DUALITY OF MODULI IN REGULAR METRIC SPACES

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1 DUALITY OF MODULI IN REGULAR METRIC SPACES ATTE LOHVANSUU AND KAI RAJALA Abstract. Gehring [3] and Ziemer [19] found a connection between the moduli of certain path families and corresponding families of hypersurfaces. In this paper we prove a generalization of this result in complete and separable Ahlfors-regular metric spaces that support a weak 1-Poincaré inequality. As an application, we obtain a new characterization for quasiconformal mappings between such spaces. 1. Introduction The modulus of a path family is a widely used tool in geometric function theory and its generalizations to R n and furthermore to metric spaces, see [5],[11] and [14]. Given 1 p < and a family Γ of paths in a metric measure space (X, d, µ), the p-modulus of Γ is defined to be mod p Γ := inf ρ p dµ, ρ where the infimum is taken over all admissible functions of Γ, i.e. Borel measurable functions ρ : X [0, ] that satisfy ρ ds 1 for all locally rectifiable Γ. If no admissible functions exist, the modulus is defined to be. The definition of modulus can be generalized considerably, as was done by Fuglede in his 1957 paper [2]. For example, instead of paths we can consider surfaces by defining the modulus with exactly the same formula but requiring the admissible functions to satisfy ρ dσ S 1 S Mathematics Subject Classification 2010: Primary 30L10, Secondary 30C65, 28A75, 51F99. Both authors were supported by the Academy of Finland, project number The first author was also supported by the Vilho, Yrjö and Kalle Väisälä foundation. Parts of this research were carried out when the first author was visiting the Mathematics Department in the University of Michigan. He would like to thank the department for hospitality. 1 X

2 2 ATTE LOHVANSUU AND KAI RAJALA for all surfaces S in the family. Here σ S denotes some Borel-measure associated to S. In our applications σ S will be comparable to a Hausdorff measure restricted to S. Our main result is concerned with Ahlfors Q-regular complete metric spaces that support a weak 1-Poincaré inequality. We also assume Q > 1. See Section 2 for all relevant definitions. Fix such a metric measure space (X, d, µ). Given a domain G X and disjoint nondegenerate continua E, F G we denote by Γ(E, F ; G) the family of rectifiable paths in G that join E and F. Similarly, we denote by Γ (E, F ; G) the family of compact sets S G that have finite (Q 1)-dimensional Hausdorff measure and separate E and F in G. By separation we mean that E and F belong to disjoint components of G S. We equip each surface S with the restriction of the (Q 1)-dimensional Hausdorff Q Q 1. measure on S G. Denote Q = The main purpose of this paper is to prove the following connection between the path modulus and the modulus of separating surfaces. THEOREM 1.1. There is a constant C that depends only on the data of X such that 1 (1) C mod QΓ(E, F ; G) 1 Q modq Γ (E, F ; G) 1 Q C, for any choice of E, F and G. This is a generalization of results of Gehring [3] and Ziemer [19], who proved that the conclusion of Theorem 1.1 holds in R n with C = 1. Theorem 1.1 holds for more general exponents in place of Q, see Theorem 3.3. As an application of Theorem 1.1 we find a new characterisation for quasiconformal maps between regular spaces. Let Y be another complete Ahlfors Q-regular space that supports a weak 1-Poincaré inequality. Recall that a homeomorphism f : X Y is (geometrically) K-quasiconformal if there exists a constant K 1 such that for every family Γ of paths in X (2) Here fγ = {f Γ}. 1 K mod Q(fΓ) mod Q Γ Kmod Q (fγ). Corollary 1.2. Let X and Y be as above. A homeomorphism f : X Y is K-quasiconformal if and only if there is a constant C, such that 1 C mod Q Γ (E, F ; G) mod Q Γ (fe, ff ; fg) Cmod Q Γ (E, F ; G) for all E, F and G as above. The constants C and K depend only on each other and the data of X and Y.

3 DUALITY OF MODULI IN REGULAR METRIC SPACES 3 See Section 6 for the proof. We remark that the only if part follows also from the recent work of Jones, Lahti and Shanmugalingam [6]. The proof of Theorem 1.1 is split into three parts. First we prove the upper and lower bounds separately in a more general setting, see Theorems 3.1 and 3.2. After this we show that regular spaces satisfy the assumptions of both of these theorems. To prove the lower bound, we follow and generalize the arguments given in [3], [19] and [13]. In contrast, the argument we use to prove the upper bound seems to be new even in the euclidean setting. In Section 7 we give examples to illustrate Theorem Preliminaries 2.1. Doubling measures. A Borel-regular measure µ is called doubling with doubling constant C µ > 1 if µ is strictly positive and finite on balls of X and satisfies (3) µ(2b) C µ µ(b) for all balls B X. Iterating (3) shows that there are constants C µ and s > 0 that depend only on C µ such that for any x, y X and 0 < r R < diam (X) with x B(y, R), ( ) s µ(b(y, R)) R (4) µ(b(x, r)) C µ. r In fact, we can choose s log 2 C µ. The space X is said to be Ahlfors Q-regular, or just Q-regular, if there are constants a and A > 0 such that ar Q µ(b(x, r)) Ar Q for every x X and 0 < r < diam (X). It follows immediately from the definitions that Q-regular spaces are doubling Moduli. Let M be a set of Borel-regular measures on X and let 1 p <. We define the p-modulus of M to be mod p M = inf ρ p dµ, where the infimum is taken over all Borel measurable functions ρ : X [0, ] with (5) ρ dν 1 X for all ν M. Such functions are called admissible functions of M. If there are no admissible functions we define the modulus to be infinite. If ρ is an admissible function for M N where N has zero p-modulus, we say that ρ is p-weakly admissible for M. As a direct consequence of the definitions we see that the p-modulus does not change if the infimum X

4 4 ATTE LOHVANSUU AND KAI RAJALA is taken over all p-weakly admissible functions. If some property holds for all ν M N we say that it holds for p-almost every ν in M. We can also use paths instead of measures; if Γ is a family of locally rectifiable paths in X we define the path p-modulus of Γ as before with mod p Γ = inf ρ p dµ, but require that X ρ ds 1 for every locally rectifiable Γ. See [18] or [1] for the definition and properties of path integrals over locally rectifiable paths. Most of the path families considered in this paper will be of the form Γ(E, F ; G) := {paths that join E and F in G}, where E, F G are disjoint continua and G is a domain in X. The modulus of Γ(E, F ; G) does not change if we consider only injective paths, see [16, Proposition 15.1]. For injective paths ρ ds = ρ dh 1, as can be seen from the area formula [1, ]. This implies that the modulus of any subfamily of A of Γ(E, F ; G) is the same as the modulus of the family {H 1 A}, so in this sense the two definitions of the modulus are equal. We will need the following basic lemma in multiple occasions. It is a straightforward combination of the lemmas of Fuglede and Mazur, see [5, p. 19, 131]. Lemma 2.1. Let M be a set of Borel measures on X and 1 < p <. Suppose mod p M <. Then there is a sequence (ρ i ) i=1 of admissible functions of M that converges in L p (X) to a p-weakly admissible function ρ of M such that for p-almost every ν M (6) ρ i dν ρ dν < and (7) mod p M = X X X ρ p dµ. Remark 2.2. Lemma 2.1 holds for the path modulus of a path family Γ with the obvious modification of replacing (6) with ρ i ds ρ ds <

5 for all Γ. DUALITY OF MODULI IN REGULAR METRIC SPACES 5 Finally, a measurable set A X is called p-exceptional if the path p-modulus of all paths that intersect A equals zero. The set A is said to have zero p-capacity if it is p-exceptional and µ(a) = 0. For more general definitions of capacities and their properties, see [7] Upper gradients. A Borel function ρ : X [0, ] is an upper gradient of a function u : X R, if (8) u((1)) u((0)) ρ ds for all rectifiable paths : [0, 1] X. If u((0)) or u((1)) equal, we agree that the left side of (8) equals. If (8) fails for a family of paths of zero p-modulus, we say that ρ is a p-weak upper gradient. The following lemma will be useful in the sequel, and will be used without further mention. It allows the use of weak upper gradients in place of upper gradients in all the relevant results used in this paper. This is Proposition of [5]. Lemma 2.3. If u : X R has a p-weak upper gradient ρ L p (X) in X, then there is a decreasing sequence (ρ k ) k=1 of upper gradients of u that converges to ρ in L p (X) Maximal functions. Suppose µ is doubling and R > 0. The restricted Hardy-Littlewood maximal function M R u of an integrable function u : X R is defined as M R u(x) = sup 0<r R B(x,r) u dµ, where u dµ := 1 u dµ. B µ(b) B The Hardy-Littlewood maximal function Mu can then be defined as Mu = sup M R u. R>0 In doubling spaces Mu is Borel measurable whenever u is, and the assignment u Mu defines a bounded operator L p (X) L p (X) for any 1 < p <, with bound depending only p and the doubling constant of X, see [5, Chapter 3.5] for details Codimension 1 spherical Hausdorff measure. Given a Borelregular measure µ, the codimension 1 spherical Hausdorff δ-content of a set A X is defined as H δ (A) := inf µ(b i ), r i i

6 6 ATTE LOHVANSUU AND KAI RAJALA where the infimum is taken over countable covers {B i } of A, and B i = B(x i, r i ) for some x i X and r i δ. The codimension 1 spherical Hausdorff measure of A is then defined to be H(A) := sup H δ (A). δ>0 By the Carathéodory construction H is a Borel-regular measure whenever µ is. In Q-regular spaces H is comparable to the (Q 1)-dimensional Hausdorff measure, provided that Q Poincaré inequalities. The space X is said to support a weak p-poincaré inequality with constants C P and λ P if all balls in X have positive and finite measure, and ( ) 1 u u B dµ C P diam (B) ρ p p dµ B for all functions u L 1 loc (X) and all upper gradients ρ of u. In the sequel we will encounter function-upper gradient pairs (v, ρ v ) that are defined only on some open and connected set G X. For such pairs the Poincaré inequality can be applied on any ball B with λ P B G, or B G if λ P = 1. To see this, let c > 1 be such that cb G and replace v with v = vχ cb and ρ v with ρ = ρ v χ B + χ X B. Then ρ is an upper gradient of v and v is locally integrable on X. 3. Main results Assume for the rest of the text that (X, d, µ) is a complete metric measure space that supports a weak 1-Poincaré inequality with constants C P and λ P. Assume also that µ is Borel-regular and doubling so that it satisfies (4) with some C µ and s > 1. Note that the doubling condition implies that X is proper and therefore also separable. By [15, Part I, II.3.11] µ is in fact a Radon-measure. Fix a domain G X and two disjoint nondegenerate continua E, F G. Denote G = G (E F ). Denote by Γ the set of all injective rectifiable paths : [0, 1] G with (0) E and (1) F. For any 1 p < denote λ P B (9) mod p Γ := mod p {H 1 Γ}. Similarly, denote by Γ the set of all compact subsets S G that separate E and F in G and have finite H-measure in G. Abbreviate (10) mod q Γ = mod q {H S G S Γ }. The requirement H(S G) < is redundant since the modulus of the family of sets with infinite H-measure is zero. Nevertheless we prefer to work with sets of finite H-measure. We denote C = C(X) if some constant C > 0 depends only on the data of X, i.e. the constants s, C µ, C P and λ P. The same symbol C

7 DUALITY OF MODULI IN REGULAR METRIC SPACES 7 will be used for various different constants. Denote p = 1 < p <. The main results of this paper are the following. THEOREM 3.1. Let 1 < p <. Then (mod p Γ) 1 p (modp Γ ) 1 p C p p 1 for each whenever mod p Γ is strictly positive and finite. The constant C depends only on the data of X. THEOREM 3.2. Let 1 < p <. Then C (mod p Γ) 1 p (modp Γ ) 1 p whenever mod p Γ(K 1, K 2 ; G) is strictly positive for any nondegenerate disjoint continua K 1, K 2 G. The constant C depends only on the data of X. THEOREM 3.3. Let X be as above and suppose in addition that X is Ahlfors Q-regular with Q > 1. Then 1 C (mod pγ) 1 p (modp Γ ) 1 p C, for every p > max{1, Q 1}. The constant C depends only on the data of X. The rest of the text is focused on the proofs of these results. Note that the conclusions in each of these results are bilipschitz invariant. Also recall that a complete metric space supporting a Poincaré inequality is C-quasiconvex for some C = C(X). Thus we may, and will, assume that X is a geodesic metric space. Note that in geodesic spaces we can choose λ P = 1. For these facts see Theorem and Remark in [5]. Theorems 3.1 and 3.2 will be proved in Sections 4 and 5, respectively. In Section 6 we show that Ahlfors Q-regular spaces satisfy the assumptions of both of these theorems for p > max{1, Q 1}, which implies Theorem Proof of Theorem 3.1 In this section we prove Theorem 3.1. Consider the sets (11) Γ j = {S Γ dist(s, E F ) > j 1 }. By applying the proof of Proposition in [5] and the general Fuglede s lemma, see [2, Theorem 3], it can be shown that (12) lim j mod p Γ j = mod p Γ.

8 8 ATTE LOHVANSUU AND KAI RAJALA Let j be so large that mod p Γ j is strictly positive and finite. Apply Lemma 2.1 to find a p -weakly admissible function ρ j of Γ j with the property mod p Γ j = dµ. Lemma 4.1. Let φ be another p -integrable, p -weakly admissible function of Γ j. Then mod p Γ j φρ p 1 j dµ. Proof. For any t [0, 1] let ω t = tφ + (1 t)ρ j. Now for any t mod p Γ j ω p t dµ with equality at t = 0. It follows that 0 d dt ω p t dµ = p (φ ρ j )ρ p 1 j dµ, t=0 G which finishes the proof. G The following theorem is the key tool in connecting the two moduli. Let s > 1 be any number such that (4) holds. Lemma 4.2. (Relative isoperimetric inequality) Let S Γ and let U be the component of G S that contains E. There are constants C = C(X, s) and λ = λ(x, s) > 1 such that G G G ρ p j { µ(b U) min, µ(b) for all balls B G. µ(b U) µ(b) } 1/s C r H( U λb) µ(λb) Proof. Given a ball B G there is a larger ball B G with B B and H( B) <. Applying Theorem 6.2 of [10] shows that B U is a so called set of finite perimeter. For sets of finite perimeter in place of U the relative isoperimetric inequality follows by combining Theorem (i) of [5] with Theorem 1.1 of [10]. Note that Lemma 4.2 requires the weak 1-Poincaré inequality. See Section 7 for examples of spaces that support a weak (1 + ε)-poincaré inequality for a given ε > 0, but no relative isoperimetric inequality. Fix Γ. The idea behind the proof of Theorem 3.1 is to construct admissible functions φ n j of Γ j that are supported close to, and then apply Lemma 4.1.

9 DUALITY OF MODULI IN REGULAR METRIC SPACES 9 Let us first construct a Whitney-type covering for. The following is an adaptation of page 103 of [5]. Let 3 n N. Denote d(x) = d(x, (X G) E F ). For any k Z let and k = {x 2 k 1 < d(x) 2 k } F n k = {B(x, d(x)/5n) x k }. Apply the 5r-covering theorem on Fk n to find a disjoint collection Gn k F k such that B 5B. B F n k B G n k Note that every Gk n is finite since is rectifiable. Thus the union of all Gk n is countable. Denote by Bn the collection of all balls 5B with B Gk n for some k Z. Given a set A X, denote by N ε(a) the set of points in X whose distance to the set A is strictly less than ε. Lemma 4.3. The collection B n has the following properties: (i) Given ε > 0 there is a δ (0, 1/2) such that B N ε (E) (resp. N ε (F )) whenever (t) B B n with t (0, δ) (resp. t (1 δ, 1)). (ii) If B 1, B 2 B n and B 1 B 2, then 1 2 r 1 r 2 2, where r 1 and r 2 are the radii of B 1 and B 2, respectively. (iii) There is a constant C = C(X) such that 1 B B n χ B (x) C for all x. Proof. Continuity of and the definition of B n imply (i). A simple application of the triangle inequality proves (ii). To prove (iii), let x and first note that 1 χ B (x) <. B B n The lower bound follows directly from the definitions. For the upper bound note that (ii) implies that there is an integer k such that the balls in B n that contain x are contained in Gk n Gn k 1, which is finite. Now let 5B 1,..., 5B N be balls in G n k that contain x with radii 5r 1,..., 5r N

10 10 ATTE LOHVANSUU AND KAI RAJALA respectively, so that r 1 r i for all i = 1,..., N. By the definition of Gk n the balls B i are disjoint, so by the doubling property and (ii) N µ(10b 1 ) µ(b i ) C(X)Nµ(10B 1 ). i=1 The same argument holds for Gk 1 n and (iii) follows. Now let S Γ. Let U be the component of G S that contains E. Let ε > 0 be such that N ε (E) U and N ε (F ) G U. Let { T n = sup t (0, 1) µ(u B) 1 } µ(b) 2 for all B Bn such that (t) B. By Lemma 4.3 (i) T n is well defined and T n (0, 1). By continuity of and the definition of T n there exist balls B i = B(x i, r i ) B n for i = 1, 2 such that B 1 B 2 and µ(b 1 U) µ(b 1 ) 1 2 µ(b 2 U). µ(b 2 ) Now let x B 1 B 2 and let i {1, 2} be the index for which r i = max{r 1, r 2 }. Let B = B(x, 2r i ). It follows from Lemma 4.3 (ii) and Lemma 4.2 applied to some s = s(x), that { µ(b U) C(X) min, µ(b) µ(b U) µ(b) } C r i H( U λb) µ(λb) for some C = C (X) and λ = λ(x). Therefore H(S λ B i ) H( U λ B i ) 1 C(X) r 1 i µ(b i ), where λ = 1 + 2λ. We conclude that the function φ n = C B B n r B µ(b) 1 χ λ B, where r B is the radius of B, is admissible for Γ. We would like to apply Lemma 4.1 to φ n and Γ, but φ n may not be p -integrable. This is why we consider the families Γ j instead. Note that if 5B B n satisfies B G k for sufficiently large k depending on j and n, then given any S Γ j µ(u B) {0, 1}. µ(b) Here U is again the component of G S that contains E. Together with the construction of φ n this implies that there is a k(j, n) Z such that φ n j = C r B µ(b) 1 χ λ B k k(j,n) B: 1 5 B Gn k is admissible for Γ j. It is p -integrable, since each G k contains only finitely many balls and G is bounded.

11 DUALITY OF MODULI IN REGULAR METRIC SPACES 11 Applying Lemma 4.1, the doubling property of µ, the definition of the Hardy-Littlewood maximal operator and (iii) gives mod p Γ j φ n j ρ p 1 j dµ G C(X) r B ρ p 1 j dµ B B n 5B C(X) B B n r B inf M C(X,G)/n(ρ p 1 j x B C(X) M C(X,G)/n (ρ p 1 j ) dh 1. )(x) Letting n and applying Fuglede s lemma [5, p. 131] we see that C(X)(mod p Γ j) 1 ρ p 1 j is weakly admissible for Γ and thus (mod p Γ) 1 p C(X)(modp Γ j) 1 ρ p 1 j L p (G) = C(X)(mod p Γ j) 1 p Letting j completes the proof. 5. Proof of Theorem 3.2 In this section we prove Theorem 3.2. Let X, G, E, F, Γ and Γ be as in Section 3. In the case G = X most of the results in this section can be found in [5]. The proofs in the case when G is a proper subdomain are more or less the same, with some simple modifications. We have included the proofs for completeness. First we construct u : G [0, ] as in [13, Section 4]. Note that since µ(g) <, mod p Γ(K 1, K 2 ; G) < for any nondegenerate disjoint continua K 1, K 2 G. Let ρ i, ρ L p (G) be the (weakly) admissible Borel functions of Γ given by Lemma 2.1. Let Γ 0 be the family of paths in G for which (13) ρ i ds i ρ ds < η η fails for some subpath η of. Then mod p Γ 0 = 0. Define u : G [0, ] by (14) u (x) = inf ρ ds, x x where the infimum is taken over all rectifiable paths Γ 0 in G that join E and x. If no such path exists, define u (x) =. Lemma 5.1. The function ρ is a p-weak upper gradient of u. In fact, the upper gradient inequality holds for the pair (u, ρ) for all paths Γ 0.

12 12 ATTE LOHVANSUU AND KAI RAJALA Proof. Let x, y G and let xy Γ 0 be a rectifiable path joining x and y. Recall the assumption that the p-modulus of any path family Γ(K 1, K 2 ; G) is positive, when K 1 and K 2 are nondegenerate disjoint continua in G. Application of this yields mod p ( xy, E; G) > 0, so there exists a path that joins E and xy so that and all of its subpaths satisfy (13). In particular u is finite at some point z on xy. Concatenating with a subpath of xy shows that u (x) and u (y) are finite and defined by (14). Now for any rectifiable path y Γ 0 that joins E and y u (x) ρ ds + ρ ds, y xy which yields the upper gradient inequality by taking infimum over the paths y. The Poincaré inequality comes into play with the following lemma. Lemma 5.2. Let U G be open and connected and suppose v : U R is locally integrable and ρ v : X [0, ] is an upper gradient of v in U that vanishes outside G. Let N U be the set of Lebesgue points of v. Then v(x) v(y) C(X) x y (M 10 x y ρ v (x) + M 10 x y ρ v (y)) whenever x, y B N for some ball B that satisfies 5B U. Proof. The case U = X is classical and proved in, for example, [5, Theorem 8.1.7]. We follow the same proof for the case of general U. Let B = B(x 0, r) satisfy 5B G. Let x B be a Lebesgue point of v. The first part of the proof of [5, Theorem 8.1.7] shows that (15) v(x) v B CrM 4r ρ v (x) for some constant C = C(X). Let y be another Lebesgue point of v in B. If r 5 x y, then applying (15) twice gives the desired result. 2 Otherwise apply (15) with B(x, 2 x y ) instead. The following lemma implies that u is measurable. Lemma 5.3. Every function on G with a p-integrable upper gradient is measurable. Proof. We follow the proof of Theorem found in [5]. Let f be a function on G with a p-integrable upper gradient g. Extend g as zero

13 DUALITY OF MODULI IN REGULAR METRIC SPACES 13 outside G and define g k = min{g, k} for each k N. Define also for each m = 0, 1,... E m = {x G Mg(x) m}. Let m 0 be the smallest number for which E m0 is nonempty. Let x 0 m E m. Define h, h k : G [0, ] with h(x) = inf 1 + g ds and h k (x) = inf 1 + g k ds, where the infima are taken over all rectifiable paths that connect x 0 and x in G. It is clear that 1 + g and 1 + g k are upper gradients of h and h k, respectively, and thus Theorem of [5] implies that h and h k are both measurable. Since X is geodesic, h k is locally C(1 + k)- Lipschitz in G, and therefore every point in G is a Lebesgue point of h k. Lemma 5.2 implies that each h k is Cm-Lipschitz on E m B for any ball B that satisfies 5B G. Note that u k u k+1 everywhere, and define a measurable function v x0 : G [0, ] to be the pointwise limit of the functions u k. By definition v x0 (x 0 ) = 0, from which it follows that v(x) < for all x m E m, since µ(g m E m ) = 0. It follows that h v x0 ; see the proof of Theorem in [5] for details. Nothing needs to be modified in our setting. Now define f : G G [0, ] with F (x, y) = inf 1 + g ds, where the infimum is taken over all paths in G that join x and y. It follows that F (x, y) v x (y) Cm x y whenever x, y E m B with 5B G. Let c = max{5, Cm + 2}. Suppose x, y E m B, where B has radius r and satisfies cb G. Let us show that for these points the paths in the definition of F (x, y) can be taken to be contained in G. Suppose there is a path in G that connects x, y and intersects G,

14 14 ATTE LOHVANSUU AND KAI RAJALA and satisfies Now on one hand 1 + g ds F (x, y) + r. 1 + g ds L 2(Cm + 1)r, and on the other hand 1 + g ds F (x, y) + r Cm x y + r 2Cmr + r. Combining these gives a contradiction. Therefore f(x) f(y) g ds 1+g ds = F (x, y) Cm x y, inf : G inf : G where the infima are taken over paths that connect x and y. This implies that f Em is locally Lipschitz and therefore measurable. Measurability of f follows. Recall that a function f : Ω R for some domain Ω X defines an equivalence class in the Newtonian space N 1,p (Ω) if it is p-integrable and admits a p-integrable upper gradient in Ω. Similarly, f N 1,p 0 (Ω) if there exists a function f N 1,p (X) such that f = f almost everywhere in Ω and f X Ω = 0 outside a set of zero p-capacity. In the following we need the fact that if f and g define the same class in N 1,p (G), then f = g in G outside a set of zero p-capacity. For exact definitions of these spaces and their further properties we refer the reader to [5] and [8]. Recall the definition of u from the beginning of the section. Lemma 5.4. The function u satisfies 0 u 1 and u F = 1 outside a set of zero p-capacity. In particular, u N 1,p (G). Proof. The idea is the same as in Lemma 4.5 of [13], but the measurability of u makes the proof easier. First note that u is finite outside a set of zero p-capacity. Recall that a set has zero p-capacity if and only if it has zero µ-measure and is p-exceptional. Consider the set A = (u ) 1 ( ) and let Γ A be the family of all rectifiable paths in G that intersect A. Using the proof of Lemma 5.1 we see that Γ A Γ 0 and therefore mod p (Γ A ) = 0. To see that µ(a) = 0 consider the sets E m as in the proof of Lemma 5.3. Since ρ L p (G), ρ is integrable on p-almost every curve Γ. On the other hand µ(g E m ) = 0, m

15 DUALITY OF MODULI IN REGULAR METRIC SPACES 15 so the family of curves in Γ that intersect m E m has positive p- modulus. Combining these facts gives that u is finite on at least one point of m E m. The final part of the proof of Lemma 5.3 implies that u is in fact finite on all points of m E m, so almost everywhere in G. Note that F A is nonempty. It follows immediately from the definition of u and Lemma 5.1 that u F A 1. Now suppose there are ε > 0 and x G such that u (x) is finite and u (x) 1 + 3ε. Then it follows from the definition of u that there exists a nondegenerate continuum α such that u > 1 + 2ε on α. Recall that µ is Borel-regular. Let A be a Borel subset of G that contains {u > 1 + ε} such that µ(a) = µ({u > 1 + ε}). Now ε 1 ρχ A is weakly admissible for Γ(E, α ; G), which implies that ρ p dµ > 0. A On the other hand if B {u 1} is Borel with µ(b) = µ({u 1}), then ρχ B is a weakly admissible for Γ by the definition of u. Now ρ p dµ < ρ p dµ + ρ p dµ ρ p dµ. B This contradicts the minimality of ρ. B We also need u to be continuous, at least outside the continua E and F. This follows from a theorem of Kinnunen and Shanmugalingam on the regularity of quasiminimizers, see [8]. Lemma 5.5. The function u N 1,p (G ) admits a continuous representative. Proof. Suppose Ω G is open. Suppose v is a function in N 1,p (G) such that v u N 1,p 0 (Ω). It follows that outside of a set A G Ω of zero p-capacity v E = 0 and v F = 1. The upper gradient inequality applied to v then implies that any upper gradient ρ v of v is weakly admissible for Γ. Thus by minimality of ρ ρ p v dµ ρ p dµ, G which implies that u is p-harmonic in the sense of [8]. Continuity of some representative of u in G is then established by the main results of [8]. Lemma 5.5 implies that u can be made continuous in G by modifying it in a set of zero p-capacity. Let u be this continuous representative. Since sets of vanishing p-capacity are p-exceptional, ρ is a A G X

16 16 ATTE LOHVANSUU AND KAI RAJALA p-weak upper gradient u. By definition u E = 0 and u F = 1 outside a set of zero p-capacity. Note that every ball must contain a point where u = u, and thus Lemma 5.4 implies that 0 u 1 outside a set of zero p-capacity. Theorem 3.2 is now a direct consequence of the following coarea estimate Proposition 5.6. Let g : G [0, ] be a p -integrable Borel function. Then (16) g dhdt C gmρ dµ (0,1) u 1 (t) G for some C = C(X). Proof of Theorem 3.2. If mod p Γ =, there is nothing to prove. Otherwise let g L p (G) be admissible for Γ. We may assume that g vanishes outside G. Note that for every t (0, 1) the set u 1 (t) separates E and F and is closed in G and by (16) H(u 1 (t)) < for almost every t. Now Lemma 5.4, Proposition 5.6 and Hölder s inequality give ( ) 1 1 g dhdt C gmρ dµ C(X)(mod p Γ) 1 p p g p. (0,1) u 1 (t) G G Taking infimum over admissible functions g gives the claim. To finish the proof of the lower bound we need to prove Proposition 5.6. Let us start with a classical estimate for Lipschitz functions. See [12, Theorem 7.7] for a euclidean version. Lemma 5.7. Let u : G R be L-Lipschitz and let A be a µ-measurable subset of G. Then (17) R H(u 1 (t) A) dt C(X)Lµ(A). Proof. Since µ is a Radon-measure, we may assume that A is open. Let δ > 0. Apply the 5r-covering theorem to find a countable collection of disjoint balls {B i } with B i = B(x i, r i ) A, 5r i δ and A i 5B i. Define a Borel function g : R [0, ] with g = i µ(5b i ) 5r i χ u(5bi ).

17 DUALITY OF MODULI IN REGULAR METRIC SPACES 17 Now for every t R we have H δ (u 1 (t) A) g(t), so by the doubling property H δ (u 1 (t) A) dt g(t) dt R R i µ(5b i ) 5r i u(5b i ) C(X)L i µ(b i ) C(X)Lµ(A). Applying the monotone convergence theorem for upper integrals finishes the proof. Proof of Proposition 5.6. By standard arguments it suffices to show (18) H(u 1 (t) A) dt C(X) Mρ dµ [0,1] for any Borel set A G. By applying a Whitney-type covering, see Proposition of [5], we may further assume that there is a ball B that satisfies A B 5B G. Continuity of u and Lemma 5.2 give (19) u(x) u(y) C(X) x y (Mρ(x) + Mρ(y)) for any x, y B. Now let B k = {x B 2 k < Mρ(x) 2 k+1 }. It follows from (19) that u Bk is C(X)2 k -Lipschitz. Let u k : X R be any Lipschitz extension of u Bk with the same Lipschitz constant. Now the monotone convergence theorem, Lemma 5.7 and the definition of A B k give [0,1] H(u 1 (t) A) dt = k [0,1] H(u 1 k (t) A B k) dt C(X) 2 k µ(a B k ) k C(X) Mρ dµ. A 6. Proof of Theorem 3.3 and Corollary 1.2 It suffices to show that Ahlfors Q-regular spaces satisfy the assumptions of Theorems 3.1 and 3.2 with p > max{1, Q 1}. Suppose X is a metric space as in Section 3, and in addition Q-regular for some Q > 1. Fix a domain G X.

18 18 ATTE LOHVANSUU AND KAI RAJALA Proposition 6.1. The modulus mod p Γ(K 1, K 2 ; G) is strictly positive, given any p > max{1, Q 1} and disjoint nondegenerate continua K 1, K 2 G. The proof of Proposition 6.1 is essentially a special case of the proof of Theorem 5.7 in [4]. We need the following lemma, which is a variant of a part of Proposition 2.17 (inequality (2.19)) in [4]. The same proof applies with trivial modifications. The (continuous) p-capacity of the triple (K 1, K 2, G) is the number cap c p(k 1, K 2 ; G) := inf ρ p dµ, ρ where the infimum is taken over all upper gradients ρ : G [0, ] of continuous functions u : G R with u K1 0 and u K2 1. Lemma 6.2. Let 1 < p < and let K 1, K 2 G be disjoint continua. Then mod p Γ(K 1, K 2 ; G) cap c p(k 1, K 2 ; Ω). for any subdomain Ω G that contains K 1 and K 2. Proof of Proposition 6.1. Fix a p > max{1, Q 1}. Let K 3 be a continuum that joins K 1 and K 2 in G. Let B be a finite cover of K = K 1 K 2 K 3 with balls of radius dist(k, G)/100 centered on K. Let Ω = B B 5B. Then Ω G and is a domain, since X is geodesic. Let u : Ω R be any continuous function with u K1 0 and u K2 1, and with a Borel measurable upper gradient ρ : Ω [0, ]. By Lemma 6.2 we are done if we can show that there is a constant c > 0 that does not depend on u or ρ such that (20) ρ p dµ c. Ω Let R > 0 be the radius given by the Lebesgue covering theorem applied on B: for every x K there is a ball B B such that B(x, R) B. Suppose that for i = 1, 2 there is a point x i K i such that (21) u(x i ) u B(xi,R) 1 3. Let (B i ) N i=0 be a collection of balls with the following properties B 0 = B(x 1, R) and B N = B(x 2, R) B i B for i 0, N G

19 DUALITY OF MODULI IN REGULAR METRIC SPACES 19 B i B i+1 for i N. Such a collection exists by connectedness of K. gradient inequality Now by the upper N 1 1 u(x 1 ) u(x 2 ) u(x 1 ) u B0 + u(x 2 ) u BN + u Bi u Bi+1. Next apply (21), the definition of B, the p-poincaré inequality (which follows from the 1-Poincaré inequality by an application of Hölder s inequality) together with Q-regularity to get N u Bi u Bi+1 i=0 N 1 3 u Bi u 5Bi+1 + u 5Bi+1 u Bi+1 i=0 N 1 3 u u 5Bi+1 dµ + u u 5Bi+1 dµ i=0 B i B i+1 N 1 C u u 5Bi+1 dµ i=0 5B i+1 ( 1/p CR 1 Q/p N ρ dµ) p. Ω This implies (20). Now suppose (21) is not true. Then for every x K 1 (22) u(x) u B(x,R) 1 3. Let B i = 2 i B(x, R). Now u Bi u(x) by continuity, so applying (22) and the p-poincaré inequality yields 1/p (23) 1 3 u Bi u Bi+1 C(X, R) (2 1 Q/p ) (B i ρ dµ) p. i i=0 This implies that there is a δ > 0, that does not depend on u or ρ, with the following property: for every x K 1 there is a ball B x = B(x, r x ) Ω such that (24) δr x ρ p dµ. B x If this was not true, (23) would give i=0 1 C(X, R)δ 1/p (2 1+(1 Q)/p ) i, i=0 i=0

20 20 ATTE LOHVANSUU AND KAI RAJALA which is a contradiction for small enough δ, since p > Q 1. Application of the 5r-covering theorem gives a disjoint and finite collection of balls B xi for some x i K 1 such that K 1 i 5B xi. Now (24) and connectedness of K 1 give diam (K 1 ) i diam (5B xi ) C(X) r xi i C(X)δ 1 ρ p dµ i B xi C(X)δ 1 ρ p dµ. This finishes the proof of Proposition 6.1. To prove Theorem 3.3, it remains to show that the assumptions of Theorem 3.1 are satisfied for any p > max{1, Q 1}. Fix such a p. Proposition 6.1 allows us to apply Theorem 3.2, which implies that mod p Γ > 0. To prove finiteness, recall the definition (11) of Γ j. Let S Γ j. Note that the mappings φ n j for n 3 constructed in the proof of Theorem 3.1 are admissible for Γ j and also p -integrable. Thus mod p Γ j <. This allows us to apply Theorems 3.2 and 3.1 with E and F replaced by the closures of N 1/j (E) and N 1/j (F ), respectively. Let Γ j be the corresponding path family. Then mod p Γ mod p Γ j for all j, since all functions that are admissible for Γ j are also admissible for Γ. Now by Theorem 3.1 (mod p Γ j) 1/p C(X)(mod p Γ j ) 1/p C(X)(mod p Γ) 1/p. Applying (12) finishes the proof. Proof of Corollary 1.2. The only if part follows directly from Theorem 1.1. To prove the if part, notice first that Theorem 1.1 shows that (2) holds for all path families Γ(E, F ; G) joining continua E and F inside G. Injecting this estimate into the proof of Theorem 4.7 in [4] shows that f is locally quasisymmetric, with constants depending only on the given data. On the other hand, Theorem 10.9 of [17] shows that locally quasisymmetric maps satisfy (2) for all path families. The required linear local connectedness and Loewner properties of X and Y are guaranteed by [9, Theorem 3.3] and [4, Theorem 5.7]. The if part follows. Ω

21 DUALITY OF MODULI IN REGULAR METRIC SPACES Counter-examples The relative isoperimetric inequality is an instrumental part of the proof of Theorem 3.1. By [10] it is equivalent to the weak 1-Poincaré inequality. Let ε (0, 1). We now construct a space X that satisfies the hypotheses of Theorem 3.3 apart from the 1-Poincaré inequality. Instead, X will support a (1 + ε)-poincaré inequality. Let K [1/4, 3/4] be a self-similar Cantor set with Hausdorffdimension 1 ε and the following property: for all x K and 0 < r < 1 H 1 ε (K B(x, r)) Cr 1 ε for some C > 0 that does not depend on r. Let Q = [0, 1] 3 R 3 and let A = [1/4, 3/4] K {0} Q. Then for any x A and 0 < r diam(q) (25) H 2 ε (A B(x, r)) Cr 2 ε for some (other) C > 0 that does not depend on r. Let Q 1 and Q 2 be two copies of the space Q. Finally, let X = Q 1 A Q 2, two cubes glued together along A. Equip X with the geodesic metric d that restricts to the metrics of the cubes in either cube, and for x Q 1 and y Q 2 set d(x, y) = inf ( x a + a y ). a A Equip X with the measure µ that restricts to the 3-dimensional Lebesgue measure on both cubes. It follows immediately from the definitions that (X, d, µ) is a complete geodesic Ahlfors 3-regular metric space. The validity of a weak (1 + ε)-poincaré inequality follows from (25) and [4, Theorem 6.15]. Now let E Q 1 A and F Q 2 A be nondegenerate continua and let G = X. Let Γ and Γ be as in Theorem 3.3. The modulus mod 3 Γ is strictly positive and finite. This can be seen by going through the proof of Proposition 6.1 again. On the other hand mod 3 Γ =, since Γ does not admit any admissible functions. To see this, note that A separates E and F in G, but has vanishing 2-measure. We conclude that X does not satisfy the upper bound of Theorem 3.3. Note that this implies that X does not support a weak 1-Poincaré inequality. This can also be deduced from, say, the main result of [10]. References [1] Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, [2] Bent Fuglede. Extremal length and functional completion. Acta Math., 98: , [3] F. W. Gehring. Extremal length definitions for the conformal capacity of rings in space. Michigan Math. J., 9: , 1962.

22 22 ATTE LOHVANSUU AND KAI RAJALA [4] Juha Heinonen and Pekka Koskela. Quasiconformal maps in metric spaces with controlled geometry. Acta Math., 181(1):1 61, [5] Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson. Sobolev spaces on metric measure spaces, an approach based on upper gradients, volume 27 of New Mathematical Monographs. Cambridge University Press, Cambridge, [6] Rebekah Jones, Panu Lahti, and Nageswari Shanmugalingam. Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally q-bounded geometry, preprint, arxiv: [7] Juha Kinnunen and Olli Martio. The Sobolev capacity on metric spaces. Ann. Acad. Sci. Fenn. Math., 21(2): , [8] Juha Kinnunen and Nageswari Shanmugalingam. Regularity of quasiminimizers on metric spaces. Manuscripta Math., 105(3): , [9] Riikka Korte. Geometric implications of the Poincaré inequality. Results Math., 50(1-2):93 107, [10] Riikka Korte and Panu Lahti. Relative isoperimetric inequalities and sufficient conditions for finite perimeter on metric spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire, 31(1): , [11] O. Lehto and K. I. Virtanen. Quasiconformal mappings in the plane. Springer- Verlag, New York-Heidelberg, second edition, Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126. [12] Pertti Mattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, Fractals and rectifiability. [13] Kai Rajala. Uniformization of two-dimensional metric surfaces. Invent. Math., 207(3): , [14] Seppo Rickman. Quasiregular mappings, volume 26 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, [15] Laurent Schwartz. Radon measures on arbitrary topological spaces and cylindrical measures. Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, Tata Institute of Fundamental Research Studies in Mathematics, No. 6. [16] Stephen Semmes. Good metric spaces without good parameterizations. Rev. Mat. Iberoamericana, 12(1): , [17] Jeremy T. Tyson. Metric and geometric quasiconformality in Ahlfors regular Loewner spaces. Conform. Geom. Dyn., 5:21 73, [18] Jussi Väisälä. Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin-New York, [19] William P. Ziemer. Extremal length and conformal capacity. Trans. Amer. Math. Soc., 126: , 1967.

23 DUALITY OF MODULI IN REGULAR METRIC SPACES 23 Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland. A.L.: K.R.: