p (B 1−δ , B 1 ) = 0. It also follows from Proposition 2.6 that X does not support a p-Poincaré inequality at x 0 . Hence the p-Poincaré assumption in Proposition 4.2 cannot be weakened if p > 1. Moreover, it also follows that it is not enough to assume that X supports global q-Poincaré inequalities for all q > p in Theorem 4.1.
When p = 1 < q we instead choose n and α so that 1 < n + α < q. In particular, X supports a global q-Poincaré inequality in this case. As above, cap 1 (B 1−δ , B 1 ) = 0 and X does not support a 1-Poincaré inequality at x 0 , showing that the Poincaré assumption in Proposition 4.2 is sharp also for p = 1.
Finally, let f (r) = µ(B r ) as in Theorem 5.1. Let q > 1 and choose n and α so that 1 < n + α < q. Then µ has the global 1-AD property and X supports a global q-Poincaré inequality. For 1 2 ≤ r < R ≤ 1, with R close to r, we see that 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−1 r r n µ(B r ) when 1 2 < r < 1.
Thus condition (d) in Theorem 5.1 fails, which shows that it is not enough to assume that X supports a global q-Poincaré inequality for some fixed q > 1 in (the last part of) Theorem 5.1.
We do not have a counterexample to the conclusion of Theorem 4.3 which supports pointwise q-Poincaré inequalities at x 0 for all q > 1.
Example 6.2. Let w be a positive nonincreasing weight function on X = [0, ∞), dµ = w dx and x 0 = 0. Assume that µ(B 1 ) < ∞. As w is nonincreasing it is easy to see that µ is doubling at x 0 . 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) with 0 ≤ a < b or B = [a, b) with a = 0 < b. In either case we have Moreover, if 0 < 1 2 R ≤ r < R, then On the other hand, if 0 < 2r < R, then Hence µ has the 1-AD property at x 0 . So far we have just assumed that w is nonincreasing, but now assume that w(x) = min{1, 1/x}. If R > 2, then by Lemma 3.1, while the right-hand sides (with r = 1 2 R) in Theorem 4.1, Proposition 4.2 and Theorem 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 0 (which also follows directly from µ(B R \ B R/2 ) = log 2) and that the reverse-doubling assumption in Theorem 4.1, Proposition 4.2 and Theorem 4.3 cannot be dropped.
Moreover, as µ(B R \ B R/2 ) = log 2 and µ(B R ) → ∞ as R → ∞, the inequality in Lemma 5.2 fails in this case (for every q ≥ 1), showing that the reverse-doubling assumption in Lemma 5.2 cannot be dropped either.