Weak capacity and modulus comparability in Ahlfors regular metric spaces

Let $(Z,d,\mu)$ be a compact, connected, Ahlfors $Q$-regular metric space with $Q>1$. Using a hyperbolic filling of $Z$, we define the notions of the $p$-capacity between certain subsets of $Z$ and of the weak covering $p$-capacity of path families $\Gamma$ in $Z$. We show comparability results and quasisymmetric invariance. As an application of our methods we deduce a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected Ahlfors $Q$-regular metric spaces.


Introduction
Modulus of path families has become an important tool in studying metric spaces with a rich supply of rectifiable paths. The existence of sufficiently many rectifiable paths, however, is not guaranteed. For instance, starting from a metric space (X, d), one sees the "snowflaked" metric space (X, d α ) with α ∈ (0, 1) carries no nonconstant rectifiable paths. Accordingly, traditional modulus techniques are insufficient in many cases.
In this paper we will study metric measure spaces (Z, d, µ) which are compact, connected, and Ahlfors Q-regular with Q > 1. This means (Z, d) is a separable metric space and µ is Borel regular. The last condition is one on the volume growth of balls: specifically, a ball B of radius r has µ-measure comparable to r Q .
We develop two rough extensions of modulus to a "hyperbolic filling" associated with a given metric space. A hyperbolic filling X = (V, E) of (Z, d, µ) is a graph with vertices that correspond to metric balls and an edge structure which mirrors the combinatorial structure of our metric space. For a useful picture to have in mind consider the unit disk model of the hyperbolic space H 2 . Here the outer circle S 1 plays the role of our metric measure space Z and the hyperbolic filling can be interpreted as a graph representing a Whitney cube decomposition of the interior. In this setting, cubes correspond to vertices and are connected by edges if they intersect. Hyperbolic fillings are Gromov hyperbolic metric spaces when endowed with the graph metric. Moreover, our original space can be identified as the boundary at infinity ∂ ∞ X = Z following a standard construction found in [BuS,Chapter 2].
Hyperbolic fillings are well defined up to a scaling parameter and a choice of a vertex set at each scale. The extensions of modulus presented below are essentially quasi-isometrically invariant; changing the given hyperbolic filling will change the quantities by a controlled multiplicative amount. This multiplicative ambiguity also appears in the modulus comparison results and hence no generality is lost by The author was partially supported by NSF grants DMS-1506099 and DMS-1162471. This work is based on the author's forthcoming thesis.
working with a fixed hyperbolic filling for each metric space. The general construction of hyperbolic fillings follows [BP] and [BuS] and is detailed in Section 3 along with some of the useful properties of such fillings.
Generalizations of modulus are not new; in [P1] and [P2] Pansu develops a generalized modulus which is adapted in [Ty]. One key advantage of these generalized notions of modulus, as here, is that proving quasisymmetric invariance is relatively straightforward after setting up the appropriate definitions.
In our definitions we will need the notion of the weak ℓ p -norm of a function with a countable domain. Let X be a countable space and f : X → C. We define f p,∞ as the infimum of all constants C > 0 such that #{x : |f (x)| > λ} ≤ C p λ p for all λ > 0. We note that in general f p,∞ is not a norm but for p > 1 it is comparable to a norm (see [BoS,Section 2]). We freely interchange the two and refer to f p,∞ as the weak ℓ p -norm of f . The use of the weak norm in the following definitions is motivated by [BoS]. We now define one of the two quantities used in this paper. We work with a compact, connected, Ahlfors Q-regular metric measure space (Z, d, µ) with hyperbolic filling X = (V, E). Both quantities are defined in a similar manner as modulus: certain functions defined on the hyperbolic filling are admissible if they give enough length to an appropriate collection of paths. Then to define the quantity in question we infimize over the p-th power of the weak ℓ p -norm of all admissible functions.
The first quantity, weak p-capacity (wcap p ), is defined both for pairs of open sets with dist(A, B) > 0 and for disjoint continua. A continuum is a compact, connected set that consists of more than one point. We use the notation dist(A, B) = inf{d(a, b) : a ∈ A, b ∈ B} for the distance between A and B. The main idea is that instead of connecting two such sets by paths lying in Z, we look at the (necessarily infinite) paths connecting A and B in the hyperbolic filling. More precisely, given A, B ⊆ Z we call a function τ : E → [0, ∞] admissible for A and B if for all infinite paths γ ⊆ E with nontangential limits in A and B we have e∈γ τ (e) ≥ 1 (see Section 3 for boundaries at infinity of hyperbolic fillings and what it means for a path to have nontangential limits). If A and B are understood we just call τ admissible. Proposition 4.8 states that when this is defined for open sets, wcap p (A, B) > 0. It is also true that for fixed sets A and B we have wcap p (A, B) → 0 as p → ∞.
We remark again here that in the definition of wcap p there is an implicit choice of a fixed hyperbolic filling and that by changing the hyperbolic filling we may change the value of wcap p by a multiplicative constant. That is, if wcap ′ p is defined as wcap p with a different hyperbolic filling, then there are constants c, C > This follows from the proof of Theorem 1.4 as hyperbolic fillings of the same metric space are quasi-isometric. For this reason we ignore the dependence on the hyperbolic filling in the statements of the theorems below. Our first main result shows that in general wcap Q is larger than the Q-modulus of the path family connecting A and B (denoted mod Q (A, B); for the definition of mod Q of a path family, see Section 2).
Theorem 1.2. Let Q > 1 and let (Z, d, µ) be a compact, connected Ahlfors Qregular metric space. Then there exists a constant C > 0 depending only on Q and the hyperbolic filling parameters with the following property: whenever A, B ⊆ Z are either open sets with dist(A, B) > 0 or disjoint continua, For a Loewner space (see Section 2 or [He,Chapter 8] for the precise definition), wcap Q is comparable to this modulus. Theorem 1.3. Let Q > 1 and let (Z, d, µ) be a compact, connected Ahlfors Qregular metric space which is also a Q-Loewner space. Then there exist constants c, C > 0 depending only on Q and the hyperbolic filling parameters with the following property: whenever Hence wcap Q is a quantity that agrees with mod Q up to a multiplicative constant, at least for path families connecting appropriate sets, on spaces with a large supply of rectifiable paths. We also prove wcap p satisfies a quasisymmetric invariance property. Given a homeomorphism η : [0, ∞) → [0, ∞), a map ϕ : Z → W is called an η-quasisymmetry if whenever z, z ′ , z ′′ ∈ Z satisfy |z − z ′ | ≤ t|z − z ′′ |, we have |ϕ(z) − ϕ(z ′ )| ≤ η(t)|ϕ(z) − ϕ(z ′′ )| where we have used the notation | · − · | to denote distance in the appropriate metric spaces.
Theorem 1.4. Let Z and W be compact, connected, Ahlfors regular metric spaces and let p > 1. If ϕ : Z → W is an η-quasisymmetric homeomorphism, then there exist c, C > 0 depending only on η and the hyperbolic filling parameters with the following property: whenever A, B ⊆ Z are either open sets with dist(A, B) > 0 or disjoint continua, We note here that the p above need not match the Ahlfors regularity dimension of neither Z nor W and that Z and W might have different Ahlfors regularity dimensions. The quasisymmetric invariance result relies on the fact that a quasisymmetry on compact, connected, metric measure spaces induces a quasi-isometry on corresponding hyperbolic fillings. A map F between two metric spaces X and Y is said to be a quasi-isometry if there are constants c, C > 0 such that for all x, x ′ ∈ X, we have and there is a constant D > 0 such that for all y ∈ Y , there is an x ∈ X such that |f (x) − y| ≤ D.
We now define the second quantity: weak covering p-capacity (wc-cap p ). As before, there is a choice of hyperbolic filling required that introduces a multiplicative ambiguity but which poses no issues for the statements of the theorems. Unlike wcap p , the quantity wc-cap p is defined for all path families in a given metric space.
The vertices V in our hyperbolic filling correspond to balls in Z: we let B v denote the ball corresponding to the vertex We call such an S a sequence of covers. We say S is expanding if for every finite A ⊆ V , we have S n ∩ A = ∅ for all large enough n.
Given τ : V → [0, ∞], we define the τ -length of a projection P of γ on S as ℓ τ,P,S (γ) = k τ (v k ). Now, let S = {S n } be an expanding sequence of covers. Given a rectifiable path γ : [0, 1] → Z, we say τ is admissible for γ relative to S if lim inf n→∞ (inf P ℓ τ,P,Sn (γ)) ≥ 1 where the infimum inf P is over all projections of γ onto S n . We say τ is admissible for γ if τ is admissible relative to S for all such S . We remark that, for a given γ, changing the parameterization does not affect the subsequent projected τ -length and we will frequently view rectifiable γ as parameterized by arclength.
The main idea is to use subsets (the covers above) of vertices of the hyperbolic filling to approximate Z in finer and finer resolution. Path families now lie on the boundary Z and are projected onto these covers in order to test admissibility of functions defined within. By infimising over all projections onto a given cover and then letting the covers "expand" to become more and more like Z, we arrive at the rough length a given function defined on the filling gives a particular path. Demanding admissibility for all rectifiable paths as with mod p and using the weak norm as with wcap p leads us to our definition. Definition 1.5. Given a collection of paths Γ in Z, we define the weak covering p-capacity wc-cap p (Γ) of Γ as wc-cap p (Γ) = inf{ τ p p,∞ : τ is admissible for all γ ∈ Γ}. With this quantity we have comparability even without the Loewner condition.
Theorem 1.6. Let Q > 1 and let (Z, d, µ) be a compact, connected Ahlfors Qregular metric space. Then there exist constants c, C > 0 depending only on Q and the hyperbolic filling parameters such that for all path families Γ, Similarly to wcap p , the quantity wc-cap p has a quasisymmetric invariance property.
Theorem 1.7. Let Z and W be compact, connected, Ahlfors regular metric spaces and let p > 1. If ϕ : Z → W is an η-quasisymmetric homeomorphism, then there exist constants c, C > 0 depending only on p, η, and the hyperbolic filling parameters such that for all path families Γ in Z we have c wc-cap p (Γ) ≤ wc-cap p (ϕ(Γ)) ≤ C wc-cap p (Γ). This quasisymmetric invariance property implies a result due to Tyson [Ty].
Corollary 1.8. Let Z and W be compact, connected, Ahlfors Q-regular metric spaces with Q > 1 and let ϕ : Z → W be an η-quasisymmetric homeomorphism. Then there exists constants c, C > 0 depending only on η, Q, and the hyperbolic filling parameters such that for all path families Γ in Z.
Indeed, this follows immediately from combining Theorem 1.6 and Theorem 1.7 above. Tyson [Ty] shows this result for locally compact, connected, Ahlfors Qregular metric spaces, but our framework with hyperbolic fillings is adapted to compact metric spaces. Williams [Wi] also derives this result; his Remark 4.3 relates the conditions in the corollary above to his condition (III) which leads to the conclusion.
Lastly we outline the structure of the paper. In Section 2 we review some preliminaries including the definition of modulus and the statement of a weak ℓ p -norm comparison result. In Section 3 we construct hyperbolic fillings and prove some of their basic properties. In particular, we sketch the proof that a quasisymmetry between boundary spaces induces a quasi-isometry between the corresponding hyperbolic fillings. In Section 4 we prove the results related to wcap p , namely Theorems 1.2, 1.3, and 1.4. In Section 5 we relate wcap p to Ahlfors regular conformal dimension. In Section 6 we prove the results corresponding to wc-cap p , namely Theorems 1.6 and 1.7.
Acknowledgements. The author thanks Mario Bonk for countless interesting discussions related to the contents of this paper; his input and guidance have been immensely valuable. The author also thanks Peter Haïssinsky and Kyle Kinneberg for stimulating conversations on the subject matter.

Preliminaries
First we define the notion of modulus of path families. Let (Z, d, µ) be a metric measure space. By a path in Z we mean a continuous function γ : I → Z where I ⊆ R is an interval. We will use γ to refer to both the path and the image of the path. For a path family Γ, we say a Borel function ρ : Z → [0, ∞] is admissible for Γ if for all rectifiable γ ∈ Γ we have γ ρ ≥ 1.
Here and elsewhere path integrals are assumed to be with respect to the arclength parameterization. We define the p-modulus of the path family Γ to be where the infimum is taken over all ρ admissible for Γ and the integral is against the measure µ. In the following we will also suppress dµ from the notation.
One may think of mod p as an outer measure on path families which is supported on rectifiable paths. The quantity mod p is meaningless in spaces without rectifiable paths such as the snowflaked metric spaces discussed in the introduction. Nonetheless, where rectifiable paths abound mod p is closely related to conformality and quasiconformality. Indeed, a conformal diffeomorphism f between two Riemannian manifolds M and N of dimension n preserves the n-modulus of path families [He,Theorem 7.10]. This invariance principle can be used to give an equivalent definition of quasiconformal homeomorphisms between appropriate spaces [He,Definition 7.12] which, while difficult to check, is quite strong. Many basic properties of modulus are detailed in [He,Chapter 7].
Often for clarity and brevity we will make use of the symbols and ≃. For two quantities A and B, that may depend on some ambient parameters, we write A B to indicate that there is a constant C > 0 depending only on these parameters such that A ≤ CB. We also write A ≃ B to indicate that there are constants c, C > 0 depending only on these parameters such that cB ≤ A ≤ CB. The exact dependencies of these constants will be clear from the context.
For some results we will make use of [BoS,Lemma 2.2]. For convenience we include the statement here.
One useful notion for a metric space to have many rectifiable paths is the Loewner condition. This relates the modulus of path families connecting nonintersecting continua, say A and B, with their relative distance We let mod Q (A, B) denote the modulus of the path family connecting A to B. We say a metric measure space (Z, d, µ) is a Q-Loewner space if there is a decreasing function φ Q : (0, ∞) → (0, ∞) such that for all such A and B we have See [He,Chapter 8] for this definition and more information on Loewner spaces. The main intuition here is that the path family connecting two continua with positive relative distance is large enough to carry positive Q-modulus; that is, there are many rectifiable paths connecting the two sets. This condition cannot be omitted from Theorem 1.3: by the quasisymmetric invariance principle for wcap Q (Theorem 1.4) we can snowflake a Q-Loewner space to construct spaces where disjoint open sets have positive wcap Q but the modulus of any path family is 0.
Given a function u on a metric measure space (Z, d, µ), we say that a Borel function ρ : Z → [0, ∞] is an upper gradient for u if whenever z, z ′ ∈ Z and γ is a rectifiable path connecting z and z ′ , we have We use the notation that if B = B(x, r), then λB = B(x, λr). For a given ball B and a locally integrable function u we set u B = 1 µ(B) B u = − B u; this is the average value of u over the ball B. We say (Z, d, µ) admits a p-Poincaré inequality if there are constants C > 0 and λ ≥ 1 such that for all open balls B in Z, for every locally integrable function u : Z → R, and every upper gradient ρ of u in Z. This definition and a subsequent discussion can be found in [HKST,Chapter 8]. The Q-Poincaré inequality is a regularity condition on our space which will follow from the Q-Loewner space hypothesis present in some of the theorems. We make use of Hausdorff content and Hausdorff measure. Given an exponent α ≥ 0 and a set E, the α-Hausdorff content of E is defined as

Hyperbolic fillings
Here we detail the construction of the hyperbolic fillings used to define wcap Q and wc-cap Q . The main idea is to construct a graph that encodes the combinatorial data of Z in finer and finer detail. As remarked in the introduction, it is useful to keep a Whitney cube decomposition of the unit disk model of the hyperbolic plane H 2 in mind with the vertices corresponding to the Whitney cubes and with edges existing between intersecting cubes.
Recall we work in a compact, connected, Ahlfors Q-regular metric measure space (Z, d, µ). Our construction of and proofs of properties involving the hyperbolic fillings of Z follows [BP, Section 2.1] almost exactly. There is a minor problem with their construction, however, which we fix by using doubled radii balls inspired a similar construction in [BuS, Section 6.1]. For completeness we include the entire construction here.
By rescaling, we may assume diam Z < 1. Let s > 1 and, for each k ∈ N 0 , let P k ⊆ Z be an s −k separated set that is maximal relative to inclusion. We call s the parameter of the hyperbolic filling. To each p ∈ P k , we associate the ball v = B(p, 2s −k ) which we will use as our vertices in our graph. We refer to k as the level of v and write this as ℓ(v). We also write V k for the set of vertices with level k. Note as diam(Z) < 1, there is a unique vertex in V 0 which we will denote O. We often write v as B v or B(v) where the level k and the center p are understood. We will occasionally make an abuse of notation, however, and write B(v, 2s −k ) referring to v as both the vertex and the center of the ball. We also use the notation r(B) for the radius of a ball.
We form a graph X = (V, E) where the vertex set V is the disjoint union of the V k and we connect two distinct v, w ∈ V by an edge if and only if |ℓ(v) − ℓ(w)| ≤ 1, and B v ∩ B w = ∅. We endow X with the unique path metric in which each edge is isometric to an interval of length 1.
For v, w ∈ V , we let (v, w) denote the Gromov product given by where O is the unique vertex with ℓ(O) = 0 and |v − w| is the graph distance between v and w. Intuitively, (v, w) is roughly the distance between O and any geodesic connecting v to w.
Then by construction there is a vertex x on level k such that 1 2 B x ∩ B v = ∅; this follows from the choice of an s −k separated set for the centers of the balls corresponding to the vertices on level k.
which is nonnegative by the triangle inequality. Thus, For the other direction, we follow the notation in [BP, Lemma 2.2] and set |v −w| = ℓ, |O − v| = m, and |O − w| = n. Let [vw] be a geodesic segment, which is formed from a sequence of balls B k for k ∈ 0, . . . , ℓ with B 0 = B v and The following lemma involves Gromov hyperbolic metric spaces. For definitions we refer the reader to [BuS, Section 2.1].
Lemma 3.2. X equipped with the graph metric is a Gromov hyperbolic space.
which is the inequality required in the definition of a Gromov hyperbolic space.
We will also work with the boundary at infinity of our hyperbolic fillings. For completeness we include the standard construction of the boundary at infinity of a Gromov hyperbolic space here and refer the reader to [BuS,Chapter 2] for some of the details as well as more background.
For our given Gromov hyperbolic space X, the points of the boundary at infinity ∂ ∞ X are equivalence classes of sequences of points "diverging to infinity". More precisely, we say a sequence of points {x n } diverges to infinity if One then may extend the Gromov product to the boundary as in [BuS]. From this one defines a metric d on ∂ ∞ X to be a visual metric if there are constants c, C > 0 and a > 1 such that for all z, z ′ ∈ ∂ ∞ X we have We now relate this construction to X and Z.
Lemma 3.3. With our constructed X above, we can identify ∂ ∞ X with Z where the original metric on Z is a visual metric.
Sketch of proof. We use We see {v n } is a sequence of vertices diverging to infinity if and only if and so not only do the diameters satisfy diam(B vn ) → 0 but the centers p n of the balls corresponding to v n also converge to a single point in z ∈ Z. This shows we can view ∂ ∞ X as a subset of Z by identifying a sequence of vertices diverging to infinity with the limit point of the centers of the corresponding balls. The other inclusion follows by considering that for each k ∈ N 0 the set of balls {B v : v ∈ V k } covers Z. Hence, for fixed z ∈ Z for each k ∈ N 0 we may choose a vertex v k with level k such that z ∈ B v k . This creates a sequence in X diverging to infinity that corresponds to z. Relation (4) above also shows our original metric on Z is in fact a visual metric with respect to X.
The metric paths in X that we are interested in travel through many vertices and are often infinite. For this reason we define a path in X as a (possibly finite) sequence of vertices v k such that for all k, the vertices v k and v k+1 are connected by an edge. Alternatively we may view a path as a sequence of edges e k such that for all k, the edges e k and e k+1 share a common vertex; the point of view will be clear from context.
We now specify what it means for a sequence of vertices v n to converge to z ∈ Z: if v n is represented by B(p n , r n ) then v n → z if and only if p n → z and r n → 0. From this we also see what it means for a path (given by a sequence of vertices {v k } k∈N or edges {e k } k∈N as discussed above) to converge to a point in Z.
In our definitions we will work with nontangential limits. Intuitively, a path approaches the boundary nontangentially if it stays within bounded distance of a geodesic. In our setting this means that the smaller the radii corresponding to vertices on a path are, the closer the centers corresponding to those vertices need to be to the limit point. We state this more precisely as a definition.
Definition 3.4. A path in X with vertices v n represented by B(p n , r n ) converges nontangentially to z ∈ Z if p n → z and r n → 0 and there exists a constant C > 0 such that for all n ∈ N we have dist(z, p n ) ≤ Cr n .
For the next lemma, we define the valence of a vertex v in a graph (V, E) as the number of edges having v as a vertex. To say a graph has bounded valence then means that there is a uniform constant C such that every vertex v ∈ V has valence at most C.
Lemma 3.5. The hyperbolic filling X of a compact, connected, Ahlfors Q-regular metric measure space has bounded valence.
Proof. Let v be a vertex with level n > 1. Let W be the vertices with level n that intersect v. We bound |W |, the cardinality of the set W . Bounds on the number of vertices adjacent to v with levels n − 1 and n + 1 follow similarly and yield the result.
Here we abuse notation and use v, w as the centers of the balls corresponding to these vertices. We note ∪ w∈W B(w, 1 3 s −n ) is a disjoint collection of balls by the separation of vertices on level n. Moreover, this collection is contained in B(v, 4s −n ). By Ahlfors regularity there are constants c, C such that for all z ∈ Z and r ∈ (0, diam(Z)] we have Hence, we see c|W | 1 3 Q s −nQ ≤ C4 Q s −nQ and so |W | is uniformly bounded above.
Our final lemma is a construction which extends a quasisymmetric homeomorphism f between two compact, connected, metric measure spaces Z and W to a quasi-isometric map F between corresponding hyperbolic fillings X and Y . As we make use of this construction explicitly in Theorem 1.7, we record the result here as a lemma as well as a sketch of the proof. The full proof can be found in [BuS,Theorem 7.2.1].
Lemma 3.6. Let Z and W be compact, connected, metric measure spaces with corresponding hyperbolic fillings X and Y . Let f : Z → W be a quasisymmetric homeomorphism. Then there is a quasi-isometry F : X → Y which extends to f on the boundary.
Sketch of proof. It suffices to define F as a map between vertices. Given x ∈ X, we see f (B x ) ⊂ W and so there is at least one vertex y ∈ Y of minimal radius (i.e. highest level) that contains f (B x ). We set F (x) = y. We show F is a quasiisometry. For the upper bound in (1) and from this deduce that the levels between, and hence the ratio of the radii of the balls B F (x) and B F (x ′ ) , must become arbitrarily large. Using a common point in B F (x) ∩ B F (x ′ ) and the quasisymmetry condition, we arrive at a contradiction. For the lower bound one considers G : Y → X defined as above but using f −1 in place of f . One then shows that the compositions G • F and F • G are within a bounded distance from the identity in each case (i.e. there is a The fact that F (X) is at a bounded distance from any point in y also follows from the compositions being within a bounded distance from the identity.

Weak capacity
Now we prove the main results involving weak capacity: Theorems 1.2, 1.3, and 1.4. Recall we consider compact, connected, Ahlfors Q-regular metric measure spaces (Z, d, µ). After the proofs of the main theorems we prove that for open A, B ⊆ Z with dist(A, B) > 0 we always have wcap p (A, B) > 0. We prove Theorem 1.2 for disjoint continua first. For this we need a technical lemma.
Lemma 4.1. Let p ≥ 1, let f ∈ ℓ p (V ), and let A ⊆ Z be a continuum. Let {B vn } be a sequence of balls corresponding to vertices v n with ℓ(v n ) → ∞. Suppose that there is a constant c > 0 such that for all large enough n we have . Then for each δ > 0 there is an N such that for all n ≥ N , there is a vertex path σ n ⊆ X that starts from v n and has limit in A which satisfies σn f (v) ≤ δ.

Proof.
We note if f has finite support then the result is immediate. Hence, we assume without loss of generality that f does not have finite support. We also may assume all balls in consideration have radius bounded above by 4 as diam(Z) ≤ 1. For ǫ > 0 and fixed n we define We then may cover K n by {B vz } z∈Kn where f (v z ) p ≥ ǫr(v z ). As the balls corresponding to these vertices have uniformly bounded diameter there is a countable subcollection of vertices G n such that z∈Kn B vz ⊆ v∈Gn 5B v and such that if v, w ∈ G n are distinct, then B v ∩ B w = ∅ (see [He,Theorem 1.2]). Hence, K n ⊆ ∪ v∈Gn 5B v . Thus, we have where f j is f restricted to vertices of level j and higher.
In the above we use ǫ = ǫ n defined by 10 ǫ n f ℓ(vn) which is possible as f ℓ(vn) p p > 0 for all n as f does not have finite support. We conclude that for large enough n we have Hence, there is a point z ∈ (A ∩ B vn ) \ K n . For this z we form a vertex path σ from v n with limit z by choosing a vertex w k for each k > ℓ(v n ) such that z ∈ B w k . As z / ∈ K n , we estimate the sum We note f Q Q,∞ ≃ τ Q Q,∞ follows from Lemma 2.1. Indeed, in this case our set J ⊆ V × E consists of vertex-edge pairs (v, e) with e having v as a boundary vertex where s v = f (v) and t e = τ (e). As X has bounded valence, Lemma 2.1 applies to give us one direction of comparability. By using edge-vertex pairs (e, v) and adjusting the definitions of the s v and t e we also get the other bound. We also note that if f Q,∞ < ∞ this means f ∈ ℓ p whenever p > Q.
Consider the functions u n : Z → R given by where we recall V n are the vertices on level n. We claim 2u n is admissible for Qmodulus between A and B for large enough n. Suppose this fails for some sequence n i . Then there is a rectifiable path γ ni connecting A and B with γn i u ni ≤ 1 2 . The endpoints of γ ni lie in balls 1 2 B v A From the definition of f it is clear that summing along the edges of these paths gives a τ -length of less than 1 16 as well. We note the hypothesis in Lemma 4.1 follows once s −ni < min(diam(A), diam(B)).
We show this violates the admisibility of τ . Indeed, if v 0 , . . . , v m is a path of vertices in V ni where γ ni passes through each 1 2 B v k and ℓ(γ ni ) ≥ 8s −ni , then where e j denotes the edge connecting v j−1 to v j . Choosing such a path with v 0 and v m corresponding to B v A n i and B v B n i and combining this path with σ A ni and σ B ni yields a path in the graph with nontangential boundary limits in A and B with τ (e) < 1, contradicting the admissibility of τ .
We have just shown that for large enough n the function 2u n is admissible for mod Q (A, B). It remains to compute u n Q Q . We have where we have used the bounded valence of our hyperbolic filling (Lemma 3.5) for the first inequality and Ahlfors Q-regularity for the second.
Thus, for specific large enough n, we see that u n is admissible for the modulus between A and B and satisfies u n Q Q τ Q Q,∞ . Infimizing over admissible τ yields the result for continua.
Now, for open sets A and B we use the same technique, but the setup is more involved: we need to work safely inside the open sets to be able to satisfy the hypothesis of Lemma 4.1. For λ > 0 define A λ = {a ∈ A : d(a, Z \ A) > λ} and define B λ similarly. Fix λ > 0 such that A λ and B λ are nonempty (all small enough λ will satisfy this). We claim for such λ we have mod Q (A λ , B λ ) wcap Q (A, B) with an implicit constant independent of λ. Indeed, as above we claim 2u n is admissible for mod Q (A λ , B λ ) for large enough n, where we recall u n is defined in equation (5). If 2u n is not admissible, then we may find a path γ n on level n with γn u n ≤ 1 2 . We note for large enough n, the endpoints v A n and v B n of γ n satisfy B v A n ⊆ A and B v B n ⊆ B. We then apply the above procedure to create a short τ -path connecting A and B which contradicts the admissibility of τ . From the norm computation above, by infimizing over admissible τ we conclude mod Q (A λ , B λ ) wcap Q (A, B) with an implicit constant independent of λ.
We show now that this implies mod Q (A, B) wcap Q (A, B). Let λ n = 1 n and, for each n, let σ n ≥ 0 be an admissible function for mod Q (A λn , B λn ) such that σ n Q Q wcap Q (A, B). By Mazur's Lemma [HKST,p. 19], there exist convex combinations ρ n of σ k with k ≥ n and a limit function ρ such that ρ n → ρ in L Q . By Fuglede's Lemma [HKST,p. 131], after passing to a subsequence of the ρ n , we may assume that for all paths γ except in a family Γ 0 of Q-modulus 0, γ ρ n → γ ρ. As the Q-modulus of Γ 0 is 0, there exists a function σ ≥ 0 with γ σ = ∞ for all γ ∈ Γ 0 and Z σ Q < ∞.
We show ρ + cσ is admissible for mod Q (A, B) for any c > 0. Let γ be a path connecting A and B. If γ ∈ Γ 0 , then γ cσ = ∞, so suppose γ / ∈ Γ 0 . Then, as A and B are open, γ connects A λn and B λn for some n. As A λn ⊆ A λ k for n ≤ k, and likewise for B, we see γ σ k ≥ 1 whenever n ≤ k. Hence, γ ρ m ≥ 1 for all m ≥ n, and so γ ρ ≥ 1. Thus, ρ + cσ is admissible for mod Q (A, B). Now, Proof of Theorem 1.3. Recall that for this direction, in addition to working in a compact, connected Ahlfors Q-regular metric measure space (Z, d, µ) we also assume that Z is a Q-Loewner space. We first assume A and B are open sets with dist(A, B) > 0. For the continua case the proof will be the same except for one detail. This is noted in the following proof and the required modifications will follow from Lemma 4.3.
If mod Q (A, B) = ∞ there is nothing to prove so suppose mod Q (A, B) < ∞. Let ρ : Z → [0, ∞] be Q-integrable and admissible for modulus. Define v : Z → R by where Γ z is the set of all rectifiable paths with one endpoint in A and the other endpoint equal to z. It is clear that ρ is an upper gradient for v; that is, given y, z ∈ Z, we have |v(y) − v(z)| ≤ γ ρ whenever γ is a rectifiable path connecting y and z. Hence, as ρ is Q-integrable, it follows from [HKST,Theorem 9.3.4] that v is measurable. Set u = min{v, 1}. We see ρ is an upper gradient for u as well.
Clearly u(a) = 0 whenever a ∈ A and, as ρ is admissible for Q-modulus, we see u(b) = 1 whenever b ∈ B. By [HK,Theorem 5.12], we note that Z supports a Q-Poincaré inequality for continuous functions. This is equivalent to a Q-Poincaré inequality for locally integrable functions by [HKST,Theorem 8.4.1]. Following [HKST,Theorem 12.3.9], first seen in [KZ], this promotes to a p-Poincaré inequality for functions which are integrable on balls for p slightly smaller than Q. Now, consider the function τ : E → R by with K to be chosen and e + , e − the balls representing the vertices of e. As before, Ke + denotes the ball with the same center and as e + and with radius Kr(e + ). We claim with appropriate K that τ is admissible. We note that for intersecting balls B ′ and B ′′ with a constant k ≥ 1 such that B ′ ⊆ kB ′′ and B ′′ ⊆ kB ′ , we have Hence, by the triangle inequality, In our graph, there is a uniform k such that the above holds whenever B ′ and B ′′ are vertices for a given edge. Thus, using the above notation where e is an edge with e + and e − as the balls representing the vertices of e, we have where K = λk arises from the Poincaré inequality (inequality (3)). Thus, if γ is a path in the hyperbolic filling with limits in A and B we have, summing over the edges e in γ, Now, γ has boundary limits in A and B. We recall that if a sequence of vertices B n along a path approaches a limit z ∈ Z = ∂ ∞ X, then the centers p n of the B n satisfy p n → z. Thus, as A and B are open, for edges sufficiently close to A we have u e+ = 0 and for edges sufficiently close to B we have u e+ = 1. We remark here that this fact is not true in the case that A and B are disjoint continua; this is where we will use Lemma 4.3. Hence, 1 γ τ (e) with constant independent of γ. Thus, for suitable c depending only on the constants λ and C from the Poincaré inequality and k above, cτ is admissible.
It remains to compute τ Q,∞ . We follow a proof similar to [BoS,Proposition 5.3]. We estimate the number of edges e with an endpoint labelled as v that belong to where α > 0. We will bound τ Q,∞ by bounding #V (α). Now, and, by the geometric structure of our graph, we have, for z ∈ Z, where B z is the ball in V (α) of smallest radius that contains z (such a ball exists for almost every z as ρ ∈ L p ).
We note that for v ∈ V (α) and z ∈ Kv we have where M denotes the (uncentered) Hardy-Littlewood maximal function (see for instance [He,Chapter 2]). Thus, Hence, where we have used Ahlfors regularity with inequality (9) to bound 1 |Bz| and the fact that Q/p > 1 to bound the maximal function as an operator on L Q/p (Z). Now, if an edge e satisfies τ (e) > α then at least one of its vertices must belong to V (α).
As X has bounded valence, there is an L > 0 such that each vertex can only occur as the boundary of at most L edges. Thus, with a constant depending only on Z and the hyperbolic filling.
We now establish a lemma required to complete the proof of Theorem 1.3 in the case of disjoint continua. We continue to work with an admissible ρ ∈ L Q (Z). For this lemma we need the following fact.
As Z is bounded it is clear we may assume the balls B z have uniformly bounded radius. Hence, we may find a disjoint collection of these balls B zi such that E ⊆ i 5B zi . Thus, We note that Z here may be replaced by an appropriate smaller ambient space Z ′ as long as we stipulate that we only consider radii r for which B(z, r) ⊆ Z ′ . Indeed, in our case we will apply this to Z ′ = c 0 B v = B(z 0 , c 0 R) with c 0 a constant that depends only on our path and the Loewner function. Thus, in this case the conclusion reads H Q ∞ (E) ≤ 10 Q η c0Bv ρ Q . We also use u from the proof of Theorem 1.3. Recall this means u = min (v, 1) where v is defined in equation (7). We also will carefully keep track of constants. We will denote the Ahlfors regularity constants as c Q and C Q . That is, for balls B ′ with radius r bounded above by diam(Z) we have c Q r Q ≤ µ(B ′ ) ≤ C Q r Q .

Lemma 4.3. Consider a sequence of balls
A rough outline of the proof is as follows: for a ball B v in the sequence we consider the set M v = {u ≥ ǫ} ∩ B v . For balls close enough to A, if M v is too large in B v we will use the Loewner condition to construct a path connecting A to M v with short ρ-length. When the ρ-length is less than ǫ this contradicts the definition of u.
Proof. Let B v = B(z 0 , R) be a ball in the sequence. As our sequence approaches nontangentially, there is a constant c 1 > 0 depending only on our sequence such that dist(z 0 , A) ≤ c 1 R. We assume B v is close enough to A that 4(1 + c 1 )R < diam(A). Fix ǫ ∈ (0, 1). Set M v = {u ≥ ǫ} ∩ B v . We then note that by setting δ = µ(Mv ) µ(Bv ) we have as u ≤ 1. We assume µ(M v ) > 0 as our conclusion holds if µ(M v ) = 0. We now relate the measure of M v with its Hausdorff Q-content.
. Now by applying Lemma 4.2 we see that the Hausdorff Q-content of the set If c0Bv ρ Q = 0 then we set η = 1 and otherwise we set Hence, Indeed, by possibly removing some of E 0 we may also assume E 0 ⊆ B 0 \ B 1 . As Z is a complete and doubling metric measure space that supports a Poincarè inequality, Z is quasiconvex (see [HKST,Theorem 8.3.2]) and hence rectifiably path connected. Thus, there is a rectifiable path β connecting x to E 0 , say β : [0, 1] → Z with β(0) = x and β(1) ∈ E 0 . We define continua E j as follows: given j > 0, let t − j denote the first time after which β does not return to B 2j+1 and let t + j denote the first time β leaves B 2j . Then set E j = β([t − j , t + j ]). We note that for each j we have E j ⊆ B 2j and diam(E j ) ≥ 1 10 diam(B 2j ). Hence, it follows that As we are in a Q-Loewner space, this means mod Q (E j , E j+1 ) ≥ ϕ(40), where ϕ is the Q-Loewner function associated to Z. As Z is Ahlfors Q-regular, it follows from [HKST,Proposition 5.3.9] that there is a constant c 3 > 0 such that when 0 < 2s < S. In particular, we can find a constant c 4 > 2 such that if Γ * (E j , E j+1 ) is the path family of rectifiable paths connecting E j to E j+1 that leaves c 4 B 2j , we have mod Q (Γ * (E j , E j+1 )) ≤ ϕ(40) 2 . Thus, the Q-modulus of the family of rectifiable paths connecting E j to E j+1 which stay inside c 4 B 2j is at least ϕ(40) 2 . In particular, this means for each j that one can find a path α j that connects E j and E j+1 , stays inside c 4 B 2j , and satisfies Here if c0Bv ρ Q = 0 then instead for each ν > 0 we can find α j (ν) such that αj ρ ≤ ν and the following argument works by choosing values of ν that are sufficiently small. Recall we also use α j to denote the image of α j . Hence, each α j is a continuum, Thus, we see Hence we may perform the same procedure as above to find paths β j connecting where we note the c 2 4 arises as α j ⊆ c 4 B 2j . From the way these paths were constructed it is clear we can extract a rectifiable path γ from j (α j ∪ β j ) connecting We note here that this is the requirement on c 0 , namely that B(x, c 2 with constant only depending on c 4 and ϕ(40) (in the case that c0Bv ρ Q = 0 we can instead make γ ρ as small as we like). Recall the definition of η given by (11) which gives

Now
, as x ∈ M v and γ is a path connecting A to x, we must have γ ρ ≥ ǫ. Thus, For fixed ǫ > 0 the right hand side tends to 0 as v → a ∈ A. Thus, we must have δ → 0 as v → a. Hence, from inequality (10) we conclude u Bv → 0 as v → a.
The above argument can be adapted to show that as B v → B we have u Bv → 1. To do this, we would instead use M v = {u ≤ 1 − ǫ} ∩ B v and argue as above that if δ = µ(Mv ) µ(Bv ) was large then there would exist a path from M to B with short ρ-length. From the definition of u this would produce a path γ connecting A to B with total length less than 1, contradicting the admissibility of ρ.
This completes the proof of the continua case by bounding below the quantity in inequality (8).
Lastly we prove Theorem 1.4, the quasisymmetric invariance property for wcap p .
Proof of Theorem 1.4. Recall Z and W are compact, connected, Ahlfors regular metric spaces and ϕ : Z → W is an η-quasisymmetry. Let X = (V X , E X ) and Y = (V Y , E Y ) be corresponding hyperbolic fillings. The proof for open sets and continua is the same, so let A and B either be open sets with dist(A, B) > 0 or disjoint continua. As ϕ −1 is an η ′ -quasisymmetry with η ′ depending on η, it suffices to show wcap p (ϕ(A), ϕ(B)) wcap p (A, B). Let G : Y → X be the quasi-isometry induced by ϕ −1 as in Lemma 3.6. We note that G maps vertices to vertices. Let D > 0 be such that for all adjacent vertices y, w ∈ Y , |G(y) − G(w)| ≤ D.
Let τ ≥ 0 be admissible for wcap p (A, B). We construct σ on E Y as follows: given an edge e ′ ∈ E Y with vertices e ′ + and e ′ − , we set where e ∼ x means e is an edge that has the vertex x as an endpoint. We show that σ is admissible for wcap p (ϕ(A), ϕ(B)). Indeed, if γ is a path in Y with limits in ϕ(A) and ϕ(B), then we construct a path in X with limits in A and B that serves as a suitable image of γ. Each vertex y ∈ γ corresponds to a point G(y) ∈ X. By our choice of D, if two vertices y and y ′ are connected by an edge in γ, then |G(y)−G(y ′ )| ≤ D. We choose a path connecting G(y) and G(y ′ ) that stays in the ball of radius D centered at G(y). By doing this for all connected vertices in γ, we produce a path γ X in X with limits in A and B. Now, by construction, where we have viewed γ as both a sequence of vertices and a sequence of edges. The last inequality follows as τ was assumed admissible for wcap p (A, B). It remains to show σ p,∞ τ p,∞ . For this we use Lemma 2.1. Our set J ⊆ E Y × E X consists of pairs (e ′ , e) for which e appears as a summand in the definition of σ(e ′ ). Following the notation from Lemma 2.1, for e ′ ∈ E Y the set J e ′ is the set of edges e that appear as a summand in the definition of σ(e ′ ). The cardinality |J e ′ | is bounded independent of e ′ as X has bounded valence. Similarly, for e ∈ E X the set J e is the set of e ′ for which e contributes to the sum in the definition of σ(e ′ ). We show that a given e can only contribute to a bounded number of such σ(e ′ ). Indeed, if (e ′ , e) ∈ J, then one of the vertices of e must lie in the D radius ball around the image under G of one of the vertices of e ′ . As G is a quasi-isometry, we see there is a constant C > 0 such that if y, y ′ ∈ V Y and |y − y ′ | > C, then |G(y) − G(y ′ )| > 2D + 1. Hence, for edges e ′′ far enough away from e ′ in Y , we must have e ′′ / ∈ J e . As Y has bounded valence by Lemma 3.5, we see |J e | is bounded independent of e.
We now show the positivity of wcap p : whenever A, B ⊆ Z are open sets with dist(A, B) > 0, we have wcap p (A, B) > 0. Here only p ≥ 1 is assumed. Recall we work with a fixed hyperbolic filling X = (V, E) with parameter s > 1. To prove this result, Proposition 4.8, we first detail a construction. For this we need the following lemma.

Lemma 4.4. There exists a constant
Proof. We write B v = B(z v , r v ). We recall Z is Ahlfors Q-regular, so there exist constants c, C > 0 such that for all z ∈ Z and r ∈ (0, diam(Z)), we have cr Q ≤ µ(B(z, r)) ≤ Cr Q . Fix small k > 0 such that and by our choice of k we see Hence, there is a point z 1 ∈ B(z v , (3/4)r v ) \ B(z v , kr v ). There is also a point z 2 ∈ B(z v , k 2 r v ). For example, let z 2 = z v . Let M be such that s −M < k 16 , where we recall s > 1 was a parameter in the construction of the hyperbolic filling. Then as the balls of level ℓ(v) + M cover Z, there must be balls B j of radius 2s −(ℓ(v)+M) with z j ∈ B j . Now, dist(z 1 , z 2 ) ≥ k 2 r v by construction. As r v = 2s −ℓ(v) , this is dist(z 1 , z 2 ) ≥ ks −ℓ(v) . Hence, the sum of the diameters of the balls 2B j is bounded by and so 2B 1 ∩ 2B 2 = ∅.
We now construct our path structure. Let v ∈ V . Let G = {0, 1} * be the set of finite sequences of elements of {0, 1}. For an element g ∈ G, we let g0 and g1 denote the concatenations of the symbols 0 and 1 to the right hand side of g. We associate elements of G to vertices in V inductively as follows: let ∅ correspond to v and, given an element g ∈ G with corresponding vertex v g , apply Lemma 4.4 to B vg to obtain B vg0 and B vg1 which correspond to g0 and g1. We also choose γ g0 (and likewise γ g1 ) to be an edge path connecting v g to v g0 with length M . To construct such a path, one can choose a point in B vg0 and select vertices corresponding to balls containing that point with levels between those of v g and v g0 . To form γ g0 one then uses the edges between these vertices.
Remark 4.5. We note from the fact that 2B 1 ∩ 2B 2 = ∅ in Lemma 4.4 that, if g, h ∈ G N , then γ g ∩ γ h = ∅ can only happen if the first N − 1 entries of g match those of h.
Definition 4.6. Given a vertex v ∈ V , we call a path structure as constructed above a binary path structure and denote it T v .
We call an edge path γ = (e k ) ascending if the levels of the endpoints of successive edges is strictly increasing. That is, if (v k ) is the sequence of vertices that γ travels through, then ℓ(v k+1 ) = ℓ(v k ) + 1 for all k. We consider functions τ : E → [0, ∞] which are admissible on ascending edge paths originating from v. That is, for such paths γ we require γ τ (e k ) ≥ 1. We claim such functions cannot have too small weak ℓ p -norm. Functions that are admissible on ascending edge paths must give at least τ -length 1 to such paths, so for these functions we have τ p,∞ ≥ 1/S(p).
Proof. We may assume τ p,∞ < ∞. Let T v be a binary path structure originating from v. For N ∈ N, let G N = {0, 1} N be the set of finite strings of elements of {0, 1} of length N . For g ∈ G N and k < N , let g k denote the element of G k which matches the first k entries of g. We also set γ ′ g = k γ g k to be the ascending edge path formed by concatenating γ g1 , . . . , γ g . The average τ -length of the paths γ ′ g , where g ∈ G N , is given by We bound this average above using τ p,∞ . For ease of notation, let a = τ p,∞ . By definition, we have #{e : τ (e) > λ} ≤ a p λ p . Hence, for j ∈ N the function τ can take values τ (e) ≥ a j (1/p) for at most j edges e. From Remark 4.5 for fixed k > 1 there are at least (2 (k−1) − 1)M distinct edges contributing to the right hand side of equation (12) belonging to paths γ f with f ∈ G ℓ where ℓ < k. With the weighting factors 2 N −k it follows that the average increases the more τ -mass is located on paths with smaller associated k values. Using these observations we conclude that to bound equation (12) above, for k > 1 and h ∈ G k we bound the sum e∈γ h τ (e) above by M a ((2 (k−1) − 1)M ) (1/p) . From this, we obtain the upper bound 1 2 We bound the first term by noting that τ (e) ≤ a for all e ∈ E. As G k has 2 k elements, our bound becomes we thus bound (12) above by aS(p) = τ p,∞ S(p). As this is a bound on the average τ -length of the paths γ ′ g , where g ∈ G N , we conclude that for each N there is a g N ∈ G N such that γ ′ gN has τ -length bounded above by τ p,∞ S(p). To complete the proof we construct a path γ from the paths γ ′ gN . For each N we have g N ∈ {0, 1} N . Hence, there is a subsequence of the g N that has the property that all strings in this subsequence have the same first element. From this subsequence, we may extract another subsequence that consists of strings that all have the same first two elements. Continuing in this manner and then diagonalizing produces an infinite sequence h ∈ {0, 1} N for which, for infinitely many k ∈ N, the first k elements of h match g k . It is clear how to associate h to an ascending path γ which, from the bounds on the τ -lengths of the paths γ ′ g k , satisfies the conclusion of the lemma.
We perform a similar analysis on a path of edges of length L. As above, if τ p,∞ = b, then τ can take values τ (e) ≥ b j (1/p) at most j times. Thus, the maximum τ -length our line can have is L k=1 b k (1/p) . This is bounded above (for p > 1) by b( L 0 1 x (1/p) dx), which is bL 1−(1/p) 1−(1/p) . For p = 1 this is bounded above by b(1 + log(L)).
We are now ready to prove positivity. Proof. We first assume p > 1. Let v, w be vertices such that B v ⊆ A and B w ⊆ B and such that ℓ(v) = ℓ(w). We connect v and w by an edge path γ contained in {x ∈ X : |x − O| ≤ ℓ(v)}, where O is the unique vertex with ℓ(O) = 0. We let L denote the length of γ. Recall T v and T w denote binary path structures originating from v and w. Now, if τ is admissible for wcap p (A, B), then τ is admissible on paths contained in T v ∪ T w ∪ γ. Let τ A , τ B , and τ γ denote the restrictions of τ to T v , T w , and γ. We then must have As τ p,∞ is larger than the norm of each of these restrictions, we see Hence, The above analysis also applies to the p = 1 case with the appropriate bound modification. In this case, this bound becomes τ 1,∞ ≥ 1 2S(1) + (1 + log(L)) .
Both cases thus yield a lower bound on τ p,∞ for admissible τ , as desired.

Ahlfors regular conformal dimension
We define a critical exponent relating to wcap which is motivated by a critical exponent defined in [BK]. This is not the first attempt to define meaningful critical exponents using hyperbolic fillings. For example, see [CP]. We write ARCdim for the Ahlfors regular conformal dimension of Z. That is, if where the infimum is taken over all θ ∈ G such that (Z, θ) is Ahlfors regular. Here (Z, θ) ∼ qs (Z, d) means that the identity map is a quasisymmetry.
Proof. From quasisymmetric invariance, it suffices to show for any p > ARCdim and any open sets A and B with dist(A, B) > 0 we have wcap p (A, B) < ∞. Fix such parameters and let θ be an Ahlfors regular metric on Z such that dim H (Z, θ) ≤ p. Let X = (V, E) be a hyperbolic filling for (Z, θ) with parameter s > 1 and consider f : V → R defined by It follows from dim H (Z, θ) ≤ p that the number of vertices on level n is bounded above by Cs np for some C > 0 and so f ∈ ℓ p,∞ (V ). Define τ : E → R by τ (e) = f (e + ) + f (e − ). Hence, τ p,∞ f p,∞ < ∞ follows from Lemma 2.1. Thus, we need only show τ is admissible. Now, if γ is any finite chain of vertices, say {v 0 , . . . , v N } (where v k is connected to v k+1 for all k), then Thus, as an infinite γ with nontangential limits in A and B has a finite subpath so τ is admissible.
In view of this inequality, one question is how does Q w relate to ARCdim? If Q w = ARCdim, then can one define a similar critical exponent that is ARCdim?
The following result shows that for some metric spaces we do have equality.
Lemma 5.3. Let (Z, d, µ) be a compact, connected metric measure space that is Ahlfors Q-regular, Q > 1, and such that there exists 1 ≤ p ≤ Q and a family of paths Γ with mod p (Γ) > 0. Then there exist open balls A and B with dist(A, B) > 0 such that for all q < Q we have wcap q (A, B) = ∞.
In such metric spaces [MT,Proposition 4.1.8] shows ARCdim(Z, d) = Q and so we have Q w = ARCdim.
Proof. From [MT,Proposition 4.1.6 (vii)] it follows that mod Q (Γ) > 0. By potentially taking subpaths, we may assume every path in Γ has distinct end points (i.e. no paths in Γ are loops). By writing Γ = ∪(Γ m ) where Γ m = {γ ∈ Γ : ℓ(γ) > 1 m }, we may assume the paths in Γ have lengths uniformly bounded from below (we need Q > 1 for this, see [MT,Proposition 4.1.6 (iv)]). By covering Z with a finite number balls of small enough radius and writing Γ as the union of paths connecting two disjoint balls with positive separation, we may assume Γ connects two open balls A and B with dist(A, B) > 0. Refining this slightly allows us to assume Γ connects A λ and B λ for some λ > 0 as in the proof of Theorem 1.2. Now suppose τ : E → R is admissible for wcap q (A, B) and satisfies τ q,∞ < ∞, where q < Q. Define f : V → R by f (v) = e∼v τ (e) and, as in the proof of Theorem 1.2, set Then, as before, for large enough n the function 2u n is admissible for Γ. Our estimate for u n Q Q is also the same as in inequality (6): As f ∈ ℓ q,∞ (V ), it follows that f ∈ ℓ Q (V ) and so v∈Vn f (v) Q → 0 as n → ∞. This shows mod Q (Γ) = 0, a contradiction. Hence, no such admissible function exists and wcap q (A, B) = ∞.

Weak covering capacity
Here we state and prove some basic properties and the main theorems involving wc-cap p . Recall we work on a compact, connected, Ahlfors Q-regular metric measure space (Z, d, µ) with Q > 1 and with a fixed hyperbolic filling X = (V X , E X ) of (Z, d, µ) with scaling parameter s > 1. We first prove that if p ≥ Q, then wc-cap p is supported on rectifiable paths. Proof. We show that for any ǫ > 0, the functions τ ǫ (v) = r(B v )ǫ are admissible for Γ ∞ and that τ ǫ p,∞ → 0 as ǫ → 0. From this it follows that wc-cap p (Γ ∞ ) = 0. Fix ǫ > 0. Let γ ∈ Γ ∞ . Let t 0 < t 1 < · · · < t m be a partition of [0, 1] such that the points γ(t k ) are distinct and k d(γ(t k−1 ), γ(t k )) > 4/ǫ. Set γ k = γ| [t k−1 ,t k ] . Let N be such that 400s −N ≤ min k d(γ(t k−1 ), γ(t k )) and m2ǫs −N < 1. Let {S j } be an expanding sequence of covers. As {S j } is expanding, for large enough j we see that the balls in S j all have radius bounded above by 2s −N . Thus, for such a j, if P k is any projection of γ k on S j with balls {B i } we have Now, let P be any projection of γ on S j . By adding the values t 0 , . . . , t m to P we obtain a partition P ′ from P and subpartitions P k of P ′ consisting of the values between t k−1 and t k . It is clear that P ′ has at most m more intervals than P . As We note ℓ τǫ,P ′ ,Sj (γ) = k ℓ τǫ,P k ,Sj (γ k ). Combining this with (13) and (14) yields As this holds for all large enough j, we conclude that τ ǫ is admissible for each γ ∈ Γ and hence for Γ ∞ . It remains to show τ ǫ p,∞ → 0 as ǫ → 0. As our hyperbolic filling has bounded valence (Lemma 3.5), we see the number of vertices with level n is comparable to s nQ up to a fixed multiplicative constant. Thus, for λ = 2ǫs −n ≤ 1 we have #{v ∈ V : τ (v) > λ} s nQ ǫ Q λ Q = ǫ Q λ p−Q λ p ǫ p λ p with implicit constants independent of n. From this the limiting behavior of τ ǫ p,∞ follows.
We remark here that the above result does not hold for p < Q. Indeed, by quasisymmetric invariance (Theorem 1.7) there are spaces with path families Γ of nonrectifiable curves for which wc-cap p (Γ) > 0 holds for some p.
This follows as if τ is admissible for Γ 1 ∪ Γ 2 , then τ is admissible for Γ 1 and if τ 1 , τ 2 are admissible for Γ 1 , Γ 2 , then max{τ 1 , τ 2 } is admissible for Γ 1 ∪ Γ 2 . With this observation and Lemma 6.1 it follows that for any path family Γ one has wc-cap Q (Γ) = wc-cap Q (Γ \ Γ ∞ ). Thus, we may assume in the following that all path families Γ consist solely of paths with finite length. Now we prove Theorems 1.6 and 1.7. We start with Theorem 1.6. Recall we work in a compact, connected, Ahlfors Q-regular metric space Z with hyperbolic filling X = (V X , E X ). We also work with a fixed path family Γ such that every γ ∈ Γ has finite length.
Proof of Theorem 1.6. We first prove wc-cap Q (Γ) mod Q (Γ). Let ρ : Z → [0, ∞] be an admissible function for the Q-modulus of Γ. As Z is compact, we may assume ρ is lower semicontinuous (this follows from the Vitali-Carathéodory theorem, see [HKST,Section 4.2]). Define τ : V → R by for v ∈ V . We show that τ is admissible for covering capacity.
Fix γ ∈ Γ. We recall we assume γ has finite length ℓ(γ) > 0. Set I = [0, ℓ(γ)]; we work with partitions of I and the arclength parameterization of γ as in the remark in the introduction. As ρ is lower semicontinuous, there is a sequence of continuous functions f n ≥ 0 such that f n increases pointwise to ρ (see [HKST,Section 4.2]). By the monotone convergence theorem γ 10f n increases to γ 10ρ and, as γ 10ρ ≥ 10, there is an N such that for n ≥ N , we have γ 10f n ≥ 7. Set f = 10f N and M = max z∈Z f (z).
We find a partition of [0, ℓ(γ)] given by 0 = x 0 < · · · < x p = ℓ(γ) with x k+1 −x k < δ 1 and find a δ 2 > 0 such that for each i, every x, y in the δ 2 neighborhood of γ i = γ([x i−1 , x i ]) satisfies |f (x) − f (y)| < ǫ. The existence of this partition and of δ 2 follow from the uniform continuity of f on Z. We also set m i to be the infimum of the values of f on the δ 2 neighborhood of γ i . We further partition each [x i−1 , x i ] as x i−1 = y i 0 < · · · < y i qi = x i such that ℓ(γ i ) − j d(γ(y i j−1 ), γ(y i j )) < ǫ Mp . Set δ 3 = 1 10 min i,j d(γ(y i j−1 ), γ(y i j )) which we may assume is positive by appropriately choosing y i j . Set δ = min( 1 3 δ 1 , δ 2 , δ 3 ). Let S = {S n } be an expanding sequence of covers. Then, as S is expanding, for large enough n it follows that r(B v ) < δ for all v ∈ S n . We work with one of these covers S n with large n which we denote S. Let P : [0, ℓ(γ)] → V be a projection of γ onto S with t 0 , . . . , t m partitioning [0, ℓ(γ)] and v 1 , . . . , v m vertices such that γ([t k−1 , t k ]) ⊆ B v k .
We have Now we group the t k into T 1 , . . . , T p where T i = {t k : x i−1 ≤ t k ≤ x i } and similarly write K i = {k : t k ∈ T i }. By our choice of δ above, we see for each i that |m i − f (γ(t k ))| < ǫ whenever k ∈ K i . Using ℓ(γ i ) − j d(γ(y i j−1 ), γ(y i j )) < ǫ Mp , we deduce i m i   j d(γ(y i j−1 ), γ(y i j ))   ≥ 6 − ǫ ≥ 4. Now, as δ ≤ δ 3 we may replace j d(γ(y i j−1 ), γ(y i j )) in the above sum with twice the sum of the radii of balls B v from our partition P with corresponding intervals intersecting ∪ j [y i j−1 , y i j ]. That is, we have i k∈Ki We note that for k ∈ K i we have Lastly we deal with the overestimation possible from having k ∈ K i for more than one i. This only happens if t k = x i for some i, which happens at most p + 1 times (recall our partition is x 0 , . . . , x p ). We note that f is bounded and that in an expanding sequence of covers we have r(B v k ) → 0 in the above sum. Thus, if f ≤ M and r(B v k ) ≤ ν(n), double counting such k adds at most (p + 1)M ν(n) to our estimate. We conclude From this we see that for large enough n the τ -length of any partition P of γ onto S n is at least 1. That is, τ is admissible for γ relative to S . As S was arbitrary, it follows that τ is admissible for γ. As this holds for all γ ∈ Γ we see τ is admissible for covering capacity. It remains to show τ Q Q,∞ ρ Q Q but this follows as in the proof of Theorem 1.3 with p = 1. We now prove the other direction, namely mod Q wc-cap Q . Let τ : V → R be admissible for covering capacity. Let As in the proof that mod Q (A, B) wcap Q (A, B) for open sets, we note that there is a subsequence σ ni with σ ni Q Q τ Q Q,∞ . Applying Mazur's Lemma to this subsequence, as in the proof of Theorem 1.2, we get convex combinations ρ k of σ ni with i ≥ k and a limit function ρ with ρ k → ρ in L Q . Similarly to that proof, by applying Fuglede's Lemma we may pass to a subsequence and assume that for for all paths γ except in a family Γ 0 of Q-modulus 0 we have γ ρ n → γ ρ. We note that ρ Q Q τ Q Q,∞ . as G(y) ∈ J y . Thus, σ p,∞ τ p,∞ and so wc-cap p (ϕ(Γ)) wc-cap p (Γ). The other inequality follows from considering ϕ −1 in place of ϕ.