Duality of moduli and quasiconformal mappings in metric spaces

We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincar\'e inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.


Introduction
A homeomorphism f ∶ X → Y between two metric spaces X, Y is said to be quasiconformal if there is a constant H ≥ such that for all x ∈ X, In metric measure spaces satisfying suitable conditions such as Ahlfors regularity and a Poincaré inequality, the study of quasiconformal mappings was begun by Heinonen and Koskela in [12] and by now the literature is extensive, see for example [3,11,13,21,25]. As in the classical Euclidean setting, there are also other notions of quasiconformality. For Ahlfors Q-regular spaces X, Y, a homeomorphism f ∶ X → Y is said to be geometric quasiconformal if there is a constant K ≥ such that whenever Γ is a family of curves in X, we have For the de nition of Q-modulus and all other concepts needed in the paper, we refer to Section 2. If both X and Y are complete and also support a Q-Poincaré inequality, the two notions of quasiconformality are equivalent, see Theorem 9.8 in [13].
A fact that has received much less attention is that quasiconformal mappings also quasi-preserve the Q Q− -modulus of certain families of surfaces obtained as "essential boundaries" of sets of nite perimeter. This result was proved in Euclidean spaces by Kelly [18,Theorem 6.6]. In the metric space setting, the theory of functions of bounded variation (BV) and sets of nite perimeter was rst developed by Ambrosio and Miranda [2,23]. The authors of the current paper together with Shanmugalingam extended Kelly's result to metric spaces in [15].
In the current paper, our main goal is to show that the converse holds as well: if a homeomorphism f quasi-preserves the modulus of families of surfaces, then it is a quasiconformal mapping. Since the analogous fact is already known to hold for families of curves, we invest most of our e orts in studying the duality of moduli of families of curves and surfaces. Speci cally, for a nonempty bounded open set Ω ⊂ X and two disjoint sets E, F ⊂ Ω, we consider the family of curves Γ joining E and F in Ω, and the family of surfaces L separating E and F in Ω in a suitable sense. Then we prove the following theorem; the precise formulation and assumptions on the sets E and F are given in Theorem 4.5. Theorem 1.1. Let < p < ∞ and suppose X is a complete metric space equipped with a doubling measure and supporting a -Poincaré inequality. For some constant C ≥ depending only on p and the space X, we have In Euclidean spaces, this was proved (with constant C = ) by Ziemer [26], and later by Aikawa and Ohtsuka who show in [1] that the same result holds for a more general weighted modulus with weights coming from the Muckenhoupt A p -class. Combining Theorem 1.1 with the characterization of quasiconformal mappings by means of the moduli of curve families, we get the following theorem.

Then f is quasiconformal.
This is given, in a somewhat more general form, in Theorem 5.1. Results similar to Theorem 1.1 and Theorem 1.2 were very recently proved in the metric space setting by Lohvansuu and Rajala [22], but their viewpoint was somewhat di erent. In [22] (similarly to [26]) the authors understood a "surface" to be a set of nite codimension one Hausdor measure separating E and F in a topological sense. By contrast, we understand surfaces to be sets of nite perimeter in the spirit of [18] and [15]. Moreover, we wish to study the problem under weaker assumptions: instead of Ahlfors regularity it is in fact enough to assume in Theorem 1.2 that the measures on X and Y are doubling and satisfy suitable one-sided growth bounds. Additionally, we do not assume the sets E and F to be closed, as was done in [22] and [26]. Working with more general sets makes it a rather subtle problem to nd the correct de nition for a "surface" that separates E and F; for this we apply the concept of ne topology, relying on results proved in [4,6,7]. Hence our arguments combine the theory of quasiconformal mappings, BV theory, and ne potential theory in metric spaces.
We say that (X, d, µ) is Ahlfors Q-regular, with Q > , if there is a constant C A ≥ such that whenever x ∈ X and < r < diam(X), we have Throughout the paper, we always assume µ to be doubling.

De nition 2.2. Let
A ⊂ X. The codimension 1 Hausdor measure of A is given by Note that a complete metric space equipped with a doubling measure is always proper, that is, closed and bounded sets are compact. Given an open set Ω ⊂ X, we write u ∈ L loc (Ω) if u ∈ L (V) for every open V ⋐ Ω; this expression means that V is a compact subset of Ω. Other local spaces are de ned analogously.
A curve is a continuous mapping from a compact interval into X, and a recti able curve is a curve with nite length. The length of a recti able curve γ is denoted by γ . Every recti able curve can be parametrized by arc-length, see e.g. [10,Theorem 3.2]. In the following de nitions, we let ≤ p < ∞; in most of the paper we will assume that < p < ∞. We say that a nonnegative Borel function ρ is p-weakly admissible for the collection M if ρ is admissible for all but a p-modulus zero collection of measures.
Mod p is an outer measure on the class of all Borel measures, see [8]. There are two types of collections of measures associated with quasiconformal mappings. Firstly, given a collection Γ of curves in X, we set Γ to also denote the arc-length measures restricted to each curve in Γ; then the admissibility condition is replaced by for recti able γ. We say that a property holds for p-almost every curve if it fails only for a curve family with zero p-modulus. Secondly, for a collection L of sets of nite perimeter in a set Ω, we consider the measures P(U, ⋅) for each U ∈ L (see the de nition given later).

De nition 2.4.
Let Ω ⊂ X be µ-measurable. Given a function u∶ Ω → R, a Borel function g∶ Ω → [ , ∞] is said to be an upper gradient of u in Ω if for every nonconstant recti able curve γ in Ω, where x and y are the endpoints of γ. We interpret u(x)−u(y) = ∞ whenever either u(x) or u(y) is in nite. A function u is said to be in the Newton-Sobolev class N ,p (Ω) if u ∈ L p (Ω) and there is an upper gradient g of u in Ω such that g ∈ L p (Ω). We let where the in mum is taken over upper gradients g of u in Ω. We say that a nonnegative µ-measurable function g is a p-weak upper gradient of a function u in Ω if (2.1) holds for p-almost every curve in Ω.
If u ∈ N ,p loc (Ω), then there exists a minimal p-weak upper gradient of u in Ω, always denoted by g u , satisfying g u ≤ g a.e. in Ω for every p-weak upper gradient g ∈ L p loc (Ω) of u in Ω; see [4,Theorem 2.25] We refer the reader to [4,14,24] for more details regarding mappings in N ,p loc (Ω).

De nition 2.5.
We say that the space X supports a p-Poincaré inequality if there exist constants C P > and λ ≥ such that for all balls B in X, all measurable functions u on X and all upper gradients g of u, Here we denote the integral average of u over B by We will assume throughout the paper that X supports a -Poincaré inequality.
De nition 2.6. For any disjoint sets E, F ⊂ X, we de ne Γ(E, F; X) to be the collection of curves in X joining E and F. We say that X is a Loewner space if there is a function ϕ∶ ( , ∞) → ( , ∞) such that whenever E and F are two disjoint, nondegenerate continua (compact connected sets) such that De nition 2.7. The p-capacity of a set A ⊂ X is given by where the in mum is taken over functions u ∈ N ,p (X) satisfying u ≥ in A. If a property holds outside a set with p-capacity zero, we say that it holds p-quasieverywhere, or p-q.e. We say that a set The relative p-capacity of two sets A ⊂ Ω ⊂ X is given by where the in mum is over all functions u ∈ N ,p (X) such that u ≥ p-q.e. in A and u ≤ in X ∖ Ω. Recall that g u denotes the minimal p-weak upper gradient of u.
We know that Cap p is an outer capacity in the following sense: for any A ⊂ X, see e.g. [4,Theorem 5.31]. If From now on, let < p < ∞.
Then the p-ne topology on X is the collection of all p-nely open sets.

De nition 2.9.
Given a nonempty open set Ω and two disjoint sets E, F, we de ne the capacity of the con- where the in mum is taken over all where Lip c (Ω) denotes the collection of Lipschitz functions that are compactly supported in Ω. If the above inequality holds for all nonnegative φ ∈ Lip c (Ω), we say that u is a p-superminimizer, and if it holds for all nonpositive φ ∈ Lip c (Ω), we say that u is a p-subminimizer.
Next we consider the theory of BV functions in metric spaces.

De nition 2.11.
For an open set Ω ⊂ X and u ∈ L loc (Ω), the total variation of u in Ω is given by It is shown in [23,Theorem 3.4] that Du is a Radon measure in Ω for any u ∈ BV loc (Ω). We call Du the variation measure of u.

De nition 2.12.
A measurable set U ⊂ X has nite perimeter in Ω if Dχ U (Ω) < ∞. We call Dχ U the perimeter measure of U and we will denote it P(U, ⋅).
De nition 2.13. We say that X supports a relative isoperimetric inequality if there exist constants C I > and λ ≥ such that for all balls B and for all measurable sets U, we have We know that when µ is doubling and X supports a -Poincaré inequality, then it supports a relative isoperimetric inequality, see for example [20,Theorem 3.3] (in a slightly di erent form, this was proved earlier in [2,Theorem 4.3]).
The noncentered Hardy-Littlewood maximal function of a function ρ ∈ L loc (X) is de ned by where the supremum is taken over all open balls containing x ∈ X.
Finally we give the de nition of quasiconformal mappings on metric spaces. Let (Y , d Y , µ Y ) be another metric space equipped with a Radon measure µ Y .
De nition 2.14. For a function f ∶ X → Y, de ne for all x ∈ X and r > .
It is known that when both X and Y are Ahlfors p-regular and support a p-Poincaré inequality, the two notions of quasiconformality are equivalent, see Theorem 9.8 in [13]. We will make use of this fact in Section 5, but we will give a self-contained proof where we only need somewhat weaker assumptions than Ahlfors regularity.
Standing assumptions: Throughout this paper we will assume that < p < ∞ and that (X, d, µ) is a complete metric measure space that supports a -Poincaré inequality, such that µ is doubling. We will use the letter C to denote various nonnegative constants that depend only on p and the space X, and the value of C could di er at each occurrence.

Background results
In this section we will gather most of the background results needed in the paper. We start with the following coarea formula for BV functions, which is stated in Remark 4.3 of [23].
We have the following "continuity from below" for families of measures; for a proof see Lemma 2.3 in [27].

Lemma 3.2.
If {L j } j∈N is a sequence of families of Borel measures such that L j ⊂ L j+ for each j, then By applying Fuglede's and Mazur's lemmas, see e.g. [13, p.19, p.131], we get the following.
We note that various results that we cite, such as the following theorem, rely on assuming the space to support a p-Poincaré inequality, but this follows via Hölder's inequality from the -Poincaré inequality that is our standing assumption. for all x ∈ X and < r < diam(X), and some constant c > . Then X is a Loewner space.
Proof. See Theorem 5.7 in [12]. Note that the so-called φ-convexity assumed in this theorem holds since under our assumptions the space is quasiconvex, meaning that every pair of points can be joined by a curve whose length is at most a constant number times the distance between the points; see e.g. [4,Theorem 4.32].
The space X is linearly locally connected in the following sense. Proof. See Remark 3.19 in [12]; note that there it is also assumed that the space is of Hausdor dimension p, but this is not needed in the proof.
Finally we give a few results concerning superminimizers; recall De nition 2.10. Let W ⊂ X be an open set. We de ne the lsc-regularization (lower semicontinuous regularization) of a function u on W by The following proposition is given as part of Theorem 8.22 in [4].

Proposition 3.7.
If u is a p-superminimizer in W, then u * is lower semicontinuous in W and u = u * p-q.e. in W.
More precisely, the fact that u = u * p-q.e. in W is given in the proof of [4,Theorem 8.22]. By (2.2) we know that u * is still a p-superminimizer.
It is a well known fact that superharmonic functions are nely continuous; this was shown in the metric space setting in [7] and [19]. Here we record this result in the following theorem, which follows by combining Proposition 7.12, Theorem 9.24(a,c), and Theorem 11.38 of [4].

Proof of Theorem 1.1
We will consider the following families of curves and surfaces; recall the concept of capacitary thinness from De nition 2.8.

De nition 4.1.
For an open set Ω ⊂ X and any disjoint sets E, F ⊂ X, we de ne Γ(E, F; Ω) to be the collection of curves in Ω joining E ∩ Ω and F ∩ Ω. We also de ne the collection of measures By an abuse of terminology, we will also talk about the sets U belonging to L(E, F; Ω). Essentially, the boundaries of U are "surfaces" that "separate" E and F in Ω, but since we do not assume E and F to necessarily be compact subsets of Ω, the choice of the correct de nition for L(E, F; Ω) becomes rather subtle. If one would employ the usual de nition where the surfaces need to stay at a strictly positive distance from E and F, it would be di cult to prove the lower bound of Theorem 1.1. On the other hand, if one allows the surfaces to "touch" E and F signi cantly, then it becomes di cult to prove the upper bound. For this reason, we allow the surfaces to "touch" E and F only at capacitary thinness points.
Throughout this section, we will abbreviate L = L(E, F; Ω) and Γ = Γ(E, F; Ω). We begin by proving the lower bound. Proof. Since E ∩ Ω and F ∩ Ω are two disjoint compact sets, we have d ∶= dist(E ∩ Ω, F ∩ Ω) > and so Mod p (Γ) < ∞; e.g. d − χ Ω is an admissible function. By [ Thus u is superminimizer in W , and analogously a subminimizer in W . Let u * be the lsc-regularization of u in W and the analogously de ned usc-regularization of u in W , and u * = u in Ω ∖ (W ∪ W ). Then by Proposition 3.7, u * is lower semicontinuous in W and, analogously, upper semicontinuous in W , and u = u * p-q.e. in Ω.
Let L be the collection of super-level sets of u * , U t ∶= {x ∈ Ω ∶ u * (x) > t}, for t ∈ ( , ). By Theorem 3.8 we have u * = in b p E ∩ Ω. Thus the sets U t ∩ W , for t ∈ ( , ), are open and contain b p E ∩ Ω, and so each set int(U t ) contains b p E ∩ Ω. Analogously, b p F ∩ Ω ⊂ ext(U t ) for all t ∈ ( , ). In conclusion we have L ⊂ L, or more precisely P(U t , Ω ∩ ⋅) is in L for every t ∈ ( , ). Thus Let ρ ∈ L p (p− ) (X) be any admissible function for Mod p p− (L ). By e.g. [4, Proposition 2.44] we know that g u, ≤ g u in Ω, where g u, and g u are the minimal -weak and p-weak upper gradients, respectively, of u in Ω. Thus also u ∈ N , (Ω). Since Lip loc (Ω) is dense in N , (Ω), see [4,Theorem 5.47], it follows that u ∈ BV(Ω) with d Du ≤ g u, dµ ≤ g u dµ in Ω. Using also the coarea formula of Theorem 3.1, we get using also (4.1). Taking the in mum over admissible ρ, we get In the case where E and F are compact, we get the lower bound also for the following smaller family of surfaces:

Proposition 4.3. Let Ω ⊂ X be a nonempty bounded domain and let E, F ⊂ Ω be disjoint nonempty compact sets. If Mod
Proof. The proof is almost the same as for Proposition 4.2; we only need to note that since E and F are compact, according to Theorem 1.1 in [17] we nd for every ε > a function u ∈ Lip loc (Ω) with ≤ u ≤ in Ω, u = in E, u = in F, and Ω g p u dµ < cap p (E, F; Ω) + ε.
Then we can consider the super-level sets {x ∈ Ω ∶ u(x) > t} for t ∈ ( , ), which all belong to L * . Now we prove the upper bound. Part of the idea for the following proof came from Lohvansuu and Rajala [22]; the authors would like to thank them for sharing an early version of their manuscript. Note that in particular, a closed set E satis es Cap p (b p E ∖ E) = .
Proof. Let x ∈ X ∖ b p E. Since Cap p (b p E ∖ E) = , by de nition of the variational capacity we get for every t > . Thus where γ is the image of γ in X. Fix i ∈ N and a recti able curve γ ∈ Γ i (assume for now that Γ i = ∅). Let Also x j ∈ N. We wish to construct an admissible function for L i,j . First we construct a Whitney covering of So F k forms a cover of γ k . Then by the -covering theorem, we can nd a pairwise disjoint subcollection Since γ k is bounded, G k is nite for each k. Letting B ∶= ⋃ k∈Z G k , the collection of ve times enlarged balls from B is a cover for γ ∖ (E ∪ F). Now for U ∈ L i,j , set We know that the above supremum is attained and T ∈ ( , γ ) since µ( B ) ≥ . Again by continuity of γ, there exists δ > such that γ(T + δ) ∈ B . Then since T + δ > T, we know that there exists a ball B ∈ B with γ(T + δ) ∈ B such that By the fact that B ∩ B is nonempty (since it contains γ(T + δ)), it is easy to check that rad(B ) ≤ rad(B ) ≤ rad(B ). Hence B ⊂ B . By using rst (4.3) and the doubling property and then the relative isoperimetric inequality of De nition 2.13, we get for some constantC > (depending only on the doubling constant) Choose K j ∈ Z so that K j < min j , dist(Ω i , X ∖ Ω) . Then for any k ≤ K j and B ∈ G k , either B ⊂ U or B and U are disjoint, which implies that Recall that G k is nite for each k. Also γ is bounded, so there exists a K ∈ Z such that G k is empty for all k ≥ K . Hence the function ϕ i,j is p (p − )-integrable. Furthermore, ϕ i,j is admissible for L i,j , since for any U ∈ L i,j , by (4.4) we have Using Lemma 3.3, pick a p (p − )-weakly admissible function ρ i,j such that Recall the de nition of the noncentered Hardy-Littlewood maximal function from (2.3). We now apply Lemma 3.4 which gives the last inequality holds because the curve γ travels at least the length rad(B) inside B, and the balls in each G k are pairwise disjoint and clearly two balls B ∈ G k and B ∈ G l can only intersect if k − l = . Now we show that ⋃ j L i,j = L. First note that Since the maximal function is a bounded operator from L p (X) to L p (X) when < p < ∞, see e.g. [4,Theorem 3.13], we get Recalling that lim j→∞ (L) = ∞. Note that (4.6) holds also if Γ i = ∅. Finally note that the sequence Γ i is increasing with

Proof of Theorem 1.2
Now we can prove Theorem 1.2 given in the introduction. We give it in the following somewhat more general form.
is another complete metric space that supports a -Poincaré inequality, such that µ Y is doubling and µ(B(x, r)) ≥ C − r p and µ Y (B(y, r)) ≤ C r p (5.1) for all x ∈ X, y ∈ Y, r > , and a constant C > . Suppose f ∶ X → Y is a homeomorphism such that for every collection of surfaces L = L(E, F; Ω) with Ω ⊂ X nonempty, open and bounded and E, F ⊂ Ω compact, we have Then f is quasiconformal with a constant depending only on C , p, and the space X.
Of course, (5.1) is satis ed in particular if X and Y are both Ahlfors p-regular; recall De nition 2.1. Also recall the de nitions of L f and l f from De nition 2.14.
Proof. As complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality, X and Y are quasiconvex, see e.g. [4,Theorem 4.32], and so for each of them a biLipschitz change in the metric gives a geodesic space (see Section 4.7 in [4]). Since the theorem is easily seen to be invariant under biLipschitz changes in the metrics, we can assume that X and Y are geodesic. We want to apply Proposition 4.3 and Proposition 4.4 to suitable sets de ned via the homeomorphism f . Fix x ∈ X and r > and let L ∶= L f (x, r) and l ∶= l f (x, r) > . Suppose also that L > C l, where C is the constant from Theorem 3.6. By choosing r su ciently small, we have L < diam(Y) (we can assume that diam(Y) > ). Since f (B(x, r)) is compact, there exists y ∈ f (B(x, r)) such that d Y (f (x), y) = L.
Let E ∶= f − (B(f (x), l)), and F ∶= f − (F * ) where F * is the maximal connected set containing y and contained in B(f (x), M) ∖ B(f (x), L C ), for some xed M ≥ C L. By Theorem 3.6 (note that here we use the upper bound in (5.1)) we have Note that balls are connected in geodesic spaces, and f is a homeomorphism, so E and F are connected. Both E and F are moreover closed, and since X and Y are proper, f and f − map bounded sets to bounded sets, and so E and F are also bounded and thus compact. Since Y is connected, the set F * and thus also the set F consists of at least 2 points and so diam(F) > . If r → then diam(E) → , and thus by choosing r even smaller if necessary, we can assume that diam(E) is less than diam(F). Note that Ω ∶= From this it is easy to see that every curve in Γ(E, F; X) has a subcurve in Γ, and so Mod p (Γ) = Mod p (Γ(E, F; X)); see e.g. [4,Lemma 1.34(c)]. Notice that f − (y) ∈ F ∩ B(x, r), and we know that x ∈ E, so dist(E, F) ≤ r. It is straightforward to show that there is some z ∈ X ∖ B(x, r) with d Y (f (x), f (z)) = l. Thus r ≤ diam(E), which we noted to be less than diam(F), and so dist(E, F) min{diam(E), diam(F)} ≤ r r = .
By Theorem 3.5 we know that X is a Loewner space (note that here we need the lower mass bound in (5.1)), and so Mod p (Γ) = Mod p (Γ(E, F; X)) ≥ ϕ( ) > , (5.3) where ϕ is the Loewner function for X. We observe that every curve in fΓ has a subcurve in the family Recall that we were assuming L > C l; in conclusion lim sup for every x ∈ X. Therefore f is quasiconformal. This is the case because it is not clear that b p E ⊂ int(f − (U)) and b p F ⊂ ext(f − (U)) for every U ∈ L(B(f (x), l), F * ; B(f (x), M + )), as would be required in the de nition of L(E, F; Ω). In other words, the image under f or f − of every "separating surface" might not be a "separating surface". It is known, at least in Ahlfors regular spaces, that a quasiconformal mapping (whose inverse is also quasiconformal) preserves the measure-theoretic interior, exterior, and boundary, see [9], [21, Theorem 6.1], and [15,Lemma 4.8]. If we knew a similar property to hold for capacitary thickness points, then the above problem would not arise. Thus we ask: • If f ∶ X → Y is a quasiconformal mapping, do we have f (b p E) = b p f (E) for every (closed) set E ⊂ X?