Sharp Besov capacity estimates for annuli in metric spaces with doubling measures

We obtain precise estimates, in terms of the measure of balls, for the Besov capacity of annuli and singletons in complete metric spaces. The spaces are only assumed to be uniformly perfect with respect to the centre of the annuli and equipped with a doubling measure.


Introduction
Capacities are intimately related to function spaces in the sense that various properties, such as quasicontinuity and Lebesgue points, of functions in such spaces are measured by capacity.Capacities also reflect metric and measure-theoretic properties of the underlying space on which they are defined.For example, it is well known that the p-capacity of a spherical condenser in R n with 0 < 2r ≤ R reflects the dimension of the space as follows, cap p (B(x, r), B(x, R)) (1.1) Capacities also play an important role in fine potential theory and appear in the famous Wiener criterion characterizing boundary regularity for various equations, such as ∆ p u = 0 (Maz ′ ya [29] and Kilpeläinen-Malý [22], with the p-capacity as in (1.1)) and the fractional p-Laplace equation (−∆ p ) s u = 0 (Kim-Lee-Lee [23], using a fractional Besov capacity from [9]).
In this paper we study Besov capacities on a complete metric space Y = (Y, d) equipped with a doubling measure ν.Analogously to (1.1), we are primarily interested in estimates for (thick) annuli, i.e. of the capacity for a ball B(x 0 , r) within B(x 0 , R) where 0 < 2r ≤ R.
Throughout the paper we assume that 1 ≤ p < ∞.We also fix a point x 0 and let B r = B(x 0 , r) be the open ball with radius r and centre x 0 .
The following are our main results.
Theorem 1.1.Assume that Y is a complete metric space which is uniformly perfect at x 0 and equipped with a doubling measure ν.Let p > 1 and 0 < θ < 1.Then for all 0 < 2r ≤ R ≤ 1  4 diam Y , with the comparison constants in "≃" independent of x 0 , r and R.
Here, cap θ,p is the Besov condenser capacity defined for bounded sets E ⋐ Ω as , (1.4) where the infimum is taken over all u such that 0 ≤ u ≤ 1 everywhere, u = 1 in a neighbourhood of E and supp u ⋐ Ω, see Definition 3.2.
The Euclidean spaces and their subsets, equipped with the Lebesgue measure or weighted measures w dx, and even singular doubling measures, are included as special cases of our results.We emphasize that we do not assume any Poincaré inequalities for upper gradients on Y (as in Gogatishvili-Koskela-Zhou [17, Section 4] and Koskela-Yang-Zhou [25]).This makes our results applicable to many disconnected spaces and spaces carrying few rectifiable curves, including fractals.
To formulate the next two results we need the following exponent sets: r R q for 0 < r < R ≤ 1 , S 0 = {s > 0 : ν(B r ) r s for 0 < r ≤ 1}, S 0 = {s > 0 : ν(B r ) r s for 0 < r ≤ 1}, These sets were introduced in Björn-Björn-Lehrbäck [5] to capture the local behaviour of the measure at x 0 .For example, for the Lebesgue measure in R n , The subscript 0 in the above definitions stands for the fact that the inequalities are required to hold for small radii.It is easily verified (see [5, Lemmas 2.4 and 2.5]) that the exponent sets can equivalently be defined using 0 < r ≤ ΘR ≤ R 0 for any fixed 0 < Θ < 1 and R 0 > 0, even though the implicit comparison constants in " " and " " will then depend on Θ and R 0 .
All of these sets are intervals.The reason for introducing them as sets is that they may or may not contain their endpoints q 0 = sup Q 0 , s 0 = sup S 0 , s 0 = inf S 0 and q 0 = inf Q 0 . (1.5) Note that q 0 < ∞ if ν is doubling, and that q 0 > 0 if Y is also uniformly perfect at x 0 (see Heinonen [19,Exercise 13.1]).When p > 1 and θp < q 0 or θp > q 0 , Theorem 1.1 provides us with exact estimates for the capacity cap θ,p (B r , B R ) in terms of ν(B r ) or ν(B R ).When p = 1, Theorem 1.1 cannot be used, but we obtain the following similar estimates for cap θ,p (B r , B R ) by using results from Björn-Björn-Lehrbäck [5], which cover all p ≥ 1.The borderline cases θp = max Q 0 and θp = min Q 0 are considered in Theorem 9.1.See also Remarks 9.2 and 9.3.Theorem 1.2.Assume that Y is a complete metric space which is uniformly perfect at x 0 and equipped with a doubling measure ν.Let 0 < θ < 1 and 0 In both cases, the comparison constants in "≃" depend on R 0 .Moreover, the lower bound in (1.6) implies p ∈ Q 0 , while the lower bound in In Ahlfors regular spaces, estimates (1.6) and (1.7) were given in Lehrbäck-Shanmugalingam [27], and used to show that Besov-norm-preserving homeomorphisms between such spaces are quasisymmetric.
In many situations it is important whether singletons have zero or positive capacity.In the following result, we characterize these cases in terms of the exponent sets S 0 and S 0 .
Theorem 1.3.Assume that Y is a complete metric space which is uniformly perfect at x 0 and equipped with a doubling measure ν.Let 0 < θ < 1.
In Anttila [1], the numbers s 0 and s 0 are called the upper and lower local dimensions of µ at x 0 , while q in Remark 9.2 is called the pointwise Assouad dimension of µ at x 0 .(See [5,Lemma 2.4] for why the definitions of s 0 and s 0 in [1] are equivalent to those in (1.5).)In [6], s 0 played a decisive role in determining the sharp integrability properties for p-harmonic Green functions and their gradients.
On R n , the spaces defined by means of the energy integral in (1.4) are often called fractional Sobolev spaces and are the traces of Sobolev spaces on sufficiently nice domains ).As such, they are suitable as boundary values for various Dirichlet problems and appear in boundary regularity results for elliptic differential equations (Kristensen-Mingione [26]).
They also play an important role in nonlocal problems, such as the fractional p-Laplace equation (−∆ p ) s u = 0.These problems have attracted a lot of attention in the past two decades, see e.g.Kim-Lee-Lee [23], Korvenpää-Kuusi-Lindgren [24] and Lindgren-Lindqvist [28], to name just a few.
Our approach to the above estimates is based on extensions of Besov functions from Y to hyperbolic fillings of Y , together with estimates from Björn-Björn-Lehrbäck [5], [6] for p-capacities associated with Sobolev spaces.More precisely, we use the comparison between Besov seminorms of functions on Y and the Dirichlet energy of their extensions to a uniformized hyperbolic filling of Y , obtained in [8].These constructions and comparisons are done in Section 5.
However, since the results in [8] only cover bounded spaces, special care has to be taken for unbounded Y .This is done in Section 7 by replacing Y with a suitably chosen bounded subset, so that the restriction of ν is still doubling.Even when Y is bounded, it is only biLipschitz equivalent to the boundary of the uniformized hyperbolic filling of Y , which would in turn put serious restrictions on the allowed radii r and R in our estimates.In Section 6 we therefore show how to replace Y by a carefully constructed enlarged space so that the involved capacities are comparable and all radii ≤ 1 4 diam Y can be treated.Along the way, in Sections 3 and 4, we prove various fundamental properties of Besov capacities in metric spaces, both for doubling and nondoubling measures, including in some cases also θ ≥ 1.Finally, in Sections 8 and 9, we prove Theorems 1.1-1.3.
As mentioned above, we use hyperbolic fillings to obtain our main results.It would be interesting to find more direct proofs.On the other hand, our technique shows that there is a direct correspondence between these results and the corresponding results for Sobolev spaces in [5] and [6].
Acknowledgement. A. B. and J. B. were supported by the Swedish Research Council, grants 2020-04011 resp.2018-04106.Part of this research was done when the authors visited Institut Mittag-Leffler in the autumn of 2022 during the programme Geometric Aspects of Nonlinear Partial Differential Equations.We thank the institute for their hospitality and support.

Preliminaries
In this section we assume that X = (X, d) is a metric space equipped with a Borel measure µ such that 0 < µ(B) < ∞ for every ball B ⊂ X.To avoid pathological situations we also assume that all metric spaces, considered in this paper, contain at least two points.
As is often customary we extend µ, and other measures, as outer measures defined on all sets.This plays a role at least in Proposition 3.3 (ii).
A metric space is proper if all closed bounded sets are compact.We denote balls in X by B(x, r) = {y ∈ X : d(y, x) < r} and let λB(x, r) = B(x, λr).
All balls in this paper are open.In metric spaces it can happen that balls with different centres and/or radii denote the same set.We will however use the convention that a ball comes with a predetermined centre and radius.
The space X is uniformly perfect at x if there is a constant κ > 1 such that B(x, κr) \ B(x, r) = ∅ whenever B(x, κr) = X. (2.1) In fact, it then follows that (2.1) holds whenever B(x, r) = X, since if B(x, κr) = X then B(x, κr) \ B(x, r) = X \ B(x, r) = ∅.The space X is uniformly perfect if it is uniformly perfect at every x with the same constant κ.This definition coincides with the one in Heinonen [19,Section 11.1], see therein for more on the history of this assumption.We do not know if pointwise uniform perfectness has been used before.Note that X is uniformly perfect with any κ > 1 if X is connected.
The measure µ is doubling if there is a doubling constant Similarly, µ is reverse-doubling at x, if there are constants C, κ > 1 such that µ(B(x, κr)) ≥ Cµ(B(x, r)) for all 0 < r < diam X/2κ. (2.2) By continuity of the measure, the estimate (2.2) holds also if r = diam X/2κ < ∞, as required in Björn-Björn-Lehrbäck [5].If µ is doubling, it is easy to see that X is uniformly perfect at x if and only if µ is reverse-doubling at x. (For necessity we can choose any κ > κ, and for sufficiency any κ > 2κ.)If µ is doubling and X is connected, then µ is reverse-doubling at every x with any κ > 1.
Throughout the paper, we write a b if there is an implicit constant C > 0 such that a ≤ Cb, and analogously a b if b a, and a ≃ b if a b a.The implicit comparison constants are allowed to depend on the standard parameters.We will carefully explain the dependence in each case.See Remarks 8.2 and 9.3 for the dependence in Theorems 1.1 and 1.2.
Sometimes, when dealing with several different spaces simultaneously, we will write B X , d X , Q X 0 , q X 0 etc. to indicate that these notions are taken with respect to the metric space X.

Besov spaces and capacities
In this section we assume that Y = (Y, d) is a proper metric space equipped with a Borel measure ν such that 0 < ν(B) < ∞ for every ball B ⊂ Y .We also assume that θ > 0, and emphasize that in this section θ ≥ 1 is allowed.
Here and elsewhere, the integrand should be interpreted as zero when y = x.
The Besov space B θ p (Y ) consists of the functions u such that the Besov norm This space is a Banach space, see Remark 9.8 in Björn-Björn-Shanmugalingam [8].
(The norm (3.1) is equivalent to the one in [8], but the norm-capacity C θ,p below exactly coincides with the one in [8].) We restrict our attention to Besov spaces with two indices (i.e."q = p").Such Besov spaces are often called fractional Sobolev spaces or Sobolev-Slobodetskiȋ spaces, although Besov spaces seem to be the most common name in the metric space literature.
When ν is also reverse-doubling (or equivalently, uniformly perfect), further equivalent definitions can be found in Gogatishvili-Koskela-Zhou [17, Theorem 4.1 and Proposition 4.1], for example that the Besov space B θ p (Y ) considered here coincides with the corresponding Haj lasz-Besov space.By [16, Lemmas 6.1 and 6.2], it is also related to fractional Haj lasz spaces, considered already in Yang [31].See these papers for the precise definitions and earlier references to the theory on R n and on Ahlfors regular metric spaces.
We are interested in two types of Besov capacities, the norm-capacity and the condenser capacity.
where the infimum is taken over all u ∈ B θ p (Y ) such that u = 1 in a neighbourhood of E and 0 ≤ u ≤ 1 everywhere.Such u are called admissible for C θ,p (E).
By truncation it follows that one can equivalently take the infimum over all u such that u ≥ 1 in a neighbourhood of E. As usual, when requiring that u ≥ 1 or 0 ≤ u ≤ 1 everywhere we mean that there is a representative of u satisfying these requirements.By E ⋐ Ω we mean that E is a compact subset of Ω.
where the infimum is taken over all u such that 0 ≤ u ≤ 1 everywhere, u = 1 in a neighbourhood of E and supp u ⋐ Ω.Such u are called admissible for cap θ,p (E).
The corresponding capacities for Sobolev spaces are called Sobolev resp.variational capacity in [2].Condenser capacities are also often called "relative".
Our main estimates remain the same (up to changes in implicit constants) when the seminorm is replaced by an equivalent seminorm.However, some of the basic properties, such as subadditivity, are not directly transferable, although the proofs often are, so we include them here.
The monotonicity (i) is trivial, while (ii) follows directly from the definition.The property (iii) follows from the fact that C θ,p is an outer capacity (by definition), i.e.
and elementary properties of compact sets, see Nuutinen [30,Section 3].As for (iv), Nuutinen [30] only obtains quasi-subadditivity since he works in a more general setting in which the countable subadditivity does not always hold.We therefore provide a proof.
Proof of (iv).We may assume that the right-hand side is finite.Let ε > 0. For each i = 1, 2, ... , choose u i admissible for C θ,p (E i ) with Letting ε → 0 completes the proof.
Again, (i) is trivial, while (ii) follows from elementary properties of compact sets since cap θ,p is an outer capacity (by definition).The proof of (iii) is similar to the proof of Proposition 3.3 (iv).
In the Ahlfors Q-regular case with p > Q > 1, these facts were stated in Costea [14] with a comment that the proof is essentially the same as in Costea [13,Theorem 3.1].His proof of (iii) uses reflexivity.Our proof is considerably shorter and also covers the case 1 ≤ p ≤ Q as well as the non-Ahlfors regular case.

Capacity estimates when ν is doubling
In this section we assume that Y is a complete metric space equipped with a doubling measure ν and that 0 < θ < 1.
Note that Y is proper, see Björn-Björn [2, Proposition 3.1].The comparison constants in this section are independent of the choice of x 0 , they depend only on θ, p and C ν , unless said otherwise.
Our next aim is to deduce the following result, which will be important later on.Note that with comparison constants also depending on κ and Θ.
We split the proof of Proposition 4.1 into two parts.We begin with the lower bound, which holds also when θ ≥ 1. Proposition 4.2.Assume that Y is uniformly perfect at x 0 with constant κ, and that 0 with comparison constant also depending on κ.
Proof.Let u be admissible for cap θ,p (B r , B R ).As B 2R = Y it follows from the uniform perfectness that there exists z ∈ B 2κR \ B 2R .Since B(z, R) ∩ B R = ∅ and d(x, y) ≤ (2κ + 2)R for all x ∈ B(z, R) and y ∈ B r , we get that Taking infimum over all u admissible for cap θ,p (B r , B R ) concludes the proof.
To prove the upper bound in Proposition 4.1 we will use the following simple lemma, which will also be used when proving Lemma 4.5.
Proposition 4.4.Assume that 0 < 2r ≤ R. Then Proof.Let u be a 3/R-Lipschitz function admissible for cap θ,p (B R/2 , B R ).The doubling property and symmetry in x and y imply that Integrating the estimate from Lemma 4.3 over Applying the last estimate with R replaced by 2r gives Proof of Proposition 4.1.This follows directly from Propositions 4.2 and 4.4, together with the doubling property.Hence, by Lemma 4.3, Letting ε → 0 completes the proof.
Note that the converse of Lemma 4.5 does not hold in general; consider e.g. a compact Y in which case cap θ,p (Y, Y ) = 0 (as u ≡ 1 is admissible) while C θ,p (Y ) ≥ ν(Y ) > 0. Nevertheless, we will prove the following characterization.The following simple observation will serve as a Poincaré type inequality.We will use it to prove Proposition 4.6 as well as Lemma 4.9 below.Lemma 4.7.If u = 0 outside a bounded measurable set Ω and K ⊂ Y \ Ω is a bounded measurable set with ν(K) > 0, then for every z ∈ K, where R = diam K + sup{d(x, y) : x ∈ K and y ∈ Ω}.
Proof.Since u = 0 outside Ω, and in particular in K, and B(x, d(x, y)) ⊂ B(z, R) for all x ∈ K and y ∈ Ω, we see that .
As an immediate consequence of Proposition 4.6 and monotonicity, we obtain the following characterization.2 diam Y , then u ≡ 1 is admissible for cap θ,p ({x 0 }, B r ) and thus cap θ,p ({x 0 }, B r ) = 0. On the other hand C θ,p ({x 0 }) > 0 if θp > 1, by Theorem 1.3.This shows that the range in (c) is sharp.
When Ω is a ball, the following result gives more precise information than Lemma 4.5.Lemma 4.9.Assume that E ⊂ B r .Then If, moreover, Y is uniformly perfect at x 0 with constant κ and Y \ B 3κr = ∅, then with comparison constant also depending on κ.Taking infimum over all u admissible for C θ,p (E) proves the first inequality in the statement of the lemma.
For the second inequality, note that every u admissible for cap θ,p (E, B 2r ) is admissible also for C θ,p (E).Next, use the uniform perfectness at x 0 to find z ∈ B 3κr \ B 3r .Lemma 4.7 with Ω = B 2r , K = B(z, r) and R = (3κ + 3)r, together with ν(B(z, R)) ν(B(z, r)), then implies that Taking infimum over all u admissible for cap θ,p (E, B 2r ) concludes the proof.
We conclude this section by comparing capacities with respect to different underlying spaces.Since the seminorm [u] θ,p is nonlocal, the sets where u vanishes cannot be ignored.
and that for a.e.x ∈ Ω, where . Then with comparison constants also depending on the implicit comparison constants in (4.1) and (4.2).
Proof.Note that u is admissible for cap

Hyperbolic fillings and capacities on them
In this section, we let Z be a compact metric space with 0 < diam Z < 1 and equipped with a doubling measure ν.Let x 0 ∈ Z be fixed.
Hyperbolic fillings will be one of our main tools when obtaining precise estimates for condenser capacities, based on results from Björn-Björn-Lehrbäck [5] and [6].We follow the construction of the hyperbolic filling in Björn-Björn-Shanmugalingam [8] as follows: Fix two parameters α, τ > 1 and let X be a hyperbolic filling of Z, constructed with these parameters.More precisely, fix z 0 ∈ Z and set A 0 = {z 0 }.Note that Z = B Z (z 0 , 1).By a recursive construction using Zorn's lemma or the Hausdorff maximality principle, for each positive integer n we can choose a maximal We define the "vertex set" The vertices v = (x, n) and v ′ = (y, m) form an edge (denoted [v, v ′ ]) in the hyperbolic filling X of Z if and only if |n − m| ≤ 1 and The hyperbolic filling X, seen as a metric space with edges of unit length, is a Gromov hyperbolic space.Its uniformization X ε with parameter ε = log α is given by the uniformized metric where d( • , v 0 ) denotes the graph distance to the root v 0 = (z 0 , 0) of the hyperbolic filling, ds denotes the arc length, and the infimum is taken over all paths in X joining x to y.We let be the completion of X ε and equip it with the measure µ β as in [8, Section 10], with Roughly, µ β is obtained by smearing out the measure ν(B(x, α −n )) to the edges adjacent to the vertex (x, n) ∈ V .Note that e ε = α and that σ, appearing in various places in [8], is in our case By [8,Proposition 4.4], Z and ∂ ε X are biLipschitz equivalent (since σ = 1) and we will therefore identify them as sets.However, the metrics are different.More precisely, by [8,Proposition 4.4], where C 1 = 1/2τ α, C 2 = 4α (l+1) /ε and l is the smallest nonnegative integer such that α −l ≤ τ − 1.
Clearly, Z is uniformly perfect at x 0 if and only if ∂ ε X is uniformly perfect at x 0 (with comparable constants κ and κ Note however that because of (5.1), if E and Ω are balls with respect to Z, they will not in general be balls with respect to ∂ ε X, which needs to be taken into account when estimating the capacity of annuli.We will need the Newtonian (Sobolev) space on X ε and its Sobolev and condenser capacities, which we now introduce, see [2] or [8] for further details.
A property holds for p-almost every curve in X ε if the curve family Γ for which it fails has zero p-modulus, i.e. there is ρ ∈ L p (X ε ) such that γ ρ ds = ∞ for every γ ∈ Γ.A measurable function g : where the left-hand side is ∞ whenever at least one of the terms therein is infinite.If u has a p-weak upper gradient in L p (X ε ), then it has a minimal p-weak upper gradient g u ∈ L p (X ε ) in the sense that g u ≤ g a.e. for every p-weak upper gradient g ∈ L p (X ε ) of u.
For measurable u : , where the infimum is taken over all p-weak upper gradients of u.The Newtonian space on X ε is N 1,p (X ε ) = {u : u N 1,p (Xε) < ∞}.
Note that functions in N 1,p are defined pointwise everywhere, not only up to a.e.equivalence classes.
The Sobolev capacity of E ⊂ X ε is where the infimum is taken over all u ∈ N 1,p (X ε ) such that u = 1 on E. The where the infimum is taken over all u ∈ N 1,p (X ε ) such that u = 1 on E and u = 0 outside Ω.For both capacities we call such u admissible.By [8,Theorem 10.3], µ β is doubling and supports a 1-Poincaré inequality on X ε , i.e. there exist C, λ > 0 such that for each ball B = B Xε (x, r) and for all integrable functions u and 1-weak upper gradients g of u on λB, where As X ε is geodesic, the dilation constant in the 1-Poincaré inequality can be chosen to be λ = 1 and moreover X ε supports a (p, p)-Poincaré inequality (i.e. ( 5 From (5.1) and (5.4) it follows that the exponent sets are the same for Z and ∂ ε X, and that, for q > 0, and similarly for the other exponent sets.Here we consider the exponent sets around x 0 ∈ Z.Moreover, if Z is uniformly perfect at x 0 , then the doubling property implies that all the exponent sets for ν and µ β are nonempty, see [5, (2.3)].Hence and similarly for the other exponents.In particular, We are now ready to estimate capacities on ∂ ε X in terms of capacities on X ε , with the aim to later translate them to capacities on the original space Z.The comparison constants in this section are independent of the choice of x 0 , and depend only on θ, p, C ν , α and τ , unless said otherwise.
Taking infimum over all u admissible for cap Xε p (E, B Xε 3R/2 ) shows that where the last comparison follows from [2, Lemma 11.22].
The following lemma controls how function values spread from Z to the hyperbolic filling.This property will be essential for obtaining a reverse estimate to Lemma 5.1.(5.1).Let U be the extension of u to X ε , given by extended piecewise linearly (with respect to d ε ) to each edge in X ε , and then by Then U ≡ b in B Xε (x, r).
Proof.Let y ∈ B X ε (x, r)\Z.Then y belongs to an edge [v 1 , v 2 ], where v 1 = (x 1 , n 1 ) and v 2 = (x 2 , n 2 ) are vertices in the hyperbolic filling.We can assume that Since also we have for all z ∈ B Z (x j , α −nj ), j = 1, 2, that using also (5.1), and thus u(z) = b by assumption.It follows from (5.7) that U (x j ) = b, j = 1, 2, and hence also U (y) = b.For y ∈ B X ε (x, r) ∩ Z, the claim follows from (5.8).
Theorem 5.3.Assume that Z is uniformly perfect at x 0 with constant κ, and that with comparison constants also depending on κ.
Proof.The " " inequality follows from Lemma 5.1, so it remains to show the " " inequality.As both capacities are outer, we may assume that E is open in Z.Let u be admissible for cap ∂εX θ,p (E, B ∂εX 2R ).Consider the extension U to X ε given by (5.7) and (5.8).It then follows from [8,Theorem 12.1] that U = u ν-a.e. in ∂ ε X and As E is open, it easily follows (e.g. from Lemma 5.2 and (5.8)) that U ≡ 1 on E.
Then, by [2, Theorem 2.15], Since ηU is admissible for cap Xε p (E, B Xε 2R ), we have In view of (5.10), it therefore suffices to estimate the last term in (5.11) using the first integral on the right-hand side.To this end, let B = B Xε 4κεR , where κ ε is the uniform perfectness constant of ∂ ε X at x 0 .We will use that where Θ is independent of U and B and only depends on ε, κ ε and C µ β .We postpone the verification of this to the end of the proof and first show how it leads us to conclude the proof.The Minkowski inequality yields where Inserting this into (5.12) and using the (p, p)-Poincaré inequality for µ β gives Together with (5.10) and (5.11)  Taking infimum over all u admissible for cap ∂εX θ,p (E, B ∂εX 2R ) shows the " " inequality in (5.9).
It remains to show that Θ > 0. By the uniform perfectness and the fact that B ∂εX 3R = Z, there is some x ∈ B ∂εX 3κεR \ B ∂εX 3R .Then u = 0 in B ∂εX (x, R) and hence by Lemma 5.2, U = 0 in B X ε (x, R/L).From this and the doubling property of µ β we see that where Θ only depends on ε, κ ε and C µ β .
Since we will be interested in the Besov capacity of annuli in Z, we next relate it to the capacity of annuli in X ε .
Theorem 5.4.Assume that Z is uniformly perfect at x 0 with constant κ.Let 0 < 2r ≤ R and L = α(1 with comparison constant also depending on κ.
Proof.We proceed as in the proof of Theorem 5.

Enlarging Y
In this section we assume that Y is a compact metric space, equipped with a doubling measure ν, and let x 0 ∈ Y be fixed.
Our aim is to embed Y into a suitable larger metric space Z.We will do this recursively, but in this section we only do the first step.
As Y is compact there is a point x 1 such that d(x 1 , x 0 ) = max x∈Y d(x, x 0 ).Let Y ′ = (Y ′ , d ′ , ν ′ ) be a copy of Y = (Y, d, ν), where we identify x 1 with its copy, but do not identify any other points.Equip Y = Y ∪ Y ′ with the measure and the metric d so that Lemma 6.1.The measure ν is doubling on Y with doubling constant C ν ≤ 2C ν and satisfies Moreover, if Y is uniformly perfect at x 0 with constant κ, then Y is uniformly perfect at x 0 with constant κ = max{κ, 2}.
Proof.That (6.1) holds follows directly from the construction.A similar formula holds if x ∈ Y ′ .It follows that ν is doubling with C ν ≤ 2C ν .
As for the uniform perfectness, let r > 0 be such that B Y κr = Y .Then κr ≤ 3d(x 0 , x 1 ) and hence r ≤ 3 2 d(x 0 , x 1 ).If r ≤ d(x 0 , x 1 ) then x 1 ∈ Y \ B Y r and thus there is The constant 2 in κ in Lemma 6.1 is optimal as seen by the following example: Let Y = [−1, 0] ∪ {1} with x 0 = 0 and x 1 = 1.In this case Y is uniformly perfect at 0 with any constant κ > 1, but Y is only uniformly perfect at 0 with constant κ ≥ 2.
From now on we call the distance d and the measure ν also on Y .The assumption of uniform perfectness cannot be dropped since cap Proof.We shall use Lemma 4.10.If Y ′ = Y , there is nothing to prove, so assume that Y and hence, using that ν ′ = ν| Y ′ is doubling by Lemma 7.1, we obtain Thus, with I(x, y) as in Lemma 4.10, On the other hand, for y ∈ A j := B Y 2 j+1 δ \ B Y 2 j δ , j = 0, 1, ... , we have d(x, y) ≃ 2 j R and ν(B Y (x, d(x, y))) ν(A j ). Hence An application of Lemma 4.10, together with Lemma 7.1 (c), concludes the proof.

Proof of Theorem
with comparison constants depending only on Θ 1 , Θ 2 , θ, p and C ν .
Proof.By the doubling property of ν, we have ν(B ρ ) ≃ ν(B r ) for all Θ 1 r ≤ ρ ≤ Θ 2 r.The statement now follows by direct calculation of the integral.
Then the following hold for 0 < 2r ≤ R ≤ R 0 , with comparison constants depending on R 0 , but independent of x 0 , r and R. Remark 9.2.If Y is unbounded, then Theorems 1.2 and 9.1 hold with R 0 = ∞ if Q 0 , q 0 , Q 0 and q 0 are replaced by r R q for 0 < r < R < ∞, , q = inf Q.
Remark 9.3.The comparison constants in Theorems 1.2 and 9.1 are independent of the choice of x 0 , but depend on θ, p, C ν , R 0 and the uniform perfectness constant κ.
In Theorem 1.2 (a) they also depend on the choice of q ∈ (θp, q 0 ) from the proof of [5, Proposition 6.1] leading to the estimate (9.2), and on the comparison constant appearing in the definition of q ∈ Q 0 .
Similarly, in Theorem 1.2 (b) the constants also depend on the choice of q ∈ (q 0 , θp) from the proof of [5, Proposition 6.1] leading to the estimate (9.3), and on the comparison constant appearing in the definition of q ∈ Q 0 .
In Theorem 9.1 the dependence is similar but with q = θp.In Remark 9.2, the dependence is instead in terms of Q and Q.
Together with Björn-Björn-Lehrbäck [6, Theorem 4.2] and the doubling property of µ β this shows that Proof of Theorem 1.2.The upper bounds follow directly from Proposition 4.4.For the lower bounds we first construct Z as in the proof of Theorem 1.1.Since the left-and right-hand sides in (1.6) and (1.7) scale in the same way, we may without loss of generality assume that 0 < diam Z < 1.