On the Sharp Lower Bound for Duality of Modulus
aa r X i v : . [ m a t h . M G ] F e b ON THE SHARP LOWER BOUND FOR DUALITY OF MODULUS
SYLVESTER ERIKSSON-BIQUE AND PIETRO POGGI-CORRADINI
Abstract.
We establish a sharp reciprocity inequality for modulus in compact metric spaces X with finite Hausdorff measure. In particular, when X is also homeomorphic to a planar rectangle,our result answers a question of K. Rajala and M. Romney. More specifically, we obtain a sharpinequality between the modulus of the family of curves connecting two disjoint continua E and F in X and the modulus of the family of surfaces of finite Hausdorff measure that separate E and F . The paper also develops approximation techniques, which may be of independent interest. Introduction
Modulus, which we define below in Section 2, is a way of measuring the richness of a collectionof curves, or more generally a collection of surfaces or even measures, and arises as the value of aconvex minimization problem. Ahlfors and Beurling showed in [1] that for a topological rectangle Q in the plane R , if Γ ( Q ) is the family of curves connecting a pair of opposite sides, and Γ ( Q )is the family of curves connecting the other pair of opposite sides, then their 2-modulus values arerelated by the following reciprocal formula:(1) Mod Γ ( Q ) · Mod Γ ( Q ) = 1 . Later, see [2, 24], this relation was generalized to families of curves and separating surfaces inEuclidean spaces of higher dimension. More recently, see [16, 12], this was extended to moregeneral metric spaces in slightly different forms, with some assumptions of the underlying space,such as doubling and the presence of a Poincar´e inequality. In the discrete setting, such inequalitieswere explored on graphs in [3].Remarkably, Kai Rajala showed in [19], that if the equality in Equation (1) is replaced by acomparability of the form(2) 1 κ ≤ (Mod Γ ( Q )) / (Mod Γ ( Q )) / ≤ κ, for some κ together with another technical assumption, then this provides a characterization ofmetric surfaces X (with locally finite Hausdorff 2-measure) which are quasiconformal to the plane.Rajala’s result extends a long line of work intent on constructing quasiconformal uniformizations inthe spirit of the classic uniformization of Riemann surfaces via conformal maps, see e.g. [4, 19] formore discussion. Recently, it was observed by Rajala and Romney [20] that the lower bound in (2)holds automatically whenever the underlying space is homeomorphic to R and has locally finiteHausdorff 2-measure. They were able to establish this fact with the specific constant κ = 8000 /π and conjectured in the same paper that the optimal constant should be κ = 4 /π . In particular, theygive an example showing κ cannot be smaller than 4 /π (see the discussion on sharpness below).The purpose of our paper is to establish this conjecture and show that it holds also in a slightlymore general context. Mathematics Subject Classification.
Primary 30L15, Secondary 30L10, 28A75, 49N15.
1N THE SHARP LOWER BOUND FOR DUALITY OF MODULUS 2
To set the notation, suppose that X is a compact metric space with finite H N -Hausdorff measure,for some N ∈ R , with N ≥
1. Suppose
E, F are two disjoint continua , i.e., nonempty, compactconnected sets, in X . Let Γ( E, F ) be the family of curves connecting E and F in X . Also, letΣ( E, F ) be the family of topological boundaries ∂U of open sets U in X , such that E ⊂ U and F ⊂ int( U c ). We think of ∂U as a surface separating E and F . Consider the corresponding familyof measures Σ H ( E, F ) consisting of all Hausdorff measures of the form H N − | ∂U that are finite. Ifwe want to consider these notions relative to a subset Q ⊂ X , we write Γ( E, F ; Q ) , Σ H ( E, F ; Q ).Throughout the paper, p ∈ (1 , ∞ ) and q is its dual exponent, namely p − + q − = 1. For our nextresult, X need not be homeomorphic to R N or any of its subsets. However, later we will specializeto such a setting. Theorem 1.1.
Suppose that X is a compact metric space, with finite H N -Hausdorff measure, forsome real number N ≥ . Set p, q ∈ (1 , ∞ ) with p − + q − = 1 . Let E, F be two disjoint continuain X , with Γ( E, F ) and Σ H ( E, F ) defined as above. Then, if Mod p (Γ( E, F )) > , the followinginequality holds (3) (Mod p Γ( E, F )) p (Mod q Σ H ( E, F )) q ≥ v N v N − , where v k := π k/ Γ ( k +1 ) , for k ≥ , and Γ (cid:0) k + 1 (cid:1) = R ∞ x k/ e − x dx is the usual Gamma function.Moreover, if Mod p Γ( E, F ) = 0 , then
Mod q Σ H ( E, F ) = ∞ . For the definition of modulus we refer to Definition (6). In particular, in the setting of Theorem1.1, we always have Mod p Γ( E, F ) < ∞ , because the constant function ρ ( x ) ≡ d ( E, F ) − is admis-sible. Note also that it is possible for Σ H to contains null-measures, in which case the modulusMod q Σ H ( E, F ) becomes infinite.As a corollary, when we restrict to planar metric spaces, we obtain the inequality conjectured byRajala and Romney in [20], which we extend also to the case p = 2. Corollary 1.2.
Suppose that Y is a metric space homeomorphic to R , which has locally finite H -measure. Set p, q ∈ (1 , ∞ ) and p − + q − = 1 . Assume Q ⊂ Y is homeomorphic to [0 , ,and hence can be thought as a quadrilateral. Let the sides of Q correspond to continua A, B, C, D (in cyclic order). Set Γ ( Q ) := Γ( A, C ; Q ) , the family of curves connecting A and C in Q , and Γ ( Q ) := Γ( B, D ; Q ) , the family of curves connecting B and D in Q . Then, the following inequalityholds: (4) (Mod p Γ ( Q )) p (Mod q Γ ( Q )) q ≥ π . To prove Corollary 1.2, we will apply Theorem 1.1 with Q = X , since X will be compact andhave finite Hausdorff H -measure. The corollary is sharp, as can be seen from the following examplealready mentioned in [19, Example 2.2]. Sharpness:
Take Y = R equipped with the ℓ ∞ -distance d (( x , x ) , ( y , y )) = max i =1 , | x i − y i | .Let Q := [0 , . We want to compute the modulus of the families of curves connecting the horizontaland vertical pairs of sides.Consider the Hausdorff measure H with respect to the metric d . The usual scaling factor v N − N in Equation (5), is equal to π/ N = 2. Since H is a translation invariant andlocally finite measure, there must be some constant c so that H ( A ) = cλ ( A ) for each Borel set A , where λ is the usual Euclidean area measure. This constant can be determined by computing H ( Q ). On one hand, we obtain H ( Q ) ≤ π/ Q by a grid of squaresof side length n − and sending n → ∞ . On the other hand, consider any countable cover A i N THE SHARP LOWER BOUND FOR DUALITY OF MODULUS 3 of Q with Q ⊂ S i A i . Each A i can be replaced by its bounding box ˜ A i , which has the samediameter in the ℓ ∞ -metric. Denote the diameter of a set E with respect to d by diam d ( E ). Then, P i diam d ( A i ) = P i diam d ( ˜ A i ) ≥ P i λ ( ˜ A i ) ≥
1. Hence, accounting for the scaling factor inEquation (5), together with the upper bound established before, we get that H ( Q ) = π/ c .We remark, that the same could have also been established by the deep result of Kirchheim [15,Lemma 6].Suppose now that ρ is admissible for the curves connecting the left to the right hand side. Byadmissibility for horizontal curves, we get Z Z ρ dλ ≥ . Thus, R Q ρ d H ≥ π . An application of H¨older’s inequality gives π ≤ (cid:18)Z Q ρ p d H (cid:19) /p (cid:18)Z Q d H (cid:19) /q = (cid:18)Z Q ρ p d H (cid:19) /p (cid:16) π (cid:17) /q and thus Z Q ρ p d H ≥ π ∀ p ∈ (1 , ∞ ) . Minimizing over ρ admissible, we get that Mod p (Γ ( Q )) ≥ π , for all p ∈ (1 , ∞ ). Conversely, theconstant function ρ ≡ ( Q ), thus Mod p (Γ ( Q )) ≤ R Q d H = π . Hence,Mod p (Γ ( Q )) = π , for all p ∈ (1 , ∞ ), and, by symmetry, Mod q (Γ ( Q )) = π as well. Thus, in thisexample Equation (4) becomes(Mod p Γ ( Q )) p (Mod q Γ ( Q )) q = (cid:16) π (cid:17) p (cid:16) π (cid:17) q = π . The proof of Theorem 1.1 rests on a co-area type estimate, analogously to [20], and a Lipschitzapproximation. However, to get the sharp constant we need a novel approximation scheme thatyields sharper bounds. See Theorem 3.5 for a precise statement. This technique was first introducedin [7], and, in the context of this paper, it yields the following result, which may be of independentinterest.
Theorem 1.3.
Let p ∈ [1 , ∞ ) with q its dual exponent. Let X be a compact metric space with finite H N -measure, for some real number N ≥ . Let E, F be two disjoint continua in X . Suppose that u is a N ,p ( X ) -function such that u = 1 on F and u = 0 on E . Then, there exists a sequence ofLipschitz functions u i ∈ N ,p ( X ) , so that u i = 1 on F and u i = 0 on E , with the property that forany Borel ρ ∈ L q ( X )lim sup i →∞ Z Z ∂ { u i The first author was partially supported by the National Science Foundationunder Grant No. DMS-1704215 and by the Finnish Academy under Research postdoctoral GrantNo. 330048. The second author thanks the Department of Mathematics at UCLA, where thisresearch started, for its generous support. The authors thank Kai Rajala and Mario Bonk fordiscussions on the topic and comments. N THE SHARP LOWER BOUND FOR DUALITY OF MODULUS 4 Preliminaries In this paper, ( X, d ) will denote a compact metric space with finite H N -Hausdorff measure, forsome real number N ≥ Y will denote a space homeomorphic to R with locally finite H -measure. The concepts of modulus, curve families and Newtonian (or Sobolev) spaces are definedin the same way for X and Y . Throughout, we will use N ≥ H N , and it is with respectto these measures that we define the Lebesgue spaces L p ( X ), for p ∈ [1 , ∞ ], and the notion ofalmost everywhere. Where needed, the norm on L p ( X ) will be denoted by k · k L p . For purposes ofnormalization, we note that the Hausdorff measure is defined as H N ( A ) := lim δ → H Nδ ( A ), where(5) H Nδ ( A ) := v N N inf (X i diam( A i ) N : A ⊂ [ i A i , diam( A i ) < δ ) where v k := π k/ Γ( k +1) , and Γ( t ) is the usual Gamma function. Also, diam( E ) = sup x,y ∈ E d ( x, y ) isthe diameter of a set E .Curves in X are continuous maps γ : I → X , defined on some compact non-empty interval I ⊂ R . Let Γ( X ) be the family of all such curves. We are interested in the families Γ( E, F ) (orΓ( E, F ; Q )) of all the curves which connect E to F (resp. in Q for some Q ⊂ X ). Throughout, E and F will be two disjoint compact connected non-empty subsets of X .Also let M ( X ) denote the family of all finite Radon measures on X . In particular, Σ H ( E, F )will consist of all finite measures H N − | ∂U , given an open set U with E ⊂ U and F ⊂ int( U c ).Here, int( A ) denotes the interior of the set A ⊂ X .Let Γ ⊂ Γ( X ) be any family of curves. We say a non-negative Borel function ρ : X → [0 , ∞ ]admissible for Γ, and write ρ ∈ Adm(Γ), if R γ ρ ds ≥ 1, for each rectifiable γ ∈ Γ( X ). Here ds represents the arc-length parametrization of γ , see [11, Chapter 5] for more details. Then, wedefine the p -modulus (for 1 ≤ p < ∞ ) of a curve family as(6) Mod p (Γ) := inf ρ ∈ Adm(Γ) Z X ρ p d H N , When p = ∞ , we take the infimum of k ρ k L ∞ . If Σ ⊂ M ( X ), then we define Mod p (Σ) by replacingthe admissibility condition with R ρ dσ ≥ σ ∈ Σ.We say that a property holds for p -almost every curve, if the set of curves Γ for which theproperty does not hold has Mod p (Γ ) = 0.If f : X → [ −∞ , ∞ ] is measurable, we call g : X → [0 , ∞ ] an upper gradient for f , if for everyrectifiable curve γ : [0 , → X we have(7) Z γ g ds ≥ | f ( γ (0)) − f ( γ (1)) | , where we interpret |∞ − ∞| = ∞ . We say that f ∈ N ,p ( X ) if f ∈ L p ( X ) and f has an uppergradient g ∈ L p ( X ). There is a function |∇ f | p , called a minimal p -weak upper gradient for which |∇ f | p ≤ g ( H N -a.e.) for each upper gradient g , and for which estimate (7) holds for p -a.e. curve.Although this notation suggests that a point-wise “gradient” ∇ f may exist, this is not necessarilythe case. However, we wish to connect this notation to the Euclidean notion, where the expressionis actually the norm of the distributional gradient. Remark 2.1. When g = |∇ f | p , for any ǫ > h : X → [0 , ∞ ] so that R h p d H N ≤ ǫ/ 2, and R γ h ds = ∞ for every curve γ Inequality (7) does not hold. Then, ˜ g = N THE SHARP LOWER BOUND FOR DUALITY OF MODULUS 5 max( |∇ f | p , h ) is an upper gradient with R ˜ g p d H N ≤ R |∇ f | pp d H N + ǫ/ 2. By Vitali-Carath´eodory,we can choose a lower semicontinous g ǫ , so that ˜ g ≤ g ǫ and R g pǫ d H N ≤ R ˜ g p d H N + ǫ/ ≤ R |∇ f | pp d H N + ǫ . With the same argument, the infimum in Definition (6) can be taken overlower semicontinuous functions. We refer the reader to [11, Chapters 4 and 5] for more details onmodulus and a proof of Vitali-Carath´eodoryWe also need the following simple case of modulus equalling capacity (see e.g. [10, Section 2.11]).We don’t wish to introduce capacity without really needing it here, so we prefer to give only a weakstatement whose proof can be given directly. Lemma 2.2. Suppose that g is admissible for Mod p (Γ( E, F )) , then there is a function u ∈ N ,p ( X ) so that u | E = 0 and u | F = 1 with |∇ u | p ≤ g almost everywhere.Proof. Define u ( x ) = min (cid:0) inf γ R g ds, (cid:1) , where the infimum is over paths γ connecting E to x . By[13, Corollary 1.10] the function u ( x ) measurable. Note that u | E = 0 by definition. Also, u | F = 1,because g is admissible. Since X is compact, we have u ( x ) ∈ L p ( X ). Next, we show that g is anupper gradient for u and this will imply that u ∈ N ,p ( X ) and |∇ u | p ≤ g .Let γ be any curve joining x = γ (0) and y = γ (1). We will show that u ( y ) − u ( x ) ≤ R γ g ds .Then, reversing the curve gives the desired bound (7). If u ( x ) = 1, the inequality is immediate sincethen u ( y ) − u ( x ) ≤ 0. Therefore, we can assume that u ( x ) = inf γ x R g ds with the infimum is takenover all the curves γ x joining E to x . Fix one such curve γ x . Define a curve γ y by concatenating γ x with γ , and parametrize it so that γ y (0) = γ x (0) ∈ E . Then, u ( y ) ≤ Z γ y g ds = Z γ x g ds + Z γ g ds. Taking an infimum over γ x yields the desired bound. (cid:3) In order to get Corollary 1.2 from Theorem 1.1, we need a general modulus statement. Lemma 2.3. Suppose that Σ , Σ ⊂ M ( X ) are two families of measures, and for each σ ∈ Σ wehave some measure σ ∈ Σ with σ ≤ σ , then Mod p (Σ ) ≤ Mod p (Σ ) .Proof. The claim follows because any Borel function admissible for Σ will automatically be ad-missible for Σ . (cid:3) We will also need the following useful topological fact. Recall that, if Y is a metric spacehomeomorphic to R , which has locally finite Hausdorff H -measure, then a quadrilateral Q ⊂ Y isa subset homeomorphic to [0 , . The homeomorphic images of the sides will be denoted A, B, C, D ,in cyclic order. Note that the notion of “opposite” edges does not depend on the orientation of theboundary. In the following, the relative boundary of a set S ⊂ Q is denoted as ∂ Q S Lemma 2.4. Suppose that Q ⊂ Y is a quadrilateral as defined above. Let U be an open set in Y such that A ⊂ U and C ⊂ int( U c ) , with H | ∂ Q U finite so that it is a measure in Σ H ( A, C ; Q ) .Then, there is a simple rectifiable curve γ connecting B to D , with γ ⊂ ∂ Q U and H | ∂ Q U ≥ H | γ . This claim is classical, but we indicate a proof for the sake of completeness. Proof. By Zorn’s lemma, there is a minimal compact set K ⊂ ∂ Q U which still separates A from C in that it does not contain any strictly smaller separating compact set. Indeed, it is enough to showthat for every chain { K j } ∞ of compact sets separating A from C with K j +1 ⊂ K j , the intersection K ∞ := ∩ j K j is also compact and must still separate A from C . Assume that K ∞ does not separate. N THE SHARP LOWER BOUND FOR DUALITY OF MODULUS 6 Then, there is a curve γ from A to C in the complement of K ∞ , i.e., with d ( γ, K ∞ ) > 0. However,there are points w j ∈ γ ∩ K j for every j and by compactness a subsequence will converge to a point w ∞ . Note that w j ∈ K i for all j ≥ i . So w ∞ ∈ K i , for every i ≥ 1. Hence, w ∞ ∈ K ∞ , which leadsto a contradiction.Moreover, this minimal separating compact set K must be connected. If not, then K could beexpressed as a union of two disjoint non-empty compact subsets K , K . Since K ∩ K = ∅ , byJaniszewiski’s theorem, see e.g. [18, p.110], either K or K must separate. However, this is acontradiction to minimality of K . Therefore, K must be a continuum.Since K is connected and has finite Hausdorff measure, then by (the argument in) [22, Lemma3.7] K is rectifiable. Thus, we can find a rectifiable curve γ : I → ∂ Q U which separates A from D .The curve γ must intersect B and D , and thus must contain a sub-curve γ | J for some J ⊂ I whichconnects B to D . By possibly removing loops, i.e. choosing the shortest curve contained in theimage of γ and connecting B to D , we can insist that γ be simple. The inclusion γ ⊂ ∂ Q U gives H | ∂U ≥ H | γ . (cid:3) We will also need the following version of Arzel`a-Ascoli’s theorem. Lemma 2.5. Assume that Z is a complete metric space and that L ∈ (0 , ∞ ) . Suppose that γ n :[0 , → Z is a sequence of L -Lipschitz curves with A t := { γ n ( t ) : n ∈ N } precompact for each t ∈ [0 , . Then, there exists a subsequence γ n k which converges uniformly to a curve γ : [0 , → Z . The proof of this version is completely classical (see for instance [21, Theorem 4.25]). However,we remark, that instead of assuming that Y is compact, we assume that the set of pointwise values A t is pre-compact. Indeed, the usual proof of first constructing γ ( t ) on a dense subset of rationalvalues t together with a diagonal argument only requires the pointwise values to be precompact.The precompactness of the sets A t , in our application, will be shown to follow from the fact that A t is close to a compact subset, except for finitely many points. This argument allows us to perform alimit process in the ambient space Z := ℓ ∞ ( N ), in which X can be embedded using the Kuratowskiembedding. Thus, Arzel`a-Ascoli argument can still be applied, despite the lack of compactness of Z . 3. Lipschitz approximation A function f : X → Y between two metric spaces is Lipschitz if sup x,y ∈ X,x = y d ( f ( x ) ,f ( y )) d ( x,y ) is finite,where d denotes the distance both on X and Y . Given a Lipschitz function f : X → Y and a subset A ⊂ X define the Lipschitz constant of F on A asLIP[ f ]( A ) := sup x,y ∈ A,x = y d ( f ( x ) , f ( y )) d ( x, y ) . Recall Eilenberg’s inequality: for any Borel set A ⊂ X and any Lipschitz function u : X → R :(8) Z ∞−∞ H N − ( u − ( t ) ∩ A ) dt ≤ v N − v N LIP[ u ]( A ) H N ( A ) . The inequality is due to [6, Theorem 1], where a relatively simple proof is presented using anupper integral. The reader can consult [8] for a discussion and further references. The version weuse combines [6, Theorem 1] with the following remark and lemma. Remark 3.1. There is a measurability consideration that is not explained in the papers citedabove, which can be bypassed by expressing the integral on the left hand-side of Equation (8) asan upper Lebesgue integral. Namely, the fact that the map t → H N − ( u − ( t ) ∩ A ) is measurable, N THE SHARP LOWER BOUND FOR DUALITY OF MODULUS 7 if A ⊂ X is a Borel set, when H N ( X ) < ∞ and u : X → R is Lipschitz. To be self-contained, wepresent here an argument for this, that suffices for our purposes. As remarked in [8, Remark 1.2],the result holds in greater generality. The interested reader may also consult the beautiful treatiseby Dellacherie related to this point [5, Chapitre VI]. Lemma 3.2.